Finite Element Methods with B-Splines

Substituting this expression into (4.5) and interchanging sums yield

where /(/) = {7 e J : i e f(j)}. The term in brackets is the proper combination of inner and outer B-splines. Using this linear combination as a new basis function, it is possible to represent all polynomials of coordinate degree < n without having to refer to the outer B-splines explicitly. As before, a weight function can be incorporated to satisfy essential homogeneous boundary conditions. We multiply [...] by u;(;c)/u;(jc,), where jc, is the center of the interior grid cell (2, C D fl supp b{. These considerations lead to the following definition [46]. 4.9 Weighted Extended B-Splines For an outer index j e J let l(j) = I + (0,..., n}m C / be an m-dimensional array of inner indices closest to j, assuming that h is small enough so that such an array exists. Moreover, denote by

the values of the Lagrange polynomials associated with 7(y) and by /(i) the set of all j with i e /O'). Then, the web-splines

form a basis for the web-space u;

4.4. Web-Splines

49

Figure 4.7. Construction of biquadratic web-splines. The construction of biquadratic web-splines is illustrated in Figure 4.7. On the left, the figure shows for an outer index j the array I ( j ) with the values efj of the Lagrange polynomials. For example, for i = j — (0, 2),

since j — I = (2, 3) and i — i = (2, 1). The appearance of many zeros is typical since the Lagrange polynomials vanish whenever jv e (£y + {0, . . . , n})\iv for some v. On the right of the figure we see the support of a web-spline and the set /(/) with the coefficients of the adjoined outer B-splines. We have also marked the point xt at the center of the grid cell <2,. Since etj = 0 for j = i + (1, —1), the linear combination £];ej(() etjbj consists only of three terms, corresponding to j —i = (1, 0), (0, !),(!, 1). As in this example, the union of the supports of the B-splines, involved in the defintion of BI (highlighted in the figure), is usually only slightly larger than the support of the inner B-spline b[. Except for positivity, web-splines inherit all essential properties of B-splines. The following facts are particularly relevant for finite element approximation. Extension coefficients: We have

and

Moreover, only -< h' ~m B-splines near the boundary need to be modified. For the majority of inner indices i, fi/ = w/w(X{) hi, as h becomes small. • Local support: The diameter of supp #, is -< h, and on each set Q n D only -< 1 web-splines are nonzero.

50

Chapter 4. Finite Element Bases • Stability: The web-splines are linearly independent and

In particular, • Approximation order: The web-space weB/, contains weighted polynomials of coordinate degree < n. Moreover,

if w and u/w are smooth. In all of the above estimates, the constants depend on the degree n, the domain D, and the weight function w (and in (4.7) also on the approximated function u). Most statements are familiar from standard spline theory and not difficult to prove. The first two properties are obvious from definition 4.9. The linear independence follows from the corresponding result for standard B-splines (theorem 4.5) since

Furthermore, by construction, web-splines can represent weighted polynomials. Hence, it remains to prove the estimates (4.6) and (4.7). This is more subtle and will be postponed to chapter 5, where we give a thorough analysis of the approximation properties of web-splines.

4.5

Hierarchical Bases

To resolve small geometric details or rapid changes in solutions, local grid refinement is essential. With B-splines such adaptive approximations are possible in a very natural way. We describe an elementary strategy, which is based on the subdivision formulas derived in section 3.5. Essentially, this technique was proposed by Kraft [58]. It is very easy to implement and well suited for handling singularities. 4.10 Hierarchical B-Splines The hierarchical spline space B£(B), corresponding to a nested sequence of domains

is spanned by the B-splines

where Kv denotes the indices k for which D D supp £>*,«,, is a subset of Dv (with nonzero measure), but is not contained in v+\.

4.5. Hierarchical Bases

51

This definition can be interpreted in the following way. We select a subset of the relevant B-splines for D with grid width h (those B-splines with D D suppbk,h0 C D\, i.e., with k e K\KQ) and replace it by B-splines of grid width h/2 via subdivision. From the relevant B-splines b^^\ on the finer grid with D n suppfr^, C D\, which include, in particular, the B-splines resulting from subdivision, again, a subset is selected and refined. This process is repeated according to the sequence of domains Dv. Figure 4.8 illustrates definition 4.10 for degree n = 1 and 3 grids. The subdomains £>2 C D\ C DO — D are displayed with different gray levels, and the lower left corners of the supports

of the B-splines in the basis are marked with circles of different sizes, according to their grid width. In this example, the spline space is refined near the corners of the domain, where we expect singularities of the solution. As is common practice, the sets Dv are unions of supports of B-splines. For example, the portion of D\ containing the reentrant corner at

Figure 4.8. Bilinear hierarchical B-spline basis.

52

Chapter 4. Finite Element Bases

x = (0, 0) is

It is easily checked that the B-splines spanning B/,(D) are linearly independent. If

we can show inductively, for v = 0, 1, . . ., that the coefficients c*,v are zero. First, we restrict x to the open set DQ,I = D0\D\. By definition, all B-splines with grid width hv, v > 0, vanish on this set. On the other hand, for each B-spline bk.hn, k € KQ, there exists a point x e DO,I where this B-spline is nonzero. Hence, by the local linear independence of the B-splines corresponding to one grid in a neighborhood of jc (cf. theorem 4.5), all coefficients Q,O must be zero. Now we repeat the argument, restricting x successively to the sets D\ 2, 1*2.3, ____ The hierarchical spline space B^(P) contains for each domain Dv all B-splines bk.hv for which the portion of their support in D is completely contained in Dv. Either such a B-spline belongs to the hierarchical basis, or, if D n supp bk.hr C Dv+\ , it can be represented as a linear combination of B-splines with smaller grid width via subdivision. Hence, as is expected, the local approximation power of B/, (D) corresponds to the level of refinement. Summarizing these observations, we have proved the following result. 4.11 Linear Independence and Local Structure The B-splines spanning B/,(B) are linearly independent and

Similarly as in section 4.3, we can define weighted hierarchical spline spaces wWh (D). All B-splines are multiplied by a weight function w which satisfies homogeneous boundary conditions. It is also possible to apply the extension technique of section 4.4 to eliminate the outer B-splines. For example, we can base the selection of the basis in definition 4.10 on the inner B-splines b,,/, u and include the corresponding web-splines 5,-^. in the hierarchical spline space. However, this is not quite sufficient to stabilize the basis. To achieve uniform stability, also with respect to the number of grid levels, wavelet techniques are required.

Chapter 5

Approximation with Weighted Splines

As was shown in section 2.5, the convergence of finite element schemes follows easily from the definition of the Ritz-Galerkin approximation. The basic estimate is Cea's inequality 2.8. It states that the error in the energy norm can be bounded in terms of the error of the best approximation from the finite element subspace. Hence, it suffices to analyze the approximation properties of the basis functions without taking the specifics of the discretized boundary value problems into account. For the finite element bases introduced in the last chapter, we obtain the familiar error bounds for piece wise polynomials. A typical estimate is where «/, are weighted approximations of degree < n from either of the spaces wB/, or w eB/, . The proofs require a number of auxiliary results, which are described at the beginning of this chapter. Essentially, the arguments are analogous to standard spline theory. Incorporating the weight function w is mostly straightforward, except perhaps for regularity issues. In section 5.1 we construct dual functions, which are used in section 5.2 to prove the stability of the web-basis. Then, we review in section 5.3 the fundamental error bound for polynomial orthogonal projections. These preliminary results provide the necessary tools for defining approximation schemes and deriving error estimates. We first show in section 5.4 that smooth functions can be approximated by weighted splines with optimal order. To obtain sharper bounds, we analyze in section 5.5 the smoothness of quotients with weight functions. With the aid of these regularity results we finally obtain in section 5.6 optimal error estimates in Sobolev norms.

5.1

Dual Functions

It would be nice if we could add "orthogonality" to the list of favorable properties of spline bases. Unfortunately, this is too much to ask, at least if we do not change the scalar product (cf. [72] for an interesting new idea). However, it is possible to construct dual bases. For web-splines, these are functions A, with

53

54

Chapter 5. Approximation with Weighted Splines

Such biorthogonal systems are important for stability questions as well as for defining local approximation schemes. For example, we can define a canonical projector

analogously to standard orthogonal expansions. For B-splines, dual bases were constructed by de Boor [16] without any restrictions on the knot sequence. We specialize his result to a uniform grid, choosing a formulation particularly suited for the generalization to web-splines. 5.1 Dual Functions For any m-dimensional cube Q't c supp ty with width Oh, there exists a function A,,- with support in Q\ which satisfies

and

const(m,n,0)

The proof is not difficult but is somewhat lengthy. Choosing Q\ as one of the grid cells in the support of bt (0 = 1) would simplify the arguments. However, we need the extra flexibility for later applications. Moreover, the dependence of the norm of A,/ on the parameters is important. To construct the dual functions A./ , we consider first the special case

with Q'Q C [0, n + l] m . Clearly, the biorthogonality needs to be checked only for the B-splines £*, which are nonzero on Q'Q. Denoting the set of the corresponding indices k by /o, the local linear independence of the B-splines (theorem 4.5) implies that

is a well-defined linear functional on H = spankelobk. Hence, by the Riesz representation theorem A. 5, there exists a function A.* = 1Z,Q € H with

In particular, (A.*, bk)o = 5o,t. To complete the proof in the special case, it remains to obtain a bound for || A.* ||o which depends only on m, n, and the width 9 of Q'0. This can be accomplished by applying the above construction only to finitely many m-cubes. As is illustrated on the left side of Figure 5. 1 for biquadratic B-splines, we partition the support of b1^ , into ((n + 1) [2/0] ) m m-cubes (dashed lines). Since each of these m-cubes have width < 0/2, at least one of them is contained in Q'Q (light gray). By choosing this m-cube Q* (dark gray) as support of A,*, the dependence of ||A.* ||o on the position of Q'0 can be eliminated.

5.1. Dual Functions

55

Figure 5.1. Construction of dual functions. The general case follows by a change of variables, as is illustrated in Figure 5.1. We define where the transformation x \-^ y = x/h — i maps the support of b, to supp&Q , and Q'f to Go- Then'

since bk(i

and Jjc = hmdy. Moreover,

as claimed. We could have given a more constructive proof of Theorem 5.1 by expressing the dual functions as linear combinations of B-splines and determining the coefficients from the linear system which represents the orthogonality conditions. However, this would have been more technical, and the explicit form of the dual functions is not relevant for our main objectives. It is important to note that the bound for the norm of the dual functions is not uniform with respect to the relative width 0 of the cubes <2-, i.e.,

Hence, we cannot construct uniformly bounded dual functions for the spline spaces B/,, h > 0, which may contain B-splines with arbitrarily small support in D. By contrast, web-splines are associated with inner B-splines, which have at least one grid cell Qi of their support contained in D. This guarantees sufficiently large support for the dual functions, and it is plausible that uniform stability can be achieved.

56

Chapters. Approximation with Weighted Splines

5.2 Weighted Dual Functions For web-splines corresponding to a weight function of order y there exist locally supported, uniformly bounded dual functions A,, i.e.,

andsupp

const

We set where A/ is the dual function described in theorem 5.1, corresponding to the specific choice

with jc, the center of the grid cell <2i C Dflsupp^, (cf. section 4.4). Since all outer B-splines bj vanish on Q, for any inner index /, B

which proves the biorthogonality of the dual functions. To establish the boundedness, we have to estimate the factor

(cf. definition 4.6 of a weight function of order y). Since the m-cube j2-, which contains the support of A,, has diameter 0^/mh and d(xi) > h/2, we note that, for x € Q'

This implies maxx€g; w(Xi)/w(x) :< 1 and yields the desired uniform bound for || A, ||0. While the arguments of this section were rather technical, the results are not. Theorems 5.1 and 5.2 allow us to compute B-spline coefficients via scalar products, very much like in orthogonal expansions. Considering, e.g., a web-spline M/, e w eB/,,

Hence, some important techniques of orthogonal expansions apply to splines.

5.2

Stability

For most finite elements the size of linear combinations of basis functions corresponds to the size of the coefficients; i.e., estimates of the norms are uniform with respect to the grid width. This fundamental property is important for the convergence of iterative methods

5.2. Stability

57

as well as for the accuracy of numerical solutions. As we already remarked, the B-spline basis for B^ is not stable in this sense. The instability caused by the outer B-splines bj with small support in D was the principal motivation for the construction of web-splines, which possess the following familiar stability property. S3 Stability For a weight function of order y linear combinations of web-splines satisfy

where the constants in the estimates depend on D, w, and n, We first show that web-splines have the same normalization as B-splines, i.e.,

To this end we recall definition 4.9:

As we noted in section 4.4, the number of indices in /(/) as well as the absolute values of the coefficients eij are bounded, uniformly with respect to the grid width. Hence, to verify the bound for the norm of Bt, it suffices to show that the quotient

is x 1 for x e supp Bi . This follows from the inequalities

which are valid since the gradient of the distance function is bounded, d(xi) > h/2, and supp BI has diameter ;< h. Because of the local support of the web-splines, the inequality (5.4) extends to arbitrary linear combinations. The crucial fact is that, at any point x in the domain, B{ (x) is nonzero for x 1 indices / only. To explain this in more detail, we write

for a grid cell Q. Moreover, we define

These definitions are illustrated in Figure 5.2. Also shown (bounded by a thick line) is the union of the supports of the B-splines appearing in the definition of Bf. In effect, I(Q) is

58

Chapter 5. Approximation with Weighted Splines

Figure 5.2. A set I(Q) (marked with dots) and the grid cells Q(i) (shaded) for bilinear web-splines. a small set of neighboring indices / with ih close to Q, and Q(i) are the grid cells adjacent to ih, which contribute to the support of /?,-. Clearly, the number of elements in both sets, which we denote by |/((2)| and |Q(/)|, respectively, is ^ 1. This follows simply because supp Bj c Xi + [—const, const]"1/z. Returning to the proof of the upper bound in theorem 5.3, the triangle and the CauchySchwarz inequality imply

Noting that we obtain by summing the above inequality over all grid cells Q and interchanging sums

Since|Q(/)| ^ 1, the right-hand side is ^ /i"1 ||C||2, proving the upper bound for || ^. : c/fl,-||o. The lower bound is derived with the aid of the weighted dual functions A,. By theorem 5.2 (cf. also (5.3)) and the Cauchy-Schwarz inequality we have

Summing this estimate over / e /, at most maxc |/(Q}\ -< 1 repetitions can occur on the right-hand side. Hence, the sum is ;< h~m \\ ^. c, B, ||Q, as desired.

5.2. Stability

59

If we assume that the weight function is smooth, we can also obtain bounds for derivatives of web-splines. The following result is the analogue of a well-known inverse inequality for polynomials. 5.4 Bernstein Inequality If w is a weight function of order y which is ^-regular, i.e., which has bounded partial derivatives up to order i and satisfies

then

for linear combinations of web-splines of degree n > I. For standard weight functions,

and the hypothesis of the theorem is satisfied with t = n. Also admissible are weight functions constructed with the R-function method (cf. theorem 4.7). Since the functions

of the standard R-function system in Table 4.1 have bounded gradient, 1-regularity is preserved by Boolean operations. Usually, any reasonably defined smooth weight function meets the requirements of the theorem because normally differentiation decreases the order of decay toward the boundary at least by 1. However, there are counterexamples, like the bivariate function

which has order 2 and bounded gradient but is not 1-regular. For the proof of theorem 5.4 it suffices to estimate the norm of a single basis function because of the local support. Applying Leibniz's rule to the definition of a web-spline yields

We estimate each factor of the summands in turn. For x e supp fi, we obtain for d(x) = dist(jc, F) the bound

since jc, has distance > h/2 from the boundary. Hence,

60

Chapters. Approximation with Weighted Splines

Moreover, since bk(x} = b1^ \(x/ h — k) and there are only •< 1 coefficients ejj,

Combining the estimates (5.5) and (5.6) yields

and we obtain the desired bound for ||fi,||v by summing over all derivatives with order |a| < v. As an application of theorems 5.3 and 5.4, we derive a bound for the condition number of the Ritz-Galerkin matrix

for web-splines corresponding to a standard weight function. Since Gh is symmetric, cond Gh is the quotient of the largest and smallest eigenvalue. These two extreme eigenvalues can be computed by maximizing and minimizing the Rayleigh quotient

The numerator is < ||u/,||j, so that r(C) ^ h~2hm by the Bernstein inequality 5.4. On the other hand, since VH vanishes on 3D, we have / ||grad Vh \\2 > \\VH \\Q by the PoincareFriedrichs inequality A. 12. Hence, r(C) >: hm by the stability theorem 5.3. This shows that which is a standard estimate for Ritz-Galerkin approximations of second-order problems.

5.3

Polynomial Approximation

Error estimates for splines are based on local polynomial approximations. Hence, we begin by studying this simpler case. As an example, we consider the formula for the remainder of the nth-degree Taylor polynomial,

We can integrate over the fixed interval [0, h] if we multiply the integrand by the characteristic function /(*, /) of the triangle { ( x , t ) : 0 < t < x < h}. Then, by Minkowsky's inequality A.I, the L2-norm of the integral on the right-hand side can be bounded by

5.3. Polynomial Approximation

61

Consequently, it follows that

Similarly, we can estimate derivatives of the error The analogous multivariate estimates play a fundamental role for finite element error analysis. They are, however, much more difficult to derive since there is no convenient integral representation of remainders for Taylor polynomials or other approximations. 5.5 Bramble-Hilbert Estimate The error of the orthogonal projection Pn onto polynomials of total degree < n on a scaled domain hD satisfies

forO 2.3).

and with | • || denoting the standard semi-norm on He (cf, section

The proof is divided into three steps. (i) As usual, the dependence on h can be eliminated by scaling the variables. If p(x) is the orthogonal projection of /(;c) onto polynomials on hD, then q(y) = p(x), y — x/h, is the corresponding projection of g(y) = f ( x ) for D. Moreover,

These two observations reduce the proof to the case h — 1 . (ii) For h = 1 we consider first the special case /z = n + 1 . Combining the estimates for 0 < v < /z, we have to show that

We assume the contrary, i.e., that there exists a sequence g

(noting that \Pnfe\n+\ = 0). Because of the compactness of the embedding Hn+\ C Hn (cf. A. 11) there exists a subsequence gt with limit g and

To reach a contradiction, we show that the limit g is a polynomial of total degree < n and, as a consequence, must vanish (\\g\\n — 0). For the first assertion we observe that, for \a\ = n + 1 and any smooth function

62

Chapter 5. Approximation with Weighted Splines

Hence, all weak derivatives of order n + 1 of g are zero. For the second assertion we express the projection Pn with the aid of a basis of orthonormal polynomials pa as

From this representation we see that (gi, pa)0 = 0 and thus also (g, pa)o = lim{|£, pa)o = 0, so that g = 0, as claimed. (iii) The remaining cases \JL < n follow easily. The projection (5.7) is a bounded operator on Hi for any £ > 0. Hence, for /z > 0,

where we used step (ii) of the proof for the last inequality. Clearly, this estimate covers all cases v < fji. The case JJL = 0 follows directly from the definition of Pn. In theorem 5.5 we can replace const(D, n) /Z M ~ U by const(m, n) diam(D) M ~ v if we only consider a family of similar domains. For example, later on we will frequently apply the estimate for m-dimensional cubes. However, we emphasize that in general the dependence on the domain cannot be omitted since irregularities of the boundary may affect the constant. Usually, the Bramble-Hilbert estimate 5.5 is stated in a slightly different form. We describe below the original version, which is convenient for analyzing interpolation and quadrature schemes. 5.6 Functionals with Polynomial Kernel If a bounded linear functional A on H" *l(D) vanishes on polynomials of total degree < n, then

In view of

this result is a direct consequence of the Bramble-Hilbert estimate 5.5 with v = n = n+l. Theorem 5.6 can be generalized slightly. Instead of linear functionals we can consider linear operators

with finite rank which annihilate polynomials of degree < n. Arguing as above, it follows that This variant of the Bramble-Hilbert estimate is useful for deriving error bounds for piecewise polynomial projections.

5.4. Quasi-lnterpolation

5.4

63

Quasi-lnterpolation

In this section we determine the approximation order of weighted splines. A direct approach, i.e., an analysis of the error of the best approximation from the finite element subspaces, would be very difficult. The main reason is that computing the coefficients c, of the basis functions involves the solution of a linear system, which obscures the dependence on the approximated function u. Fortunately, there is a much simpler technique, referred to as quasi-interpolation. The coefficients c, are computed separately and depend only on values of w in the support of Bf . This greatly facilitates the derivation of error estimates. Local support and polynomial precision imply that web-splines approximate with the same order as local polynomial projections. Of course, since w eB/, C w B/, , the results for web-splines hold for weighted splines as well. The canonical quasi-interpolant for web-splines is the standard projector

whic h was already mentioned in connection with the construction of the dual functions A, in section 5.1. Since by theorem 5.2, B^ and A, are biorthogonal, PhBi = Bj. This implies, in particular, that the standard projector reproduces weighted polynomials, which are contained in the web-space w eB/, . Moreover, because of the local support of Bt and the uniform estimate for the norm of A,, Ph is locally bounded. We summarize these two important properties in the following theorem. 5.7 Standard Projector For any polynomial p of coordinate degree < n

Moreover, if w is an l-regular weight function of order y , then, for any grid cell Q,

where Q' — Ui6/(0supp #, C D is the union of the supports of all web-splines which are nonzero on Q H £>, The second assertion still requires a proof. As we already remarked, the result is a direct consequence of the properties of fi, and A,. By definition (5.1) of Ph, the norm of on Q D D can be bounded by

The first factor of the summands is •< h m/2 ||w||o,e, by theorem 5.2 and the second -< hm/2h~v by the Bernstein inequality 5.4. Since the number of summands is •< 1 and Qi C Q', we obtain the desired upper bound. We will now determine the approximation order of the standard projector P/,. As is expected, the local error coincides with that of polynomial approximations. However,

64

Chapter 5. Approximation with Weighted Splines

we have to take the boundary conditions, which are described by the weight function, into account. Since a linear combination of web-splines has the form

we can approximate a function u with full order only if u = wv with sufficiently regular v. This explains the hypothesis in the following theorem. 5.8 Approximation Order If u; is an ^-regular weight function of order y and v = u/w is smooth on Z>, men

for v < min(l,«). This imph'es, in particular, that web-splines w eB* and weighted splines wH&h have the optimal approximation order. The estimate is proved by analyzing the error separately on each grid cell Q. To this end we define Q' as in theorem 5.7 and denote by Q a smallest w -dimensional cube containing Q'. Moreover, we let v = £v be a smooth extension of v to Rm, as described in A.13. The key argument is to relate the local error to that of the projector Pn onto polynomials on Q. Since Ph reproduces weighted polynomials and uses only information from within the domain D where v = v,

the last inequality following from theorem 5.7. Moreover, since we assume that w has bounded derivatives up to order £ > v,

Finally, by the Bramble-Hilbert estimate 5.5,

Here we see the necessity of the extension v. The shape of the sets Q n D will in general depend on h, which affects the constant in the error estimate for the polynomial projection. This is why we defined Pn on the larger m-cube Q. Combining the above estimates, we have shown that

Summing this inequality over all grid cells Q completes the proof in a by now familiar fashion. The point is again that the cubes Q overlap only •< 1 times. Theorem 5.8 establishes the standard error bound

5.5. Boundary Regularity

65

for weighted splines of degree < n. However, to keep the arguments simple, we did not make the required regularity precise. This will be done in section 5.6, where we discuss the important special case of a standard weight function in more detail. While from a practical point of view the resulting sharp estimates are perhaps not needed, the theory would not be complete without them. In particular, the interplay of regularity and approximation order is crucial for the analysis of multigrid methods in chapter 7.

5.5

Boundary Regularity

As we saw in section 5.4, the accuracy of weighted spline approximations to a function u depends on the regularity of the quotient v = u/w. While theorem 5.8 does not make precise assumptions about the smoothness, examining the proof in more detail shows that the arguments require v e Hn+]. This is slightly more restricitive than the necessary assumption u e Hn+l since the division by w causes a loss of regularity near the boundary even if w is smooth. For a standard weight function, we will derive in the next section a sharp error bound. It is based on the following auxiliary regularity result [46, 45]. 5.9 Regularity of Quotients If w is a standard weight function and u = wv, then, for any subdomain D' c D with distance 8 to the boundary,

Moreover,

The first inequality (5.8) is a direct consequence of Leibniz's formula for the derivatives of the product u = wv. Rearranging the terms appropriately, we have

Dividing by w and recalling that w is smooth and equivalent to the distance to 3D, the desired estimate follows. The proof of the second inequality (5.9) is based on the following univariate estimate. 540 Regularity of Univariate Quotients

This estimate follows from the identity

66

Chapter 5. Approximation with Weighted Splines

Differentiating (t — l)-times and taking norms, we obtain with the aid of Minkowsky's inequality A.I

Changing variables in the inner integral, the right-hand side can be bounded by f completing the proof with const = 2 The univariate estimate 5.10 captures the essential part of inequality (5.9). It is generalized to several variables by a standard localization technique. With a smooth partition of unity,

we can write u as a sum of finitely many functions with small support:

(using, e.g., B-splines: Xk = t>k). As is easily seen, it suffices to prove (5.9) for each of the functions x\>u = wxvv. This local estimate implies

i.e., the desired global bound. Hence, it remains to consider functions u with small support. If the distance of supp u from 3D is > 0, the estimate (5.9) is trivial. Therefore, we assume that the support of u intersects the boundary and that the portion of 3D in supp u is nearly flat. Clearly, this can be accomplished by a suitable choice of the functions /„. As depicted in Figure 5.3, we require that, with an appropriate choice of coordinates, supp u is contained in the strip

where \fr is a smooth function with i/r(0) = 0. We first conclude from theorem 5.10 that the inequality (5.9) is valid for DO = [0, s]m if the boundary is flat (TJS = 0) and w(x) = t. We simply observe that tangential derivatives do not affect the estimate. For a curved boundary, we map the strip D^ to the m-cube DO with the transformation

Denoting by «, v, ... functions with respect to the transformed variables, the estimate (5.9) follows from the inequalities

5.6. Error Estimates for Standard Weight Functions

67

Figure 5.3. Local regularity estimate. The first and last inequalities hold because Sobolev norms remain bounded after a smooth change of variables. For the second inequality we note that

and observe that multiplication with the smooth quotient s/w is a bounded operation on Hl. This completes the proof of theorem 5.9. We could derive more general boundary estimates for higher order weight functions by iterating the inequalities (5.8) and (5.9). For example,

yields the appropriate estimate for a second-order weight function. However, higher order essential boundary conditions do not occur very frequently. Therefore, we have preferred to discuss only the slightly less technical case of a standard weight function.

5.6

Error Estimates for Standard Weight Functions

In this section we derive an optimal error bound for the projector

onto the web-space ioeB/, c wB/,. This estimate, proved in [46, 45], is based on the regularity results of the previous section. It sharpens theorem 5.8 by requiring only minimal smoothness of the approximated function. As is customary in approximation theory, we call such a result a Jackson inequality in memory of beautiful classical error estimates for trigonometric projection operators.

68

Chapters. Approximation with Weighted Splines

5.11 Jackson Inequality If w is a standard weight function, then

for any function u € Hk which vanishes on 3D. The proof is similar to that of theorem 5.8. In particular, we use the symbol ~ for extensions of functions to R m , write Q' for the union of all web-spline supports overlapping Q fl D, and denote by Q a smallest m-dimensional cube containing Q'. Again, we split the error on a grid cell Q in the form

where Pn is the orthogonal projection onto polynomials of degree < n on Q. Considering for simplicity only the slightly more difficult case I > 0, theorem 5.7 implies the local estimate

At this point, a more subtle argument is necessary. We cannot simply bound ||u;/||£ in terms °f \\f\\t but have to exploit that w is small near the boundary. By Leibniz's rule

Since the relevant derivatives of w are bounded,

for any subdomain D' of D. Applying this estimate to (5.10), the right-hand side can be bounded by a constant times

We claim that this expression is

To this end we distinguish two cases, depending on the distance 8 = dist((2, 9D) of Q to the boundary. (i) 8 < h: Since diam(<2') ^ h and w is equivalent to the distance to 3D, we have maxQw < h. Moreover, by the Bramble-Hilbert estimate 5.5,

5.6. Error Estimates for Standard Weight Functions

69

Using this inequality with v = £, 0, i — 1, and IJL = k — 1, it follows that r •< s\ . Of course, we can replace Q D D and Q' by the larger set Q. (ii) 8 > h: In this case, Q C D, and

Applying (5.11) with v = £, 0, 1 — 1, and //, — k, we obtain

by the first estimate (5.8) of theorem 5.9. Summarizing, we have shown that

Summing this estimate over all grid cells Q, we obtain in the usual way the global bound

This completes the proof in view of the boundedness of the extension operator £ : v h-> v for Hk~l (D) (cf. A. 13) and the second estimate (5.9) of theorem 5.9. From a practical point of view the sharp error estimate of theorem 5. 1 1 is not always essential. For smooth problems the simpler version of theorem 5.8 is adequate for most applications. The order of approximation is usually limited by the degree of the finite elements rather than the precise differentiability of the data. For problems with singularities, caused, e.g., by corners or edges of the domain or discontinuities of coefficients of the differential operator, this is not the case. However, such problems require more sophisticated techniques (e.g., hierarchical refinement) if full approximation order should be maintained. Despite these remarks, there are several instances where an optimal Jackson inequality is crucial. An example is the Aubin-Nitsche duality principle 2.9, yielding a bound for the L 2 -error. 5.12 Error of Ritz-Galerkin Approximations

Let w be a standard weight function and u^ the Ritz-Oalerkin approximation from w or w&h of an HQ -elliptic problem

Moreover, assume that the dual problem has the standard regularity; ie,f the solution #* for theright-handside /* satisf I/* lo.

where The proof involves nothing more than specializing the hypotheses of theorem 2.9. With H = HQ , H* — Li.eh — /*, and V/, either of the spaces w eB/, or if B/,, we have

70

Chapter 5. Approximation with Weighted Splines

and the infimum can be bounded by

by the Jackson inequality 5.11 and the regularity assumption. The precise error bound of theorem 5.11 is also needed for establishing the gridindependent convergence rate of multigrid methods. Here, we use the Jackson inequality to relate the accuracy of approximations on different grids (cf. section 7.4).

Chapter 6

Boundary Value Problems

In this chapter we discuss the approximation of typical boundary value problems with the spline spaces introduced in chapter 4. All examples fall into the general framework described in section 2.4: the differential equation for the solution u has a weak formulation as a variational problem where H is a Hilbert space which incorporates the boundary conditions. The Ritz-Galerkin approximation Uh = £],-e/ M; #,- is obtained simply by replacing u by M/, and v by the basis functions B^. Although conceptually very straightforward, this approach covers a wide range of applications. In section 6.1 we discuss the general inhomogeneous Poisson problem as a typical example for essential boundary conditions. Such constraints have to be incorporated into the finite element subspaces and thus require a weight function. In contrast, natural boundary conditions are automatically satisfied by solutions of the variational problems and thus permit a much simpler approximation. This is illustrated in section 6.2 for the computation of incompressible flow. Section 6.3 combines both cases and considers more generally second-order differential operators with variable coefficients. In section 6.4 we describe the plate problem as an example of a fourth-order equation. Finally, sections 6.5 and 6.6 are devoted to linear elasticity, which is the classical and perhaps most important engineering problem in finite element analysis.

6.1

Essential Boundary Conditions

As a typical model problem we consider Poisson's equation with inhomogeneous Dirichlet boundary conditions: This basic boundary value problem describes many physical phenomena. The classical membrane problem was already mentioned in section 2 (cf. Figure 2.1). Other examples include heat conduction, electrostatic and magnetic potentials, and fluid flow. 71

72

Chapter6. Boundary Value Problems

To derive a variational formulation, we first eliminate the inhomogeneous boundary conditions by setting where u e HQ (D) and g is an extension of g to all of D. Then we multiply the differential equation by v e HQ and integrate by parts. This yields

as the associated variational problem. Applying the Lax-Milgram theorem 2.7, we obtain the following existence result. 6.1 DirichJet Problem The inhomogeneous Dirichlet problem (6.1), corresponding to the bilinear form

and the linear functional

has a unique solution

we can use any of the spaces tuB/,, w eB/,, and u;B/, (D), described in chapter 4, with a weight function of order 1 which vanishes on all of dD. The standard choice are web-splines (cf. definition 4.9). The simpler weighted splines tuB/, are adequate for small systems, where stability considerations do not play a significant role. Hierarchical refinement is recommended when the solution has singularities. To compute the right-hand side of the Ritz-Galerkin system we have to define an extension g of the boundary data. If no exact analytic expression is available, we can use a linear combination of B-splines

defined by minimizing fdD \g — gh\2- The set dK contains all indices for which b^ is nonzero on 3D. If the domain is defined via the R-function method, we can apply transfinite

6.1. Essential Boundary Conditions

73

interpolation techniques as described in [78]. These methods allow us to handle other types of boundary conditions as well and yield exact representations of the data. If the boundary, the weight function, and the data /, g are smooth, web-splines provide full approximation order. By Cea's inequality 2.8 and Jackson's inequality 5.11,

with eh = u — Uh- For the L2-error we gain a factor h by the Aubin-Nitsche duality principle 2.9. Since a is symmetric, the dual problem

is a homogeneous Poisson problem with right-hand side eh. Hence, ||w*||2 :< lkfcllo» and theorem 5.12 yields For both the H1- and the L2-norm, the convergence rate is optimal. To illustrate the above estimates, we consider a test problem with a known solution which permits the computation of exact error data. As shown in Figure 6.1, we choose a domain D which is bounded by the algebraic curve

Moreover, we set g = 0 and define the right-hand side / so that

Figure 6.1. Solution u ofPoisson's equation for test data.

74

Chapter 6. Boundary Value Problems

solves— AM = /. This problem is slightly more complicated than examples with elementary boundaries (rectangles, ellipses, etc.), to rule out accidental simplifications in the numerical tests. Figure 6.2, visualizes the error of web-spline approximations, using the polynomial w, which defines 3D, as a natural weight function. In the top two diagrams, we plotted \\6h Ho and \\£h II i versus the grid width h for degrees n = 1 , . . . , 5 (markers O, A, D , iV , $ ). The bottom diagrams show the numerically computed convergence rates

As expected, ro ^ n + 1 and r\ % «, since the solution is infinitely differentiable. The example confirms that using a higher degree pays off, in particular since the dimension of web-spaces is essentially independent of the degree. For example, for h = 2~3, the /^-error of quintic approximations is roughly by a factor 107~3 smaller than for linear web-splines.

Figure 6.2. Logarithmic errors log ||e/, ||o, log ||e/, || i (top) and convergence rates (bottom) for web-spline approximations of degrees 1, . . . , 5 and grid width h = 2~^, IJL = 0, . . . , 5.

6.2. Natural Boundary Conditions

6.2

75

Natural Boundary Conditions

While essential boundary conditions have to be incorporated into the solution space, natural boundary conditions are automatically satisfied by weak solutions. As an example, we consider the Neumann problem

where 31- denotes the normal derivative, i.e., d^u = £gradw with £ the outward unit normal to 3D. Despite the simplicity of the equations there are some subtle points. From the identity

we see that the data must satisfy the compatibility condition

The necessity of this condition will also become apparent from the physical interpretation in the example discussed below (cf. Figure 6.3). Moreover, a solution is only determined up to an additive constant. These facts have to be taken into account in the associated variational problem. As usual, we derive the weak formulation by multiplying with a test function v and integrating by parts,

Having not assumed that i; vanishes on 3D, the boundary conditions for u appear naturally on the right-hand side; i.e., we can define

Since the restriction to the boundary is a continuous map from Hl(D) to L2(3D) (cf. A. 10), A. is bounded: Obviously, the bilinear form gradwgradi; is also bounded on Hl. A slight difficulty is that a is not bounded from below since it vanishes on constants. This problem is resolved by working with the subspace

Using that the projection

1 onto constants is zero on //]_, we have

76

Chapter 6. Boundary Value Problems

by the Bramble-Hilbert estimate 5.5 and with | • |i denoting the Sobolev semi-norm (cf. section 2.3). Hence, the ellipticity of a on H]_ follows, and we can apply the Lax-Milgram theorem 2.7.
The Neumann problem (6.2), corresponding to the bilinear form

and the linear functional

has a unique solution u € H[, provided mat the compatibility condition (6.3) is satisfied and / and g are square integrable. We note that the compatibility condition was not needed to establish the hypothesis of the Lax-Milgram theorem. It is necessary in order to conclude that a sufficiently regular solution of the variational problem solves the boundary value problem (6.2). If (6.3) is satisfied, A. vanishes on constants, so that

Choosing v e HQ proves that — Aw = /, since in this case the variational equations are identical to those for Poisson's problem. Hence,

which implies d±u = g since all test functions v e Hl are admissible. The Ritz-Galerkin approximation can be computed with any of the spaces B/,, eB/, (web-splines with w = 1), and B/,(P); a weight function is not necessary. We can ignore that these spaces are not contained in //}, i.e., that the constraint /D w/, = 0 is not satisfied. Considering, e.g., web-splines (with w = 1), we solve the semidefinite system

for Uh = X!/e/ M( Bi • Th£ compatibility condition guarantees that A. vanishes on constants, so that a solution exists. Subtracting a constant from Uh does not affect the bilinear form, which involves only gradients. Hence, we can select a solution with the proper normalization. As an application we consider the flow of an incompressible fluid in a channel, as is illustrated in Figure 6.3. If the velocity field V has a potential, i.e., if

then conservation of mass implies

6.2. Natural Boundary Conditions

Figure 6.3. Streamlines (top) and velocity through a channel with two islands.

77

(bottom) of incompressible flow

Moreover, the flow rate at the boundary is

In the example of the figure, g equals Ujn > 0 (— i>in) at the left (right) vertical boundaries and is zero at the horizontal and inner boundaries. Here, the compatibility condition fdD g = 0 assures that the total mass of the fluid is conserved (inflow = outflow). The numerical computation of the streamlines is very sensitive. Since the RitzGalerkin approximation does not satisfy the natural boundary condition exactly, some of the numerically generated streamlines might intersect the boundary. Of course, as the grid width becomes small, the percentage of such cases diminishes. The data for Figure 6.3 were computed using quadratic nonweighted web-splines. As is illustrated in Figure 6.4, only relatively few inner quadratic B-splines (marked at the lower left corners of their supports) are coupled with outer B-splines. Hence, eB/, (the web-space with if = 1) is almost a standard spline space. This simplifies the assembly of the Ritz-Galerkin matrix since for the majority of entries precomputed values can be used.

78

Chapter 6. Boundary Value Problems

Figure 6.4. Extended inner B-splines b, ( J ( i ) ^ 0). For a smooth solution, as in the above example,

by Cea's inequality 2.8 and theorem 5.8. This requires a minor additional argument since, strictly speaking, we have to assume fD uh = 0. To remove this constraint, we let u/, be the best approximation to u from eB^ in Hl and define an approximation vh = VH — Povh € //} by subtracting the orthogonal projection onto constants. Then we have

by Jackson's inequality 5.11. Moreover, since PQU = 0,

by the Bramble-Hilbert estimate 5.5. Combining both estimates proves that the //'-error is :< hn also for the constraint approximation vh . The same argument shows that the Aubin-Nitsche duality principle 2.9 applies. There, we have to consider the auxiliary problem

with eh = u — uh. Elliptic regularity implies ||«*||2 ^ \\eh\\o, so that

where we argue as above to remove the constraint on the finite element subspace. Hence, the L2-error is of order O(hn+]).

6.3. Mixed Problems with Variable Coefficients

6.3

79

Mixed Problems with Variable Coefficients

As a generalization of the two basic problems discussed in the previous section we consider the boundary value problem

where A(JC) is a symmetric positive definite matrix, which is smooth on D, ao and a, are bounded nonnegative functions, F is a subset of 3D with nonzero (m — 1) -dimensional measure, and £ is the outward unit normal for 3D. For simplicity we have assumed that the essential boundary conditions on F are homogeneous. The reduction to this case has been discussed for Poisson's equation in section 6. 1 . We recall that we do not distinguish between row and column vectors. This means we write

for vectors £ and 77. Since the matrix A is symmetric, no ambiguity arises. A brief comment about the assumptions on the sign of ao and a is in order. Their relevance becomes apparent by considering the univariate problem For both ao < 0 and a < 0, there are choices of the parameters for which the boundary value problem has no solution (e.g., (ao, a) equal to (— n2/4, 0) or (0, — 1)). Hence, without restrictions on the coefficient functions the analysis becomes much more subtle. Returning to the discussion of the general boundary value problem (6.4), we derive a variational formulation in the usual way. First, the essential boundary conditions are incorporated into the solution space, which we choose as Then, integrating by parts, we obtain

since v vanishes on F. Replacing the integrand of the boundary integral by (g — au)v leads to the following result. 63 General Second-Order Elliptic Problem The boundary value problem (6.4) corresponding to the bilinear form

and the linear functional

has a unique weak solution u e H£ if f and g are square integrable.

80

Chapter 6. Boundary Value Problems

The assumptions on the coefficients and the data, specified at the beginning of this section, are just right to guarantee ellipticity as required by the Lax-Milgram theorem 2.7. Since A, ao, and a are bounded,

Therefore,

by the Cauchy-Schwarz inequality and because ||M ||o,3D ^ II" II i by the boundedness of the trace operator (cf. A. 10). For the lower bound we note that

This implies

and for u € //r the right-hand side is >; ||M||^ by the Poincare-Friedrichs inequality A. 12. For the boundedness of A, the second requirement of the Lax-Milgram theorem, the assumptions / e L2(D), g e Lz(dD\r), are clearly sufficient. It would be necessary that these functions belong to the dual spaces of // r (D) and // r (D)|9 D \r, respectively. For numerical and practical considerations, specifying these minimal regularity requirements is rarely of importance. As an example, we consider the heat conduction problem depicted in Figure 6.5. Four pipes with radius r and distance 2r are enclosed in a cylindrical isolation with radius 5r. They have a constant temperature u = M p jpe, and the heat flux at the cylinder boundary satisfies

where ua\T < «pipe is the outer temperature. Assuming that the isolation material is homogeneous, AM = 0 in the simulation region. Hence, we obtain a special case of the boundary value problem (6.4) with A the unit matrix, aQ = Q,a = y/r, f = 0, g = au^T, and F the union of the circular boundaries of the four pipes. Because of the apparent symmetry we have to compute the solution only for a quarter of the domain. To this end we add the boundary condition 3 ± w = 0 at the horizontal and vertical artificial boundaries. Figure 6.5 visualizes the temperature distribution u(x) for y/r = 1. The values of Mpipe, « a ir> and r are not relevant. We can change them by scaling and translating the solution and the variables. A convenient choice is

For these values the level curves in Figure 6.5 correspond to u = —0.1, . . . , —0.7.

6.3. Mixed Problems with Variable Coefficients

81

Figure 6.5. Temperature distribution in a cylindrical isolation of four pipes. We computed a numerical solution with web-splines, using

as a natural weight function. To judge the accuracy, we determined the relative error of the artificial boundary condition in the maximum norm,

The left-hand diagram in Figure 6.6 shows log e as a function of h = 1~* for /JL = 0 , . . . , 5, using as in section 6.1 the markers O, A, D, "fr, & for the degrees n = 1, . . . , 5. On the right we have listed the convergence rates Q, which were estimated from the overdetermined equations The numerical results indicate thatthe approximation order is optimal, i.e., e(n, h) = O(hn). This does not follow from the standard finite element error analysis, which uses only the Sobolev norms || \\i. Boundary estimates and convergence rates in the maximum norm are much harder to obtain. We included one numerical example (not confirmed by the theory) to illustrate that weighted spline approximations generally perform very well regardless of the type of error measure. The simplicity of obtaining existence of weak solutions and error estimates for RitzGalerkin approximations is deceptive. Let us mention an important theoretical aspect which has been neglected when discussing numerical approximations: regularity of solutions. As

Chapters. Boundary Value Problems

82

Figure 6.6. Error of the normal derivative (left) and estimated convergence rates (right). is well known, even moderate smoothness of solutions cannot be taken for granted. For the boundary value problem under consideration, the Lax-Milgram theorem just asserts that u has first-order partial derivatives in LI- This is not sufficient to conclude that the differential equation or the natural boundary conditions are satisfied, not even in a weak sense, i.e., pointwise almost everywhere. To accomplish this we need that u e H2. Then, for v e H

since integration by parts is justified. Choosing test functions v which vanish on all of 3D, we see that [...] = 0, a.e. Moreover, since the restrictions of test functions to the boundary are dense in L,2(dD\r), it also follows that {...} = 0, a.e. This completes the cycle. From a classical solution we have obtained the variational formulation via integration by parts. Elegant (but soft) functional analysis yields the existence of weak solutions. After improving the regularity of solutions (the hard part; cf., e.g., [59]) we arrive back at the pointwise interpretation of the differential equation.

6.4

Biharmonic Equation

Many engineering applications have a natural variational formulation. An example is the clamped plate problem, illustrated in Figure 6.7. A transverse force with normalized density / is acting on a plate of width d, which is fixed in horizontal position at the boundary. The equilibrium position u(x\ , x^) is determined by minimizing the potential energy

83

6.4. Biharmonic Equation

Figure 6.7. Clamped plate. over all u e HQ. In the first integral, v = A/(2(A + />i)) € (0, 1/2) is the Poisson coefficient of the plate, determined from the Lame constants A. and /z [89]. Because HQ is the closure of smooth functions u with compact support in D,

Hence, the energy functional simplifies to

Of course the constant v cannot just disappear. It is hidden in the normalization of /. The actual force per unit surface area is

where E = /u(3A + 2/z)/(A. + JJL) is the Young modulus. The ellipticity of the bilinear form a on HQ is easily established. As usual, only the lower bound requires a more elaborate argument. By (6.5)

Moreover, by the Poincare-Friedrichs inequality A. 12

for u e HQ, so that \\u||2 ^ |w| 2 . In view of (6.6) this proves the lower bound for a. As a consequence we obtain the following result.

Chapter 6. Boundary Value Problems

84

6.4 Clamped Plate Hie boundary value problem, associated with the bilinear form

and the linear form

with / € La, has a unique solution u € /f0. The weak formulation of the plate problem corresponds to the biharmonic boundary value problem as is seen in the usual way by integrating by parts. For the Ritz-Galerkin approximation

we can use any of the spaces wB/,, w e B/j, or tuB/,(D) with a weight function of order 2. Figure 6.8 shows a numerical solution, computed with web-spaces of degree 2 and grid width h = 1/4 (dim weB/, = 220). In this example the plate has the form of a disc of radius 2 with a rectangular gap and is pulled down by its own weight (/ = —const). Because of the simple geometry the R-function method suggests itself for the construction of the weight function w. To this end, we have to represent the domain D in terms of elementary

Figure 6.8. Displacement (magnified) of a clamped plate under gravitational force.

6.4. Biharmonic Equation

85

sets, described by simple inequalities. For example, we can write D — DI Pi #2,3 with

With the aid of R-functions the Boolean operations are translated into expressions which combine the elementary weight functions wv (cf. section 4.3). Using the standard R-function system of Table 4.1, we obtain

Up to scaling, w = w^ models the qualitative shape of the displacement u fairly accurately. Hence, we obtain good results even with relatively low-dimensional spline spaces. The numerically estimated relative errors for the displacement u of the example in the figure are

respectively. For a 2-regular weight function, for which u/wis smooth on D, the error in the energy norm is of optimal order,

by Cea's inequality 2.8 and theorem 5.8. One can also show that the L2-error is of order O(hn+}} for n > 2. However, this requires a generalization of Jackson's inequality 5.11, which we proved only for a standard weight function vanishing, by definition, linearly on 3D (cf. the remark at the end of section 5.5). For the plate problem stability of the finite element basis is particularly important. Because of the fourth-order differential operator the condition number of the Ritz-Galerkin matrix grows like h~4 for standard subspaces. For bases, which are not uniformly stable with respect to the grid width, cond Gh becomes prohibitively large. For example, for cubic approximations uh e wB/, of a clamped circular plate with radius 1,

Already for h = 1/8 the MATLAB "cond" command [63] returns the value "inf"; i.e., the Ritz-Galerkin matrix is considered singular in double precision arithmetic. In contrast, the condition numbers for the stabilized web-spaces (uh e u;eB/,) are considerably smaller. In the left diagram of Figure 6.9 we have plotted log(condG/,) versus the logarithm of the dimension d of the web-space for the degrees n = 2, . . . , 5 (markers A, D , "fr, &). By comparing with the auxiliary dashed line (cond = const d2 x /z~ 4 ), we see that cond G/, does not grow faster than predicted by the theory. The right diagram shows the number of iterations required by the ssor-preconditioned conjugate gradient method for a relative accuracy < le — 10. As expected, the computational work remains acceptable also for small grid widths. In all cases the computing times are small compared to the time for assembling the Ritz-Galerkin system (< 2 seconds on a PC with Pentium 800 processor). If inhomogeneous boundary conditions are specified in (6.7), we can reduce the problem to the homogeneous case, as discussed in section 6.1. We subtract from the solution u a

Chapter 6. Boundary Value Problems

86

Figure 6.9. Logarithmic plots of the condition number condGh and the number of peg-iterations versus the dimension of the web-space u^B/,. function g which satisfies the boundary conditions. Then, the difference is a solution of the homogeneous problem. If an explicit analytic extension is not available, we can construct a spline extension g/, of the second-order boundary data. A simple procedure is to minimize

over linear combinations

6.5

of B-splines with some support on 3D.

Linear Elasticity

Models in structural analysis have been the starting point for developing finite element methods [90, 3]. In this application standard finite elements have a natural physical interpretation as small building blocks of elastic materials. Simulations in elasticity still represent today a very important branch of finite element analysis. The basic physical model is illustrated in Figure 6.10. An elastic solid, occupying a volume D, is fixed at a portion F of its boundary and subject to a volume force on D and a boundary force on 3D\F with densities (f\ , /2, fo) and (g\ , g2, £3), respectively. These forces result in a small deformation of the solid, described by a displacement M(JC) e M3 of the material points jc e D. Typically u is very small, and large distortions indicate excessive forces. The deformation is determined by minimizing the total energy

over all admissible displacements

.5. Linear Elasticity

87

Figure 6.10. Elastic deformation (magnified) of a dam under lateral pressure and g ravitational force. with //jl the subspace of functions in H1 which vanish on f (cf. section 6.3). Here, a is the stress and s the strain tensor, and we used the notation

We refer to [60] for a derivation of the energy functional and merely give the definition of the two symmetric tensors involved:

where trace e = £1,1 + £2,2 + £3,3- The constants A. and /x are the Lame coefficients, which describe the elastic properties of the material. The second identity, which relates the stress and strain tensor, is known as Hooke's law, valid for an isotropic, homogeneous material. Writing the first part of the energy functional (6.8) in the form

we see that it corresponds to a symmetric bilinear form a (u, u). Since the integrand involves only first-order derivatives, it is straightforward to establish the boundedness of a in the space (//p)3, which is equipped with the product norm

Chapter 6. Boundary Value Problems The derivation of the lower bound, necessary for ellipticity, is much harder. It involves Korn's inequality

proved, e.g., in [65] (cf. also the original work of Kom [56, 57]). Moreover, using that u vanishes on F, we can eliminate the term / MM on the right-hand side with the aid of the Poincare-Friedrichs inequality A. 12. Having established the hypothesis of the LaxMilgram theorem 2.7, we obtain the following familiar result.

6.5 Elasticity Problem The variational problem corresponding to the bilinear form

and the linear functional

with

3

2(D)

and g e L2(BD\r)3 has a unique solution (MI, u2, «a) €

Integrating the variational equations

by parts (recalling the definition ek^ = ^(dkUf + d^Uk)), it follows that

for a sufficiently smooth solution u (£ denotes the outward normal). Because of the symmetry of a, the sums of the expressions in brackets are equal in both integrals. For example, the first integral simplifies to

Moreover, arguing as in the last paragraph of section 6.3, both integrals must vanish separately. Hence, we can read off the boundary value problem associated with the quadratic form (6.8), which is referred to as the Lame-Navier system. We have assumed that the elastic solid is fixed at a surface F c 3D. However, this need not be the case. The entire boundary could be subject to forces and deformed.

6.6. Plane Strain and Plane Stress

89

Both the variational formulation and the differential equation remain valid. However, the displacement u is not unique and compatibility conditions must be satisfied. Similar problems were encountered for the basic Neumann problem discussed in section 6.2. Assembling the Ritz-Galerkin system is slightly more complicated than for scalar problems. Therefore, we discuss it in more detail. Using web-splines as the basis for the finite element subspace, each component uv of the vector u is approximated separately by a linear combination

where J5,|r = 0. Equivalently, we can write

where Hence, the Ritz-Galerkin system GU = F has block structure. The block (k, i) of G is the (3 x 3)-matrix with entries

and the fcth block of the vector F is the 3-vector with components

Since the basis functions Biv are nonzero in only one component, the tensors a and e simplify slightly. For example,

and

according to the definitions (6.9) and (6.10).

6.6

Plane Strain and Plane Stress

In many applications of linear elasticity a reduction of the dimension is possible. Examples are long objects with constant cross section, thin plates, and discs. Two common simplifications assume that either the strain or the stress components are zero in one coordinate direction. We discuss each case in turn.

90

ChapterG. Boundary Value Problems

The plane strain model applies to objects with a constant horizontal cross section D which are subject to horizontal forces and have no vertical displacement. These assumptions imply so that Hooke's law (6.10) simplifies to

We can write the horizontal components of this identity in vector form by introducing the notation Expressing also the Lame constants in terms of the Poisson ratio v e (0, 1/2) and the Young modulus E > 0 via

we obtain the two-dimensional stress/strain relation

With these definitions, specializing theorem 6.5 yields the following variational formulation for the plane strain model. 6.6 Plane Strain Model If / and g are square integrable densities of horizontal forces, applied to an elastic object with constant cross section D and vertical displacement u$(x\ , JC2> = 0, then

is determined by

where e' We note a minor technical point. In the product a : e the mixed term o\ ,221,2 appears twice; i.e., the product of the two tensors equals

(°r3,^£3,£ = 0^,3^,3 = 0 for plane strain). This is the reason for the modification £ — > • £ ' in the bilinear form. Alternatively, we could have doubled the third diagonal element of \L strain-

6.6. Plane Strain and Plane Stress

91

Figure 6.11. Displacement (magnified) for a tunnel with constant cross section. As an example, Figure 6.11 visualizes the displacement field (x\, Jt2) i-> (MI , M 2 ) for a tunnel under normal boundary forces. A numerical solution was computed with cubic web-splines for the parameters E = 30 * 109N/m2 and v = 0.2 (beton). The two fixed boundary parts F were modeled with the canonical weight function w(x) = x2. We now turn to the description of the other possible simplification mentioned at the beginning of this section. The plane stress model applies to objects with a uniform vertical thickness which is small compared to the horizontal dimensions. Moreover, it is assumed that o does not depend on *3 and

i.e., that the stress tensor has only horizontal nonzero components. As for the plain strain model, we obtain a two-dimensional version of Hooke's law (6.10). First we see that

However, e3,3 is in general nonzero, which means that small vertical deformations are possible: We can determine £3,3 from the equation 03 3 = 0, which yields

Substituting this formula into (6.10) and replacing the Lame constants by the Poisson ratio

92

Chapter 6. Boundary Value Problems

Figure 6.12. Displacement (left, magnified) and principal stress (right) of a rotating excentric disc. and the Young modules, a short computation yields the identity

for the horizontal components q_ = (a\,\, 02,2, a\,i) of the stress tensor. The variational formulation for the plane stress model is identical to that for plane strain, except that the matrix <2 strain ls replaced by <2stress- Rather than repeating theorem 6.6 almost verbatim, we state an equivalent version, using the characterization of weak solutions as minimizers of the energy functional. 6.7 Plane Stress Model If / and g are square integrable densities of horizontal forces, applied to an elastic object with small constant vertical thickness, and if cr has only horizontal components, then the first two components (u i , «2> of the displacement minimize

over all u e (H^)2, where Figure 6.12 shows as an example a rotating steel disc (E = 212*10 9 N/m 2 , v = 0.28), fixed at a slightly eccentric axis with radius r r . The centrifugal force with ||/|| = const r yields a deformation which, for the boundary of the disc, is indicated by a dashed line. The level curves {amax = const} on the right visualize the distribution of the principal stress, which is the largest eigenvalue of the matrix a. As is expected, the largest stresses occur close to the side of the axis with points farthest from the center.

6.6. Plane Strain and Plane Stress

93

Once more, we check for this example, the accuracy of the numerical approximations. Since weighted splines are continuously differentiable for n > 2, we can compute the pointwise residuals for the Lame-Navier boundary value problem (6.11):

where g = 0 for the rotating disc. This would not be possible for standard finite element approximations which usually merely belong to Hl. Figure 6.13 shows the logarithmic errors for a normalized right-hand side (||/||o = 1) and degrees n = 2, . . . , 5 (markers A, D, "fr, $). We expect that ep<je = O(hn~l) since the residuum of the partial differential equation involves second-order derivatives. This is confirmed in the example. Actually, the numerically estimated rates are slightly better, as one sees by measuring the slopes in the diagram. Similarly, the norms e^c of the residuum of the natural boundary condition are in good agreement with the optimal estimate O(hn).

Figure 6.13. Li-norm of the residuum for the differential equation (left) and the boundary condition (right) versus the grid width h = rr2~ M , /j, — 0, . . . , 3.

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Chapter 7

Multigrid Methods

For uniform B-spline bases the use of multigrid techniques suggests itself. Such algorithms provide the most efficient iterative solvers for large Ritz-Galerkin systems. The solution time is proportional to the number of unknowns, hence, asymptotically optimal. While for classical iterations like successive overrelaxation or conjugate gradients the convergence rate deteriorates as the grid width becomes small, multigrid schemes reduce the error by a grid-independent factor in each step. The corresponding mathematical theory is fascinating and has long been an area of very active research. As is outlined in this chapter, the general multigrid theory applies without major modifications to web-splines. We first discuss in section 7.1 a trivial univariate model problem to illustrate the main multigrid ideas: smoothing and coarse grid correction. Then we describe and analyze a standard multigrid method for the web-spaces w eB. In section 7.2 we construct canonical grid transfer operators, and in section 7.3 we discuss the recursive implementation of an iteration step. Sections 7.4 and 7.5 are devoted to the convergence proof. It is based on the smoothing property of Richardson's iteration and the sharp error estimate for quasi-interpolants.

7.1

Multigrid Idea

To illustrate the basic multigrid idea, we consider in this section the univariate model problem

This trivial example already exhibits some typical features of finite element discretizations. Moreover, the components of a multigrid algorithm can be explained in the simplest possible setting. The standard finite element approximation

with hat-functions bt = bl(-/h — i) (cf. section 3.1 and Figure 7.2) is computed from the 95

96

Chapter 7. Multigrid Methods

tridiagonal Ritz-Galerkin system

with gkj = fQ b'jb'k and fa = fQ b ^ f . Proceeding as for realistic multidimensional problems, we solve this system by an iterative method. We choose Richardson's scheme, which updates an spproximation U % U* of the coefficient vector by subtracting a multiple of the residual: The parameter y is chosen as 4/ h = IJGUoo, so that the tridiagonal iteration matrix

has eigenvalues in (0, 1). As is typical for the iterative solution of elliptic equations, Richardson's method is extremely slow because the largest eigenvalue

of £ — y~*G tends to 1 as the grid width becomes small. However, we can accelerate the convergence by exploiting that the discretizations corresponding to different grid widths are related; they approximate the same continuous problem. The obvious way is to use the solution «/, as an initial guess for computing w/,/2- Surprisingly, a more substantial improvement is obtained with the opposite procedure, i.e., by passing information from fine to coarse grids. To explain this key multigrid idea, we analyze the error reduction

of Richardson's iteration in more detail. As is illustrated on the left of Figure 7.1, highly oscillating error components are strongly damped. The diagram shows the modification of a randomly chosen initial error eu = £^ c, &, with coefficients C = U — U* by two Richardson steps (thin lines). Already the error ev of the approximation V after two steps has become slowly varying. This smoothing property of Richardson's iteration is made more precise with the diagram on the right of the figure. The top two graphs show the reduction factors Q(9), 9 = n, 2n, . . . , (\/h — !)TT, for the Fourier basis ee, which interpolates the sine-functions, i.e.,

As is apparent, the convergence for low frequencies is poor (£#(#) ^ 1 for 9 = TT). Such frequencies dominate the error after a few steps, as is the case for the error of V in the example:

7.1. Multigrid Idea

97

Figure 7.1. Errors after Richardson iterations and coarse grid correction (left) and reduction factors for the Fourier basis {sin(9vh)}o
by considering the associated variational problem. To this end we assume that the vector R corresponds to a function r, i.e.,

This means that W* are the coefficients of the Ritz-Galerkin approximation to

Clearly, the coarse grid system should approximate the same problem, so that

where bj = bl (-/(2h) — /) are the hat-functions on the coarse grid (cf. Figure 7.2). Since the Ritz-Galerkin approximations are close, a correction based on W leads to a substantial improvement of the vector V. It remains to discuss the grid transfer. We must express R in terms of R since we do not want to determine the function r explicitly. Moreover, we have to compute from W an

98

Chapter 7. Multigrid Methods

Figure 7.2. Subdivision of hat-functions. update W % W* on the fine grid. Both tasks are easily accomplished by subdividing the coarse grid hat-functions. As is illustrated in Figure 7.2,

Hence, it follows that

where the matrix

contains the subdivision coefficients. Subdivision is also used to transfer the coarse grid correction W back to the fine grid, i.e.,

Finally, the correction P W ~ W* = V — U* is subtracted from the current approximation on the fine grid: This completes one step U -> V —>• W of the so-called two-grid iteration. For the example on the left of Figure 7.1, the L2-norm of the error ew after the coarse grid correction (thick line) is smaller by more than 90% than H^JIo- To reach the same accuracy would require 250 additional Richardson iterations. The good performance of the two-grid iteration is confirmed by the convergence factors Qr(0) for the Fourier basis, shown on the right side of the figure. For all frequencies 9, Qj(9} is less than 1/3. Despite the substantial improvement it seems that we have not gained much, since the coarse grid system must be solved exactly. However, a recursive application of the correction strategy suggests itself. We interpret one iteration step U —> W as an algorithm

7.2. Grid Transfer

99

Figure 7.3. Program structure of the v-cycle (left) and convergence rates of the Richardson and multigrid iteration (right). which improves an approximation U of the solution U* — G ' F of the Ritz-Galerkin system. Then, instead of solving the coarse grid system G W = R, we define

i.e., we compute an approximation using the null vector as a starting juess. Only if the grid width is sufficiently large, we determine the exact solution W = G"1 R. This leads to a special case (v-cycle, ft — 1) of the basic multigrid algorithm 7.3, which will be described in section 7.3. The program structure of the v-cycle is shown on the left side of Figure 7.3. Starting on the finest grid, we smooth the error with Richardson iterations (steps 5) and project the residual to the coarser grid (step /?). Then, we repeat the steps s and p recursively until we reach the coarsest level /z max > where we compute the correction exactly (step c). Finally, we successively update the approximations on the finer grids (step M). The diagram on the right side of the figure compares the convergence rates of the vcycle (thin lines) with that of Richardson's iteration (thick line). For /zmax = 1/2, we show the error reduction factor Q as a function of the finest grid width h — 2~e for v = 1, 2, 3 smoothing steps. While QR —»• 1 for Richardson's iteration, the rates QV remain < 1/2 for all variants of the v-cycle. Hence, we obtain a first indication of the grid-independent error reduction of multigrid algorithms. This remarkable property will be rigorously proved at the end of this chapter in a very general context.

7.2

Grid Transfer

The finite element bases described in chapter 4 are ideally suited for multigrid techniques because of their uniform structure. As for the trivial model problem, subdivision provides a natural grid transfer operator. For the weighted spline spaces wB, this is straightforward. The univariate subdivision formula 3.9 and definition 4.1 imply

100

Chapter 7. Multigrid Methods

Figure 7.4. Relevant biquadratic B-splines bk and bi on the fine and coarse grid (left) and subdivision coefficients sa multiplied by 2nm = 16 (right).

Substituting kv <— kv — 21v and expanding the product yield the following identity, which relates the B-spline bases on different grids. 7.1 Multivariate Subdivision The relevant B-splines bt with grid width 2h can be expressed as linear combinations

k

= b\ h), where we set (n+l) = 0 for IJL < 0 or //, > n + 1.

As in the previous section, the symbol ~ refers to the coarse grid. For example, bt, t e K, are the relevant B-splines bnk 2h- Moreover, U are the coefficients of a spline u = ^£ ii(bi on the coarse grid. This notation avoids an excessive use of the subscripts h and 2h. Figure 7.4 illustrates the grid refinement in the two-dimensional case. On the left, we show for two consecutive grids with width h and h = 2h the grid positions of the relevant biquadratic B-splines, marked with small and large circles, respectively. The linear combination in the subdivision formula 7.1 involves all B-splines bk with supp bk C supp bg. In the example, these are 16 B-splines, if suppbi C D. The corresponding coefficients sa are shown on the right side of the figure. As usual, the values are associated with the lower left corner of the B-spline supports.

7.2. Grid Transfer

101

A linear combination of coarse grid B-splines bk is represented on the fine grid as

i.e., the coefficients are related by

Moreover, as for the trivial model problem in section 7.1, a residual is transferred to the coarse grid according to

where A. is the linear functional determining the right-hand side of the Ritz-Galerkin equations. Clearly, multiplying with a weight function w does not affect these formulas. Hence, the transformation rules are valid also for the spline space u>B. The grid transfer for web-splines is slightly more complicated. Because of the modification of B-splines near the boundary, the space w eB with grid width 2h is in general not a subspace of w eB. Hence, not all web-splines BI possess an exact representation on the fine grid. However, the standard projector P/,, defined in section 5.1, provides a natural grid transfer operator; i.e., we approximate B^ by

Fortunately, this does not require any detailed knowledge about the dual functions A,. To obtain explicit formulas we recall that

By applying the subdivision formula 7.1, we see that the web-spline coefficients of the approximation Ph BI of Bt on the fine grid are

We note that the summation can be restricted to the inner indices k e / since A; is supported on an interior grid cell; hence its product with any outer B-spline vanishes. The same argument shows that

Chapter 7. Multigrid Methods

102

Therefore, the sum over k reduces to a single term, corresponding to k — i, which yields the following explicit form of the projection. 7.2 Grid Transfer for Web-Splines The projection of a coarse grid web-spline is

Accordingly,

and R = P'R is the approximation of a residual on the coarse grid.

Figure 7.5. Band width of the grid transfer projection matrix.

7.3. Basic Algorithm

103

The matrix P is sparse with band width depending on the degree n and the domain D, but not on the grid width h. This follows because the coefficients s,-2/t are nonzero only if / e / n (2k + {0, . . . , n + \}m} and the number of elements in J(i) is ^ 1. Geometrically, an entry pif is ^ 0 if supp bf C supp bj for j — t or for some j with e^j ^ 0. Figure 7.5 shows the band width of the subdivision matrix for a biquadratic spline space. We listed the number of nonzero entries of P with column index I at the grid points i(2h), I e I. This is only done for the boundary indices where P does not have the standard band width (n + 2)m = 42 corresponding to the subdivision of a B-spline bt. We see that the modifications near the boundary do not substantially increase the band width. While there are up to 10 elements in /(£), there are fewer inner B-splines on the fine grid with supp hi C supp bj near the boundary. Moreover, a B-spline bt may be associated with several indices j.

7.3

Basic Algorithm

The basic components of a standard multigrid algorithm are very simple. In addition to the grid transfer operators, described in the last section, we just need an iterative scheme which strongly damps oscillating error components. As for the trivial model problem, discussed in section 7.1, we choose Richardson's iteration because this is most convenient for the convergence analysis. For a Ritz-Galerkin system

one Richardson step is defined by

with

as the standard relaxation parameter. Since y is an upper bound for the spectral radius, the eigenvalues of the iteration matrix (E — y~{G) for symmetric positive definite G lie in the interval [0, 1). Hence, Richardson's iteration converges. But the rate is in general very poor. For smooth error components, the reduction factor tends to 1 as the grid width becomes small. With all basic components at hand we can now describe one step of a multigrid iteration. It uses a sequence of grids with widths

and is conceptually identical to the procedure for the trivial model problem.

104

Chapter 7. Multigrid Methods

7.3 Multigrid Algorithm

One step of the multigrid iteration, which improves an approximation U & U* = G~l F, is defined by the program

else

end where a and £ denote the number of smoothing and coarse multigrid iterations, respectively. Let us comment once more on the heuristic for the multigrid strategy. The error after a Richardson steps is Supposedly, its slowly varying components dominate. Therefore, the residual equation

can be very well approximated on the coarse grid. This means that we consider approximate solutions of the form P W % V — U*, which are obtained as projections from the coarse grid. Since we have

Hence, G % P'GP,sothat is the appropriate approximation of (7.2) (noting that GPW % R). Solving (7.3) directly (2h = /Zmax) or approximately with ft steps of a multigrid iteration with starting guess 0, we expect that Therefore, the correction V -^ W = V — PW should lead to a substantial improvement. Figure 7.6 illustrates the grid transfers of Algorithm 7.3 for ft = 2. As suggested by the recursive pattern, this variant of the multigrid iteration is called a w-cycle. It is slightly more robust than the v-cycle, shown in Figure 7.3, and, as we will see in section 7.5, also easier to analyze.

7.4. Smoothing and Coarse Grid Approximation

105

Figure 7.6. W-cycle of a multigrid iteration with four levels. A fixed choice of the parameters a and ft is convenient for theoretical purposes such as the convergence analysis of the next two setions. However, when implementing the multigrid iteration, it is much more efficient to control the grid transfer dynamically. One should stop the smoothing iterations if the convergence becomes slow and transfer the correction back to the fine grid if the residual has been sufficiently reduced. For details, we refer to the excellent paper [20] by Brandt, who describes how to tune multigrid algorithms to optimal performance. We conclude our description of the multigrid algorithm with an operation count. To this end we recall that the matrices P and G have uniformly bounded band width. Hence, all statements in the recursive algorithm 7.3 require ;< h~m operations, i.e., an amount proportional to the number of unknowns. Denoting by a(h) the number of operations for one multigrid step with grid width h, we have

Neglecting the operations for computing the exact solution on the coarsest grid, this yields

if ft < 2m. Thus, if ft < 4 in two dimensions and ft < 8 for three variables, the computational work for a multigrid step is equivalent to that of a fixed number of Richardson iterations. Due to the reduction of the grid size, the recursive calls cause only a moderate increase in complexity.

7.4

Smoothing and Coarse Grid Approximation

We will now give an analytical explanation for the smoothing effect of Richardson's iteration and the accuracy of the coarse grid correction. These results about the two key multigrid components are fundamental for the convergence proof of the next section. In their most basic form they require regularity of the variational problem and stability of the finite element basis. Therefore, we assume that the domain D is smooth and discuss only approximation by web-splines with a standard weight function (cf. definition 4.6). Moreover, in order

106

Chapter 7. Multigrid Methods

to focus on the essential aspects of the theory rather than to aim for great generality, we consider Poisson's equation as a typical model problem. We recall from section 2. 1 that

and

Because of the local support of the web-splines, the Ritz-Galerkin matrix G has uniformly bounded band width. Moreover, since ||#,||i -< hm/2~l by Bernstein's inequality 5.4. To analyze the smoothing effect, we estimate the error V — U* after a steps of Richardson's iteration. As for any convergent scheme with symmetric iteration matrix, However, we can obtain a more precise estimate in a norm associated with the Ritz-Galerkin matrix G.

7.4 Smoothing of Richardson's Iteration The error after a Richardson steps U -> V satisfies

where U, V are approximations of U* = G

F.

If we interpret the multiplication with the Ritz-Galerkin matrix as a second-order difference operation, we expect the factor h~2\ the factor hm just reflects the normalization of the web-splines. The important fact is the division by a + 1, which quantifies the smoothing effect of the iteration. The inequality is proved by estimating the eigenvalues /c, of the symmetric matrix which describes the transition from U - t/* to G(V - U*). By (7.4), #, = A.,/y e (0, 1] for any eigenvalue A./ of G. Hence, we obtain

establishing the desired bound since y :< hm 2. We now turn to the analysis of the coarse grid correction. The following estimate shows that there is only a small discrepancy between solutions on consecutive grids.

7.4. Smoothing and Coarse Grid Approximation

107

7.5 Error of the Coarse Grid Correction

If

where jj, #*, w*, r are the web-splines corresponding to the coefficient vectors We prove this estimate by interpreting v — u* and M* as approximations to the solution of an auxiliary Poisson problem

To this end we define r] e LI by

By Bernstein's inequality 5.4,

i.e., the right-hand side of (7.5) defines indeed a continuous linear functional on L 2 . Hence, the Riesz representation theorem A.5 is applicable. From the definition (7.5) of rj it is easily deduced that We just set ty = BI = PhBi for the first identity and note that a(v — M*, B{) is the /th entry of the vector G(V — [/*). The second identity follows from

Therefore, v — u* and u* are both Ritz-Galerkin approximations to (p. By the standard error estimate (2.9),

This completes the proof since, with q = P

and by the stability theorem 5.3 and the boundedness of the quasi-interpolant Ph. We already see how theorems 7.4 and 7.5 fit together. The error after smoothing is bounded in a norm which can be viewed as a discrete analogue of || • ||2- This quantifies the damping of oscillating components and permits a sufficiently accurate coarse grid correction. Combining both estimates leads to a grid-independent multigrid convergence rate, as will be described in the next section.

Chapter 7. Multigrid Methods

108

7.5

Convergence

The truly remarkable feature of multigrid iterations is the uniform bound on the convergence rate Q, which is independent of the grid width h. The first such algorithm was discovered by Fedorenko [35] after a long history of research on iterative elliptic solvers (cf., e.g., [91]). Table 7.1 lists some of the most well-known schemes for the standard 5-point discretization of Poisson's problem on the unit square. We note the substantial qualitative improvement of the error reduction factors Q for the iteration matrices. While the Jacobi or the GauBSeidel method require — l/log£> x h~2 iterations to reduce the error by a decimal digit, for multigrid techniques, like the MGR-algorithm, a fixed number of steps suffices. This huge performance gain becomes even more dramatic when combined with the continuing improvement of computer hardware. After commenting briefly on some of the fascinating classical research on iterative methods, we now state the basic multigrid theorem. As in the last section, we continue to use Poisson's equation with homogeneous Dirichlet boundary conditions as a model problem, and we consider web-spline approximations with standard weight functions. With these assumptions we have the following result [48]. 7.6 Multigrid Convergence For ft = 2 coarse grid iterations,

for one multigrid step U -> W. Hence, for a sufficiently large number a of smoothing iterations, the convergence rate Q of the w-cycle is less than 1, uniformly with respect to the grid width h. For the proof of this result we will frequently use the stability of the web-basis and the boundedness of the standard projector, i.e., the estimates

Year

Spectral radius Q

Jacobi [53]

1845

cos(Tth)

GauB-Seidel [39, 81]

1832

COS2(7Th)

Successive overtaxation (SOR) [92, 93]

1950

1 — sin(jr/2) 1 + sin(;r/z)

Fast Fourier transform (FFT) [28]

1965

direct method -< h~2\\ogh\ flops

Multigrid reduction method (MGR) [73]

1979

< 0.096 . . .

Algorithm

Table 7.1. Classical Poisson solvers.

7.5. Convergence

109

As usual, Q = {
where U* = G"1 R is the exact solution of the coarse grid system. Since P/j reproduces the web-spline v — M*, we can bound the error after one iteration step U -> W by

Moreover, by combining theorems 7.5 and 7.4 of the previous section, we obtain

where we have also used that ||r||0 x hm/2\\R\\ and R = G(V - U*). Substituting (7.7) into (7.6) yields the desired bound Q ^ l/(or + 1) for the two-grid convergence rate. If more than two grids are used, the solution of the coarse grid system is approximated by a recursive application of the multigrid iteration. We merely compute an approximation w to it*. While this presumably increases the error of the correction, it is plausible that the bound on the convergence rate for the two-grid iteration persists. We prove this via induction on the grid level; i.e., we assume that the result is valid for all grids with grid width > 2h with a constant Q to be chosen later. To bound the convergence rate for grid width h, we split the error of the approximation W = V — P W after one multigrid step in the form

The first norm on the right is the error after a two-grid step, which is

as we have shown before. In the second norm on the right the vector W is the result of ft = 2 multigrid iterations on the coarse grid, starting from the zero vector as initial approximation to U*. Hence, by the uniform boundedness of the projection P and the induction hypothesis, this norm is

Finally, we note that

Chapter 7. Multigrid Methods

110

To estimate the norm d\, we use (7.7), and for di x hm/2 \\V-U* \\ we note that Richardson's method reduces the norm of the error. Combining the last three displayed inequalities, it follows that

Hence, we obtain the recursion

for the multigrid convergence rate, where € refers to the grid level. To complete the proof of theorem 7.6, we have to show that

for sufficiently large a. Since the linear system on the coarsest grid is solved exactly, QO = 0. Moreover, the worst case bound is obviously obtained if all inequalities in (7.8) are sharp. The qualitative behavior of the resulting quadratic recursion is illustrated in Figure 7.7. Clearly, the largest value of c/(a + 1) for which the sequence Qe remains bounded corresponds to the limiting case when the parabola touches the diagonal, i.e., when its slope equals 1 at the fixed point: 1 = c 2^oo- Substituting this into the fixed point equation yields

for the critical value of a. Assuming now that a + 1 > 4c2, the sequence of rates behaves as in the right diagram of the figure. Hence,

Figure 7.7. Divergence (left) and convergence (right) of the recursion (7.8).

7.5. Convergence

111

is the smallest fixed point. This confirms that the asymptotic behavior of the convergence rate is not affected by the number of grids. The proof shows once again that finite elements are ideally suited for multigrid techniques. The simplicity of the analysis, which essentially follows [6, 19], becomes particularly apparent when compared to the very first convergence proofs in [5, 35], which treated the more difficult case of difference equations.

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Chapter 8

Implementation

Finite element methods with splines do not require any mesh generation and thereby eliminate a difficult and time-consuming preprocessing step. The construction of basis functions is comparatively fast, in particular if natural weight functions are available. Once the basis is assembled, the course of a finite element simulation parallels that of mesh-based techniques. This is illustrated in Figure 8.1, which shows the components of a typical finite element algorithm and, in more detail, the steps of the construction of a web-basis. The regular grid is a definite advantage in all stages of the computations, as will become apparent from the details provided in this chapter. Moreover, extensive software for manipulating B-splines is available and can be utilized for computing and visualizing the finite element approximations.

Figure 8.1. Steps of a finite element simulation (left) and for the construction of a web-basis (right). 113

114

Chapters. Implementation

In section 8.1 we give a brief introduction to the Bezier form for curves and surfaces. This boundary representation is a particularly convenient interface to geometric modeling systems and, of course, combines well with B-spline approximations. Section 8.2 discusses the classification of grid cells, which is essential for all basis constructions and also needed for integration routines. In section 8.3 we describe the evaluation and differentiation of weight functions, considering R-functions as well as numerically constructed distance functions. The final two sections are devoted to routine finite element procedures: numerical integration and matrix assembly.

8.1

Boundary Representation

B-splines have become a widely accepted standard for geometry description. Hence, splinebased finite element methods can be naturally integrated in CAD/CAM systems. In particular, we can take advantage of accurate boundary representations rather than having to resort to piecewise linear approximations of standard mesh-based techniques. There are many variants of piecewise polynomial geometry models. For finite element approximation the Bezier form [51, 32] is most convenient. It facilitates numerical computations since software for manipulating polynomials and solving polynomial equations can be utilized. Moreover, any spline model, such as trimmed nonuniform rational B-spline representations (NURBS) [34], can be converted to this basic form. The Bezier form was introduced by Bezier [11] for automatizing the design of car bodies. He recognized that Bernstein polynomials yield parametrizations with very intuitive geometric properties. 8.1 Bezier Curve A rational Bdzier curve with control points cv e Mm and weights a)v > 0 is parametrized by

where fi"(t) = (")(1 — t)n vtv are the Bernstein polynomials of degree n. Figure 8.2 illustrates some important properties of Bezier curves, which follow readily from the definition (cf. [33, 51]). • Endpoint interpolation: The curve starts at the first and ends at the last control point, i.e., Moreover, the control polygon, formed by CQ, . . . , c,,, is tangent to p. • Convex hull: The curve p lies in the convex hull of the control points CQ, . . . , cn. • Influence of weights: Increasing a weight a>v pulls the curve toward the control point c y . If tov = 1 for all v, the denominator ^ v a>v fi"(t) is identically equal to 1, and p is a polynomial parametrization.

5.1. Boundary Representation

115

Figure 8.2. Conic sections, represented by quadratic Bezier curves (left), and a quintic Bezier curve with convex hull of its control polygon (right).

Because of the first two properties the shape of a Bezier curve is modeled fairly accurately by the control polygon. This fact has been the principal motivation for using the Bezier form in geometric design. Varying the weights provides additional design flexibility. However, more importantly, the rational form permits the exact representation of conic sections as quadratic Bezier curves. In the example on the left side of Figure 8.2 with

and a)Q = 0)2 = 1, the quadratic Bezier curve represents a circular arc if a)\ = l/\/2, a parabola if co\ = 1 (dashed line), and part of a hyperbola if a>\ > 1. Chapter 6 already contains several examples of domains modeled by Bezier curves. If the boundary cannot be represented exactly, spline approximations are used and converted to Bezier form. This ensures that the curves join smoothly. As for B-splines we can form tensor products of Bernstein polynomials to obtain bivariate representations. 8.2 Bezier Surfaces A rational Be*zier surface with control points cVtfi € R3 and weights parametrized by

with The properties of Bezier curves generalize in an obvious way. In particular, the endpoint interpolation property implies that the control points cv^ with v or /x equal to

Chapter 8. Implementation

116

Figure 8.3. Bezier surfaces with boundary curves and control polygons, representing a three-dimensional domain D. either 0 or n correspond to the boundary of the Bezier surface. For example,

since ^(0) = 1 and /?^(0) = 0 for /z > 0. This property is very convenient for joining Bezier surfaces. If neighboring surfaces share the common boundary control points, exact continuity of the piecewise Bezier surface is ensured. Figure 8.3 shows an example of a Bezier boundary representation, consisting of eight biquadratic rational Bezier surfaces (having nine control points each and sharing the control points of the highlighted boundary curves) and a disc at the bottom of the object. The upper half is modeled with four triangular surfaces, which have a multiple control point at the tip; i.e., for each surface all three control points corresponding to the top boundary coincide. Including such degenerate representations increases the flexibility for modeling surfaces with a complicated topological structure. This is particularly important for NURBS surfaces with many trim curves.

8.2

Classification of Grid Cells

The classification of grid cells is an important preprocessing step. Identifying the cell types is necessary for the construction of the spline bases as well as for integration routines. As described in section 4.4, we distinguish between inner, outer, and boundary cells Q, depending on whether Q c D, QC\ D — 0, or the interior of Q intersects the boundary (cf. Figure 4.6). The difficult part is to determine the boundary cells. The distinction between

.2. Classification of Grid Cells

117

Figure 8.4. Determining two-dimensional boundary cells from grid line intersections (black solid markers), local extrema (squares), and auxiliary test points (circles). inner and outer cells can then be made by applying a standard in/out test (cf., e.g., [68]) to a single point in each cell. The two-dimensional case is straightforward. There are a number of conceivable strategies for identifying the boundary cells. For example, as is illustrated in Figure 8.4, we can determine the intersections of 3D with the grid lines and local extrema, i.e., points of 3D with minimal or maximal horizontal or vertical component. Then, the boundary cells can be identified with the aid of the following observation. 8.3 Characterization of Planar Boundary Cells The interior of a planar boundary cell contains at least one local extremum of 9 D or a segment between two consecutive intersections of BD with grid lines. An algorithm based on this characterization is easily implemented. We first determine horizontal and vertical (linear) boundary segments and record one point for each such segment on a list of test points. Then we add the isolated extreme points to this list. Finally, we determine the grid line intersections and choose one boundary point between consecutive intersections as further test points. Now we just have to check to which cells the test points belong (ignoring points on grid lines). This strategy is illustrated in Figure 8.4, which shows possible patterns of 3D within a boundary cell. In each case isolated local extrema are marked with squares and additional test points with circles. The two examples on the left illustrate typical situations. For the third cell, a local extremum lies on a grid line. This exceptional case shows that we cannot ignore tangential grid line intersections since the set of test points may otherwise be insufficient. While covered by the algorithm, the fourth case should not be permitted in practice. If a hole of the domain is completely contained in a grid cell, the grid width is too large. It is unlikely that the finite element approximation will be sufficiently accurate. In fact, one should require that, except for corners, the tangent direction of 3D changes only moderately within a grid cell. For boundaries in Bezier form the above procedure requires one to compute roots of low-degree polynomials. While up to degree 4 explicit formulas exist, some instabilities may be caused by almost tangential intersections. However, this can be avoided by choosing a grid which maximizes the distance from extreme points. The generalization of the above methods to three dimensions is possible but not straightforward. There exist considerably more topological possibilities. Hence, we prefer an alternative approach, which is particularly suited for Bezier representations.

118

Chapter 8.

Implementation

Figure 8.5. Bezier surface with bounding boxes (left) and classification strategy for grid cells (right). As is illustrated in Figure 8.5, we enclose the Bezier surface in bounding boxes, corresponding to a uniform partition of the parameter domain. The midpoints p(sv, / M ) of these boxes (marked with dots) lie on the surface, and the width is determined with the aid of bounds on the derivatives of the parametrization. 8.4 Bounding Box If dt,s and dt,t (£ = 1,2,3) are bounds for the absolute value of the derivatives dspt and 3tpt, then the surface piece

is contained in the box with center p(s, t) and width de,s8s + dt,t8t in the £th coordinate direction. This result is a simple consequence of the integral form of the mean value theorem:

Taking absolute values, the Ith component of the left-hand side is

Hence, the distance from the center of the box is at most half the specified width.

8.3. Evaluation of Weight Functions

119

The strategy for identifying the boundary cells is illustrated schematically on the right side of Figure 8.5. If 8S and 8t are chosen small enough, most grid cells which intersect the Bezier surface will contain one of the points p(sv, ?M) and hence are easily identified. For example, we may choose

Then, according to theorem 8.4, the surface pieces are contained in boxes of width h/2, i.e., half the width of the grid cells. As for the Bezier surface in the figure the boxes form a narrow strip, which describes the position of the surface fairly accurately. Therefore, only relatively few grid cells will intersect one of the bounding boxes but not contain any of the points p(sv, ? M ). In the example on the right of the figure this is the case for the bottom grid cell. Such grid cells have to be analyzed by a different method. For example, we can use a constrained optimization algorithm to check whether the distance in the maximum norm of the cell midpoint to the surface is < h/2. Of course, we can also first repeat the bounding box strategy locally, with smaller step sizes 8S and 8t . Bounds on the partial derivatives of polynomial Bezier parametrizations p(s, t) are easily determined. As for linear combinations of uniform B-splines we just have to form differences of the control points (cf., e.g., [51]). The control points of dsp and dtp are the arrays

of dimension n x (n + 1) and (n + 1) x n, respectively. Hence, by the convex hull property, we can define

It is also possible to derive more precise bounds which take the signs of the partial derivatives into account. Alternatively, we can determine bounding boxes directly via the convex hull property and implement a classification scheme based on de Casteljau's subdivision algorithm. However, the description of such more sophisticated—and slightly more efficient — techniques would require more advanced knowledge of Bezier representations.

8.3

Evaluation of Weight Functions

In section 4.3 we introduced two classes of weight functions w: R-functions and blended distance functions. As we will see in this section, both types can be evaluated efficiently. We begin by describing a basic localization technique. For complicated domains it is convenient if the values w(x) depend only on boundary parts close to x for any point x e D. This reduces the complexity of computations and also simplifies the definition of weight functions. A simple solution is provided by the following theorem. Local weight functions are blended with the aid of a positive partition of unity (provided, e.g., by B-splines).

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8.5 Local Weight Functions Let F be the union of smooth components Ff of the boundary of D and denote by b'k the relevant B-splines with grid width h'. Moreover, assume that wk are continous functions which, for some 8 > 0, satisfy

with dk,t = dist(supp b'k, F^). Then,

is a weight function of order 1 for F. The application of this theorem is straightforward. We can define weight functions separately on the B-spline supports suppb'k = kti + [0, n + \]mh' (usually h' ^> h). On each support only neighboring boundary components have to be considered, i.e., parts F^ of F with distance < 8. Ifdkj > 8 for all t, we can just set Wk = 1. Formally this is implied by the convention dist(;t, 0) = 1. For the proof of theorem 8.5 we first observe that the assumptions about the local weight functions imply that

This is intuitively clear since boundary components F^ with dk.t > 8 should not matter for determining the distance near the boundary. Indeed, if the distance to x is attained at such a boundary part, then the right-hand side is > 8. Hence, u>k(x) ;< (diamD)dist(jc, F)/5, establishing the nontrivial estimate of (8.1). To complete the proof, we note that convex combinations preserve the estimate, so that

as claimed. Figure 8.6 illustrates the construction of theorem 8.5. We see that, if 8 is chosen small enough, only near a corner is more than one curve Ff involved in the construction of the functions Wk- The supports of the corresponding B-splines, which intersect two of the three boundary strips of width <$, are marked with black squares at their lower left corners. B-splines marked with white squares are multiplied either by w^ = 1, like the one with highlighted support in the middle of the domain, or by a weight function for a single boundary curve. The localization has a similar effect in three dimensions. The weight functions w\^ typically correspond to three boundary components near a corner and to two boundary components near an edge. It is also apparent from the figure that the construction of theorem 8.5 needs to be applied to relatively few weight functions only. If the weight functions wk coincide for all B-splines which are nonzero on a grid cell Q, then ^ b^Wk = Wt on Q. In the example this is the case for the shaded grid cells since we have used local weight functions with common

.3. Evaluation of Weight Functions

121

Figure 8.6. Construction of local weight functions with biquadratic B-splines. plateau, i.e., Blending different weight functions is required only on very few grid cells near the corners. We now turn to the evaluation of weight functions. This also involves the computation of gradients. As we already remarked in section 4.3, software for automatic differentiation can be used. However, the implementation of special purpose routines is not difficult and makes the finite element codes self-contained. We sketch below basic algorithmic components for computing R-functions as well as blended distance functions. A weight function w, constructed with R-functions, can be evaluated recursively. Starting from predefined weight functions

we compute leading to w = wp. The R-functions rt correspond to Boolean operations (cf. Table 4.1) and have one or two previously defined weight functions as arguments (in the case of one argument the dependence on w^ is ignored). The gradient of w can be evaluated simultaneously by differentiating the recursion. By the chain rule we have

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Chapter 8. Implementation

where 9^ denotes the partial derivative with respect to the kth variable. Hence, we can successively compute

Higher derivatives can be evaluated analogously, but are needed only for fourth- and higher order problems. The standard R-functions for union and intersection are

with the positive square root corresponding to the eperation U and–s to U. Accordingly, the recursion for the gradients becomes

which does not require much additional computational work. While the pure R-function method yields explicit expressions for weight functions, blended distance functions have to be evaluated numerically. The basic step is to compute the distance from smooth boundary parts for x in a sufficiently narrow strip rs (cf. Figure 8.7). Once the distance is determined, combining d with other functions (also R-functions) and computing derivatives present no problems. The width 8 of the boundary strip must be chosen small enough to avoid singularities of d. In particular, opposite boundaries of the domain must be separated, so that for any x G rs there is a unique closest point p e F. With these assumptions, the distance can be determined locally with the aid of the familiar orthogonality condition. We first discuss the two-dimensional case and assume that the curve segment F has a regular parametrization (typically in Bezier form)

Figure 8.7. Distance to a smooth boundary component F.

8.3. Evaluation of Weight Functions

123

Then, as is illustrated in Figure 8.7, the tangent vector at the point p(t) closest to x is orthogonal to the segment [jc, p ( t ) ] , i.e.,

This is a nonlinear equation for the parameter t which has a unique solution if the width of the strip r$ is sufficiently small. The gradient of d(x) = \\x — p(t)\\ requires no further computations. It is equal to the curve normal:

This is intuitively clear since the gradient represents the direction of steepest increase. A precise explanation is given in a moment, when we discuss the three-dimensional case. Moreover, we will also comment on the numerical solution of the orthogonality equation. The distance to a surface F can be determined in a completely analogous fashion. Here, the parametrization has the form

and we assume that the cross product d\p x 82 p is nonzero and oriented as the outward normal %(t\ , 12). With these assumptions we have the following characterization. 8.6 Distance Function For points x in a sufficiently narrow boundary strip T& the distance

is determined with the aid of the orthogonality relations

and gradd? The solvability of the nonlinear system (8.2) can be established with the aid of the Kantorowitsch criterion. We omit the standard argument and merely confirm that grad d = — £. To this end we consider the bijective transformation

By the inverse function theorem, the gradient of the distance d = yj, with respect to (jti , Jt2, XT,) is the last row of the inverse of the Jacobi matrix

To prove that the entries (3, 1), (3, 2), (3, 3) of (/')"' equal — £, we have to check that

This follows since the scalar products £ d\ p, % 82/7, £ d\ % , £ 82^ vanish; the first two because the partial derivatives of p span the tangent plane at p(t), and the last two because 3V (£ £) =

Chapter 8. Implementation

124

The nonlinear system (8.2) can be solved efficiently. If F is represented in Bezier form, p and dvp have rational components. Hence, by multiplying with the common denominator, the orthogonality relations can be transformed to a system of two polynomial equations (for boundary curves to a single polynomial). Hence, algebraic and semi-algebraic software for finding polynomial roots can be applied. Alternatively, we can use Newton's method. In particular, for degree > 2, this is faster since very good start values are available. For assembling Ritz-Galerkin matrices weight functions have to be evaluated at a dense grid of points, used in the quadrature formulas. Processing these points sequentially, one or two Newton steps usually suffice.

8.4

Numerical Integration

To assemble Ritz-Galerkin systems we need to compute integrals over subsets of the domain D and its boundary. This is done by summing the contributions from each grid cell Q\ i.e., the integrals have the form

where

Table 8.1. GauS parameters for [0, 1].

8.4. Numerical Integration

125

degree Legendre polynomial, and the weights yv are the integrals of the associated Lagrange polynomials. Turning to the discussion of the domain integral fQnD (p, let us first dispose of a simple case. For a grid cell Q = ih + [0, l]mh which does not intersect the boundary, we can just use the tensor product GauB formula. For example, for m = 3,

where t' = th + (tv, t^, ta}h are the transformed GauB nodes. We emphasize that for small h most integrals fall into this simple category. The number of boundary grid cells is ^ h' ~m. Hence, the techniques described below pertain only to an increasingly small percentage of integrals. For boundary grid cells we cannot apply a tensor product formula. Setting

Figure 8.8. Subdivision of a two-dimensional boundary cell and integration over smoothly deformed rectangles.

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126

we can apply the tensor product formula in a straightforward way. We obtain

where ^ and r^ denote the transformed GauB nodes for the intervals [a, /3] an respectively. This is illustrated in Figure 8.8, which shows how the GauB points are mapped from the reference square to the two subsets Of course, the boundary of Q D D is usually not given in the form (8.3). However, the quadrature formula requires only isolated points on the boundary, which are easily generated. For example, if 3D is a spline curve (x\, X2) = ( p \ ( s ) , /?2 (•?))» then f ( t ' v ) and g(t'v) are obtained by solving the polynomial equation t'v = p\(s), which yields f ( t ' v ) = P2(s/} and g(t'v) = P2(sg) for the appropriate roots Sf and SK. Even for high degree, when the roots have to be computed by some iterative method, this is not time consuming. Since neighboring values are calculated, good starting values are available and generally few steps suffice. In three dimensions, the shape of the intersection Q D D can be rather complicated. Still, there are a number of typical cases, which can be reduced to integration over smoothly deformed cubes with few subdivisions. Figure 8.9 shows several examples. In each case, the transformations to the standard reference cube [0, I]3 are straightforward. For example, the prism-shaped intersection in the middle of the figure can be represented as

Accordingly, the integral

nD

after a change of variables

There are, however, more difficult cases. We are confronted with corners and edges of 3D within the grid cell Q and complicated boundaries caused by almost tangential intersections.

Figure 8.9. Integration over subdivided boundary cells.

8.5. Matrix Assembly

127

As mentioned above, the percentage of such types of intersections is small. However, they have to be dealt with, at least if high accuracy should be maintained in all parts of the simulation region. We propose to subdivide irregularly shaped sets Q D D into tetrahedral cells (moderately deformed tetrahedra) with a special purpose meshing algorithm. This is a very flexible approach, analogous to isoparametric finite element approximation, which also covers simple situations. For the purpose of integration, the geometric quality of the local mesh is not important. Moreover, we do not have to represent the boundary exactly. In fact, we will normally use isoparametric elements [24]. This means that the cells of the mesh are images of tetrahedra under polynomial maps which are close to the identity. Once such a tetrahedral subdivision of Q n D is generated, the numerical integration presents no problems. We simply transform standard formulas for the three-dimensional simplex [88].

8.5

Matrix Assembly

The assembly of finite element matrices for B-splines is similar to that for mesh-based elements. There are, however, some simplications due to the regular grid. Regardless of whether we use the spaces u;B or weB, the approximations are weighted splines

where each coefficient Uk is associated with a grid point kh. As is illustrated in Figure 8.10, it is convenient to enlarge the set K of relevant indices, so that the range of summation is an array K'. When evaluating w/,, we can set the irrelevant coefficients, which correspond to the points marked with small squares, equal to 0 and restrict the arguments of w and bk to D. Whenever possible, we keep this simple structure throughout the computations. In particular, for parallelizing algorithms this is preferable to working with unstructured index sets.

Figure 8.10. Partitioning of index sets for bilinear spline approximations.

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Chapters. Implementation

We consider first the assembly of the Ritz-Galerkin matrices for the spaces if B. Since the bilinear form «(-, •) and the linear form A. are defined in terms of integrals, they can be computed by adding the contributions from each grid cell Q, i.e., Q

This yields the following algorithm. 8.8 Ritz-Galerkin System for Weighted B-Splines

Hie matrix G and the right-hand side F of the Ritz-Galerkin system for the spaces wB can be assembled with the following algorithm:

end end

end

Here we assume that the array K' D K corresponds to proper matrix indices; i.e., the lower left corner is labeled by k = (1,1,...). Since the support of each B-spline bk consists of (n + l)m grid cells, the two innermost loops have a fixed length. All pairs of the (n + l) m B-splines, which overlap the grid cell Q under consideration, have to be processed. The supports of two B-splines bk and bt intersect only if k — I e {— n, . . . , n}m. Therefore, the matrix G has at most (2n + l) m entries for each row index k and can thus be stored in a 2m -dimensional array with indices in

for scalar problems. Keeping irrelevant entries with indices k e K'\K as well allows us to conform to the simple geometric structure induced by the tensor product basis functions. To apply standard iterative schemes, we can enlarge the Ritz-Galerkin system by setting

This yields a solution with zero coefficients for irrelevant indices. The stabilization of the weighted spline basis wbk, k & K, which leads to the webspace can be viewed as a parameter condensation and preconditioning procedure. Hence, it can be accomplished with a simple modification of the Ritz-Galerkin system for wM. To describe this in more detail, we rewrite definition 4.9 as

129

8.5. Matrix Assembly Then, by linearity,

and

. This proves the following transformation rule.

8.9 Ritz-Galerkin System for Web-Splines The Rit2-<3detfcm systems OU = F andKGeU = eF for the weighted spline spaces and tyeB are related by where

For assembling the system e G e t/ = e F it is best to give up on the correspondence between matrix entries and grid points. Instead, G and e G are stored with the aid of index lists in a standard sparse format. To this end we number the inner indices i, marked with dots in the example of Figure 8.10, from 1 to |/| and the outer indices j, marked with circles, from |/| + l t o | / | + |/|. For the matrix E a column-oriented storage scheme is convenient since the complementary index sets

are easier to describe than the sets /(/) of column indices. As is illustrated in Figure 8.11, the first block of the matrix E, corresponding to the inner indices /, is a diagonal matrix containing the weights \/w(xi). The second block

Figure 8.11. Partitioning and sparsity pattern of the coupling matrix E for the example in Figure 8.10.

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Chapters. Implementation

of dimension |/| x \J\ contains the coupling coefficients. For each outer index j G J the corresponding column contains the coefficients

and hence has at most (n + l) m entries. There are fewer entries if the array e,--j, i e /(y), contains zeros. For the matrix in Figure 8.11 this is the case for 27 indices j. Only less than 50% of the columns in the second block have the maximal number of (n + \)m = (1 + I) 2 nonzero entries. As the grid width becomes small, the transformation described in theorem 8.9 does not substantially affect the sparsity pattern of G. The ratio \J |/|/| is of order O(h}. Hence, most of the rows of e G still have < (2n +1)7" nonzero entries. Asymptotically, the required storage for the Ritz-Galerkin systems is ^ h~m for all weighted spline spaces.

Appendix Most of the basic theory of finite element methods with splines can be developed without advanced mathematical tools. We just need a few concepts and results from functional analysis, which are listed below. With the exception of the extension theorem A. 13 for Lipschitz domains, proofs can be found in any of the standard textbooks. For example, a short exposition to Hilbert spaces is contained in the book by Rudin [75]. The book by Adams [1] gives a comprehensive treatment of Sobolev spaces. A reader interested in the historical background may also want to consult the original work of Hilbert [41], von Neumann [64], and Sobolev [83, 84, 85]. A.I Minkowsky Inequality. If / is a continuous function with values in a Banach space (complete, normed vector space), then

This estimate can be interpreted as a continuous analogue of the triangle inequality and is proved by approximation with step functions. A.2 Cauchy-Schwarz Inequality. product, then

If \\u\\ = ^/(u, u] is a norm induced by a scalar

with equality if and only if u and v are linearly dependent. A.3 Bilinear Form. A bilinear form a (•, •) on a vector space is linear in each variable, i.e.,

for all scalars r v , sv. If a is symmetric and a(u,u) > 0 for u ^ 0, then a induces a scalar product and \\u \\a = -^/a(u, u) defines a norm. A.4 Hilbert Space. A Hilbert space H is a complete vector space with a norm || • || induced by a scalar product {•,•). These few structural assumptions have many consequences. We need, in particular, the following characterization of best approximations from a closed subspace V. 131

Appendix

132

For any u e H there exists a unique t>* e V with

which is determined by the orthogonality condition

familiar from elementary geometry. A.5 Riesz Representation. Any bounded linear functional X on a Hilbert space // can be represented in the form where 7£ is an isometry onto H, i.e., a bijective linear operator with A.6 Domain. We use the term domain for a bounded open set D in Em with Lipschitz boundary. This means that 3D can be covered by finitely many open balls D, so that, by choosing appropriate coordinates y -o- x,

for a Lipschitz-continuous function
Figure A.I. Lipschitz domain (left) and nonadmissible boundary with cusps (right).

Appendix

133

A.7 Square Integrable Functions. The Hilbert space L2(D) is the closure of the set of smooth functions on D with respect to the norm

associated with the scalar product (/, g)o = fD f g . Alternatively, L2(D) consists of all measurable functions /, for which |/|2 is Lebesgue integrable. The space L2(9D) of square integrable functions on the boundary of a domain is defined analogously. We simply integrate with respect to the boundary measure in the definition of the scalar product and the norm. A.8 General Sobolev Spaces. Denote by LP(D) the measurable functions u on D with

for 1 < p < oo and

I M OOI < const < oo for almost every x e D for p = oo. Then, for t e N, the Sobolev space W*(D) consists of all functions with partial derivatives of order < I in LP(D}. In particular,

Via interpolation [7] it is possible to extend this definition to fractional orders of differentiability as well. A.9 Integration by Parts. If dvf and / are absolutely integrable on D and 3D, respectively, then

where £ is the outward unit normal. Specific choices of / yield various classical theorems of vector analysis. Summing over v with / = u\, 112, ... yields GauB's theorem:

Similarly, with / = dvu v, we obtain Green's formula,

where d^u — t; grad u denotes the normal derivative. This identity is often used for u, v e HQ (D), when the boundary integral on the right-hand side is zero. A. 10 Trace Operator. The restriction to the boundary is a bounded operator from H}(D) to L2(3D). More precisely, since boundary data are not well defined in the Lebesgue

134

Appendix

sense, the restriction operator has to be extended from smooth functions, which are dense inH}(D). We note that there is a sharper version of this theorem, but it involves Sobolev spaces of fractional order, which have a more technical definition and are not needed for the objectives of this book. A.ll Compact Embedding. Any bounded sequence in // £+1 has a convergent subsequence in Hi. This is a typical statement, which relates smoothness to compactness. It is a (very) special case of Sobolev's famous embedding theorem, which states that

A.12 Poincare-Friedrichs Inequality. (m — 1) -dimensional measure, then

If u vanishes on a set F c 3D of positive

An important consequence is that the semi-norm | • | \ is equivalent to the norm || • || i on //ji = {u 6 Hl : u = 0 onF} and, in particular, on HQ (\u\i x \\u\\\). A.13 Extension Operator. There exists a bounded linear extension operator E : He(D) —> and

Moreover, we can assume that the extensions vanish outside a domain The operator 8 was constructed by Stein [86], improving an earlier result by Calderon [22], where the extensions were not independent of the order of differentiability t. The main difficulty is to cope with the minimal smoothness of the boundary of D as specified in A. 6. Requiring only Lipschitz continuity does not allow simple local transformation techniques.

Notation and Symbols General N, Z, R || • || :<,>:, x cond const(-) 3V, da diam dist supp

Natural numbers, integers, and reals 2-norm for vectors and matrices Inequalities up to constants, 1 Condition number of a matrix Constant, depending on parameters, 1 Partial derivatives, 1, 2.3 Diameter of a set Distance function Support of a function B-Splines

bn bnk h bi, b j , bk Bi Wh(D) wsWh(D} Wh (D) eij, E = {eitk} / € /, j e /, k e K h A.,-, A, PI, Q=lh + [0,l]mh w

Univariate B-spline of degree n, 3.2 Scaled translated tensor product B-spline, 4.1 Inner, outer, and relevant B-spline, 1, 4.4 Web-spline, 4.4 Splines on a bounded domain, 4.2 Web-space, 4.4 Hierarchical splines, 4.5 Extension coefficients and matrix, 4.4, 8.5 Inner, outer, and relevant indices, 1, 4.4 Grid width Dual and weighted dual function, 5.1 Standard projector, 5.1 Grid cell Weight function, 4.3

135

136

Notation and Symbols Finite Elements a(-, •) Cl A d1D c Mw D dD div G/, grad //o (£>) c // £ (£>) (••> •)£» II • I I £ » I • \t H[, //p L2(D) X Q Uh % u U = {«,} U = {Hi}

Bilinear form, 2.4 £-times continuously differentiable functions, 2.2 Laplace operator Normal derivative Bounded domain, A.6 Closure of D Domain boundary Divergence Ritz-Galerkin matrix, 2.1 Gradient Sobolev spaces, 2.3 Sobolev scalar product, norm, and semi-norm, 2.3 Constrained Sobolev spaces, 2.3, 6.2, 6.3 Square integrable functions, A.7 Linear functional Quadratic form, 2.1 Approximation of exact solution Coefficients of a Ritz-Galerkin approximation Coefficients of a coarse grid approximation, 7.2 General Sobolev spaces, A.8

Bibliography [1] R. A. Adams: Sobolev Spaces, Academic Press, New York, 1978. [2] J. H. Ahlberg, E. N. Nielson, and J. L. Walsh: The Theory of Splines and Their Applications, Academic Press, New York, 1967. [3] J. H. Argyris: Energy Theorems and Structural Analysis, Butterworth, London, 1960. [4] J. H. Argyris: Recent Advances in Matrix Methods of Structural Analysis, Pergamon Press, Elmsford, NY, 1964. [5] N. S. Bakhvalov: On the convergence of a relaxation method with natural constraints on the elliptic operator, USSR Comp. Math, and Math. Phys. 6 (1966), 101-135. [6] R. E. Bank and T. Dupont: An optimal order process for solving finite element equations, Math. Comp. 36 (1981), 35-51. [7] J. Berg and J. Lofstrom: Interpolation Spaces, Springer-Verlag, New York, 1976. [8] S. Bernstein: Demonstration du theoreme de Weierstrass, fondee sur le calcul des probabilites, Comm. Soc. Math. Kharkow 13 (1912-1913), 1-2. [9] P. Bezier: Definition numerique des courbes et surfaces I, Automatisme XI (1966), 625-632. [10] P. Bezier: Definition numerique des courbes et surfaces II, Automatisme XII (1967), 17-21. [11] P. Bezier: Numerical Control—Mathematics and Applications, John Wiley & Sons, New York, 1972. [12] W. Bohm: Inserting new knots into B-spline curves, Computer-Aided Design 12 (1980), 199-201. [13] C. de Boor: On uniform approximation by splines, J. Approx. Theory 1 (1968), 219235. [14] C. de Boor: On calculating with B-splines, J. Approx. Theory 6 (1972), 50-62. [15] C. de Boor: Package for calculating with B-splines, SIAM J. Numer. Anal. 14 (1977), 441^72. 137

138

Bibliography

[16] C. de Boor: A Practical Guide to Splines, Springer-Verlag, New York, 1978. [17] C. de Boor and G Fix: Spline approximation by quasi-interpolants, J. Approx. Theory 8 (1973), 19-^5. [18] C. de Boor, K. Hollig, and S. Riemenschneider: Box Splines, Springer-Verlag, New York, 1993. [19] D. Braess, M. Dryja, and W. Hackbusch: A multigrid method for nonconforming fediscretizations with application to non-matching grids, Computing 63 (1999), 1-25. [20] A. Brandt: Multi-level adaptive solutions to boundary value problems, Math. Comp. 31 (1977), 333-390. [21] I. G Bubnow: Rezension iiber die mil dem Schurawski-Preis ausgezeichneten Abhandlungen von Prof. Timoschenko, Sammlung des Instituts fur Verkehrswesen, Heft 81,SPB, 1913. [22] A. P. Calderon: Lebesgue spaces of differentiate functions and distributions, Proc. Symp. in Pure Math. 4 (1961), 33^9. [23] P. de Casteljau: Courbes et surfaces a poles, Technical report, Andre Citroen Automobiles SA, Paris, 1959. [24] P. G Ciarlet: The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [25] P. G Ciarlet and J. P. Lions (eds.): Handbook of Numerical Analysis, North-Holland, Amsterdam, 1991. [26] R. W. Clough: The finite element method in plane stress analysis, Proc. 2nd ASCE Conf. on Electronic Computation, Pittsburgh, 1960. [27] E. Cohen, T. Lyche, and R. Riesenfeld: Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics, Computer Graphics Image Proc. 14(1980), 87-111. [28] J. W. Cooley and J. W. Tukey: An algorithm for the machine calculation of complex Fourier series, Math. Comp. 19 (1965), 297-301. [29] R. Courant: Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc. 49 (1943), 1-23. [30] R. Courant and D. Hilbert: Methods of Mathematical Physics, John Wiley & Sons, New York, 1953. [31] M. G Cox: The numerical evaluation of B-splines, J. Inst. Math. Appl. 10 (1972), 134-149. [32] G E. Farm (ed.): Geometric Modeling: Algorithms and New Trends, SI AM, Philadelphia, 1987.

Bibliography

139

[33] G. E. Farin: Curves and Surfaces for Computer Aided Geometric Design, Academic Press, New York, 1988. [34] G. E. Farin (ed.): NURBS for Curve and Surface Design, SIAM, Philadelphia, 1991. [35] R. P. Fedorenko: The speed of convergence of one iterative process, USSR Comp. Math, and Math. Phys. 4 (1964), 227-235. [36] A. Fuchs: OptimierteDelaunay-TriangulierungenzurVernetzunggetrimmterNURBSKorper, Dissertation, Universitat Stuttgart, Stuttgart, 1999. [37] A. Fuchs: Almost regular triangulations of trimmed NURBS-solids, Engineering with Computers 17 (2001), 55-65. [38] B. G. Galerkin: Sta.be und Flatten; Reihen in gewissen Gleichgewichtsproblemen elastischer Stdbe und Flatten, Vestnik der Ingenieure 19 (1915), 897-908. [39] C. F. GauB: Carl Friedrich Gauft, Gesammelte Werke, Vol. IX, 278-281. Translated by G.E. Forsythe, Math. Tables Aids Comput. 5 (1961), 255-258. [40] A. Griewank: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Frontiers in Applied Mathematics 19, SIAM, 1999. [41] D. Hilbert: Grundziige einer allgemeinen Theorie der Integralgleichungen, Teubner, Stuttgart, 1912. [42] K. Hollig: Multivariate splines, in Approximation Theory, C. de Boor (ed.), AMS Short Course Lecture Notes 36 (1986), 103-128. [43] K. Hollig: Grundlagen der Numerik, MathText, Zavelstein, 1998. [44] K. Hollig: Finite element approximation with splines, in Handbook of Computer Aided Geometric Design, G. Farin, J. Hoschek, and M.S. Kim (eds.), Elsevier, 2002, 283-308. [45] K. Hollig, U. Reif, and J. Wipper: Error estimatesfor the web-method, in Mathematical Methods for Curves and Surfaces: Oslo 2000, T. Lyche and L. L. Schumaker (eds.), Vanderbilt University Press, Nashville, TN, 2001, 195-209. [46] K. Hollig, U. Reif, and J. Wipper: Weighted extended B-spline approximation of Dirichletproblems, SIAM J. Numer. Anal. 39 (2001), 442^62. [47] K. Hollig, U. Reif, and J. Wipper: Verfahren zur Erhohung der Leistungsfahigkeit einer Computereinrichtung bei Finite-Elemente-Simulationen und eine solche Computereinrichtung, Deutsche Patentanmeldung DE10023377A1 (2001). [48] K. Hollig, U. Reif, and J. Wipper: Multigrid methods with web-splines, Numer. Math. 91 (2002), 237-256. [49] K. Hollig, U. Reif, and J. Wipper: Process for increasing the efficiency of a computer infinite element simulations and a computer for performing that process, United States Patent Application, US2002/0029135A1 (2002).

140

Bibliography

[50] K. Ho-Le: Finite element mesh generation methods: A review and classification, Comput. Aided Design 20 (1988), 27-38. [51] J. Hoschek and D. Lasser: Fundamentals of Computer Aided Geometric Design, A. K. Peters, Natick, MA, 1993. [52] K. H. Huebner, E. A. Thornton, and T. G Byrom: The Finite Element Method for Engineers, John Wiley & Sons, New York, 1994. [53] C. G J. Jacobi: Uber eine neue Auflosungsart der bei der Methode der kleinsten Quadrate vorkommenden linearen Gleichungen, Astronomische Nachrichten 22:523 (1845), Spalten 297-306. [54] L. W. Kantorowitsch and W. I. Krylow: Ndherungsmethoden der Hoheren Analysis, VEB Deutscher Verlag der Wissenschaften, Berlin, 1956. [55] H. Kardestuncer (ed.): Finite Element Handbook, McGraw-Hill, New York, 1987. [56] A. Korn: Die Eigenschwingungen eines elastischen Korpers mil ruhender Oberfldche, Akad. der Wissenschaften Munich, Math.-Phys., Kl. Berichte 36 (1906), 351^01. [57] A. Korn: Uber einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen, Bulletin Internationale, Cracovie Akademie Umiejet, Classe des sciences mathematiques et naturelles (1909), 705-724. [58] R. Kraft: Adaptive und linear unabhdngige multilevel B-Splines und ihre Anwendungen, Dissertation, Universitat Stuttgart, Stuttgart, 1998. [59] O. A. Ladyzhenskaya and N.N. Ural'tseva: Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. [60] L. D. Landau and E. M. Lifshitz: Theory of Elasticity, 3rd edition, Pergamon Press, Elmsford, NY, 1986. [61] J. M. Lane and R. F. Riesenfeld: A theoretical development for the computer generation and display ofpiecewise polynomial surfaces, IEEE Trans. Pattern Anal. Mach. Intellig. 2 (1980), 35-45. [62] J. Mackerle: Finite Element Methods: A Guide to Information Sources, Elsevier Science Publishers B.V., Amsterdam, 1991. [63] Math Works, Inc.: The Student Edition ofMATLABfor Prentice-Hall, Englewood Cliffs, NJ, 1992.

MS-DOS Personal Computers,

[64] J. von Neumann: Allgemeine Eigenwerttheorie Hermitescher Integraloperatoren, Math. Ann. 102 (1930), 49-131. [65] J. A. Nitsche: On Korn's second inequality, RAIRO J. Numer. Anal. 15 (1981), 237248. [66] G D. Nowottny: Netzerzeugung durch Gebietszerlegung und duale Graphenmethode, Dissertation, Universitat Stuttgart, Stuttgart, 1999.

Bibliography

141

[67] S. J. Owen: A survey of unstructured mesh generation technology, in Proceedings, 7th International Meshing Roundtable, Sandia National Laboratories (1998), 239-268. [68] F. P. Preparata and M. I. Shamos: Computational Geometry, Springer-Verlag, New York, 1985. [69] L. B. Rail: Automatic Differentiation, Lecture Notes in Comput. Sci. 120, SpringerVerlag, New York, 1981. [70] J. W. S. Rayleigh: On the theory of resonance, Trans. Roy. Soc. (London), A161 (1870), 77-118. [71] J. W. S. Rayleigh: The Theory of Sound, 2nd edition, London, 1894 and 1896. [72] U. Reif: Orthogonality ofB-splines in weighted Sobolev spaces, SIAM J. Math. Anal. 28(1997), 1258-1263. [73] M. Ries, U. Trottenberg, and G. Winter: A note on MGR methods, Linear Algebra Appl. 49 (1983), 1-26. [74] W. Ritz: Uber eine neue Methode zur Losung gewisser Variationsprobleme der mathematischen Physik, J. Reine Angew. Math. 135 (1908), 1-61. [75] W. Rudin: Functional Analysis, McGraw-Hill, New York, 1973. [76] V. L. Rvachev and T. I. Sheiko: R-functions in boundary value problems in mechanics, Appl. Mech. Rev. 48 (1995), 151-188. [77] V. L. Rvachev, T. I. Sheiko, V. Shapiro, and I. Tsukanov: On completeness of RFM solution structures, Comp. Mech. 25 (2000), 305-316. [78] V. L. Rvachev, T. I. Sheiko, V. Shapiro, and I. Tsukanov: Transfinite interpolation over implicitly defined sets, Comput. Aided Geom. Design 18 (2001), 195-220. [79] I. J. Schoenberg: Contributions to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math. 4 (1946), 45-99 and 112-141. [80] L. L. Schumaker: Spline Functions: Basic Theory, Wiley-Interscience, New York, 1980. [81] Ph. L. Seidel: Uber ein Verfahren, die Gleichungen, auf welche die Methode der kleinsten Quadrate ftihrt, sowie lineare Gleichungen uberhaupt, durch sukzessive Annaherung aufzulosen, Abh. Bayer. Akad. Wiss. 11 (1874), 81-108. [82] V. Shapiro and I. Tsukanov: Meshfree simulation of deforming domains, Comput. Aided Design 31 (1999), 459^71. [83] S. L. Sobolev: On some estimates of families of functions having square-integrable derivatives, Dokl. Akad. Nauk SSSR 1 (1936), 267-270. [84] S. L. Sobolev: Sur un probleme limite pour les equations polyharmonics, Mat. Sb. (N.S.) 44 (1937), 467-500.

142

Bibliography

[85] S. L. Sobolev: Applications of Functional Analysis in Mathematical Physics, Transl. Math. Monographs 7, AMS, Providence, RI, 1963. [86] E. M. Stein: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970. [87] G Strang and G J. Fix: An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973. [88] A. H. Stroud: Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs, NJ, 1971. [89] R. Szilard: Theory and Analysis of Plates, Prentice-Hall, Englewood Cliffs, NJ, 1974. [90] M. J. Turner, R. W. Clough, H. C. Martin, and L. C. Topp: Stiffness and deflection analysis of complex structures, J. Aeronaut. Sci. 23 (1956), 805-823, 854. [91] R. S. Varga: Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962. [92] D. M. Young: Iterative Methods for Solving Partial Differential Equations of Elliptic Type, Ph.D. Thesis, Harvard University, Cambridge, MA, 1950. [93] D. M. Young: Iterative methods for solving partial differential equations of elliptic type, Trans. Amer. Math. Soc. 76 (1954), 92-111. [94] O. C. Zienkiewicz: Origins, milestones and directions of the finite element method, Arch. Comput. Methods Engrg. 2 (1995), 1-48. [95] O. C. Zienkiewicz and R. I. Taylor: Finite Element Method, Vol. I-III, Butterworth & Heinemann, London, 2000.

Index BEZIER, 1 curve, 114 surface, 115 bilinear form, 131 elliptic, 16 POISSON, 17 boundary condition essential, 2, 71 inhomogeneous, 85 mixed, 79 natural, 75 boundary value problem abstract, 16, 17 biharmonic, 84 DIRICHLET, 71,72 elasticity, 88 mixed, 79 NEUMANN, 75, 76 plane strain, 90 plane stress, 92 POISSON, 7, 71 variable coefficients, 79 bounding box, 118 BRAMBLE-HILBERT estimate, 61,62 BRANDT, 105

algebraic curve, 73 approximation order spline, 21 web-spline, 64, 73, 78 ARGYRIS, 1,11 AUBIN-NITSCHE duality principle, 21 B-spline bicubic, 38 classification, 47 convolution, 35 cubic, 30, 33 definition, 25 derivative, 25, 29 extended, 3 hierarchical, 50 inner, 47 linear independence, 32, 42, 52 multivariate, 37 outer, 47 partial derivative, 39 properties, 26, 38 recursion, 27 relevant, 40 representation of polynomials, 30,42 scalar product, 35, 36 segments, 28 subdivision, 32 tensor product, 37 weighted, 3, 43 weighted extended —>• web-spline,4 BERNSTEIN inequality, 59 polynomial, 2, 114 best approximation, 19, 132

CAD/CAM, 1 CAUCHY-SCHWARZ inequality, 131 CEA inequality, 20 CLOUGH, 1 CLOUGH-TOCHER element, 11 compatibility condition, 75 condition number, 60, 85 cone condition, 132 conjugate gradients, 85 143

Index

144

control polygon curve, 115 surface, 116 COURANT, 1 DEBOOR, 1,27 DE CASTELJAU, 2, 119 DIRICHLET problem, 72 distance function, 123 domain, 5, 132 dual function B-spline, 54 weighted, 56 elasticity, 88 embedding, 134 energy functional, 9 error correction, 107 finite element method, 20, 21, 73, 78,85 quasi-interpolant, 68 smoothing, 97, 106 extension, 134 FEDORENKO, 108 FEM (see also finite element method), 1 FFT, 108 finite element algorithm, 113 approximation, 8, 16 basis function, 2 CLOUGH-TOCHER, 11 history, 1 LAGRANGE, 11 linear, 9 properties, 12 standard, 1 FIX, 1 flow, 76 GALERKIN, 1 GAU6 formula, 124 GAUB-SEIDEL iteration, 108

grid cell, 3, 117 refinement, 100 transfer, 102 hat-function, 1, 10, 25, 98 HILBERT space, 131 history, 2, 108 HOOKE's law, 87 index classification, 127 inner, 3, 47 outer, 3, 47 relevant, 40 integration boundary cells, 125 by parts, 133 JACKSON inequality, 68 JACOBI iteration, 108 KANTOROWITSCH, 3, 43, 123 KORN inequality, 88 KRYLOW, 3, 43 LAME coefficients, 83, 87 LAME-NAVIER system, 88 LAX-MILGRAM theorem, 17 MARSDEN identity, 30, 42 MARTIN, 1 MATLAB, 85 matrix assembly B-spline, 128 web-spline, 129 membrane, 8 MGR algorithm, 108 MINKOWSKY inequality, 131 model problem, 7, 20, 21, 71, 95, 106 multigrid algorithm, 104 convergence, 108 correction, 107 smoothing, 106 NEUMANN problem, 75, 76 notation, 5, 100

Index orthogonal expansion, 56 orthogonality relation, 19 partition of unity, 31 plate, 83, 84 POINCARE-FRIEDRICHS inequality, 134 POISSON equation, 106 problem, 7, 12, 71 ratio, 83, 90 solvers, 108 projector, 54, 61, 63 quadratic form, 9 quasi-interpolant, 54, 63 R-function differentiation, 121 method, 3, 43, 44, 84 system, 45 RAYLEIGH, 1 regularity, 20,65,81 RICHARDSON iteration, 96, 103 RIESZ representation, 132 RITZ, 1 RITZ-GALERKIN approximation, 8, 10, 16, 72, 76, 84,89 RVACHEV, 3, 43 SCHOENBERG, 1,23,29 SOBOLEV space, 13, 15, 133 SOR, 108 sparsity pattern, 129 spline approximation, 21 cardinal, 29, 34 cubic, 24 definition, 23 history, 1 linear, 25 multivariate, 40 web- —>• web-spline, 48 square integrable, 133 stability, 3, 57

145

strain plane, 90 tensor, 87 STRANG, 1 stress plane, 91 tensor, 87 subdivision B-spline, 32 grid cell, 125 multivariate, 100 spline, 34 TOPP, 1 trace, 133 triangulation, 10 TURNER, 1 v-cycle, 99 variational problem —> boundary value problem, 17 w-cycle, 105 weak derivative, 14 web-spline approximation order, 64 definition, 48 idea, 4 normalization, 57 properties, 49 stability, 57 weight function algebraic, 44 definition, 43 local, 120, 121 numerical, 46 regular, 59 signed, 44 standard, 43 YOUNG modulus, 83, 90