A p-version finite element method for nonlinear elliptic variational inequalities in 2D

Numer. Math. (2007) 105:457–480 DOI 10.1007/s00211-006-0035-0

Numerische Mathematik

A p-version finite element method for nonlinear elliptic variational inequalities in 2D Andreas Krebs · Ernst P. Stephan

Received: 14 December 2004 / Published online: 15 November 2006 © Springer-Verlag 2006

Abstract This article introduces and analyzes a p-version FEM for variational inequalities resulting from obstacle problems for some quasi-linear elliptic partial differential operators. We approximate the solution by controlling the obstacle condition in images of the Gauss–Lobatto points. We show existence and uniqueness for the discrete solution up from the p-version for the obstacle problem. We prove the convergence of up towards the solution with respect to the energy norm, and assuming some additional regularity for the solution we derive an a priori error estimate. In numerical experiments the p-version turns out to be superior to the h-version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation. Mathematics Subject Classification 65N35 · 65K10 · 65N22 1 Introduction The objective of this article is the design, analysis, and implementation of a p-FEM for the treatment of nonlinear variational inequalities which correspond to obstacle problems for second order quasi-linear elliptic partial differential operators. Variational inequalities play an important role in the modeling of practical problems, for example in mechanics of elastic and elasto-plastic bodies

A. Krebs (B) Institut für Mathematik, Brandenburgische Technische Universität Cottbus, 03046 Cottbus, Germany e-mail: [email protected] E. P. Stephan Institut für Angewandte Mathematik, Universität Hannover, 30167 Hannover, Germany e-mail: [email protected]

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[11] and in geometry of minimal surfaces [12]. Using the calculus of variations, a variational inequality can be written mathematically as a minimization problem on a closed convex subset K of a Banach space V (cf. Theorem 1). Usually, V is a Sobolev space and K is defined by equality and inequality constraints which must be satisfied by the functions from V. In this paper we approximate the solution by controlling the obstacle condition in images of the Gauss–Lobatto points. We deduce existence and uniqueness for the discrete solution up from the p-version for the obstacle problem. We prove the convergence of up towards the solution with respect to the energy norm, and assuming some additional regularity for the solution we derive an a priori error estimate which corresponds to Falk’s result for the h-version [7]. In numerical experiments the p-version turns out to be superior to the h-version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness. Variational equations can be solved approximately by searching the solution on h-, p-, and hp-discrete subspaces of Sobolev spaces called finite element spaces. Here, the h-version FEM achieves the convergence of the approximate solution in the Sobolev space by mesh refinement, whereas the p-version FEM achieves the convergence by increasing the polynomial degree of the discrete subspace. Particularly when the solutions to variational equations are substantially smoother than H 2 () (cf. [2, 3]) or only piecewise smoother the p-version is superior to the h-version approach. Intuitively, variational inequalities seem to be comparable to variational equations, when we decompose the domain into the contact zone where the solution u is equal to the obstacle function ψ almost everywhere and into the free zone where u is greater than ψ. Then, u is governed by the material laws of a variational equation on the free zone. On the contact zone, we have the regularity of the obstacle. Unfortunately, there is a twofold problem with this intuitive approach: firstly, we do not know the contact zone in advance, i.e., we cannot tailor the mesh according to the boundary of the contact zone. Secondly, the crossover of the solution from the contact to the free zone determines the regularity of the solution significantly. Nevertheless, Kinderlehrer and Stampacchia succeeded in showing H 2 ()-regularity for the solutions of elliptic and locally elliptic obstacle problems (cf. [12, Theorem IV.2.5, Theorem IV.3.6, Theorem IV.4.3]). This paper is organized as follows. In Sect. 2 we recall a model obstacle problem, its corresponding variational inequality, and a criterion for existence and uniqueness of a solution u on a closed convex subset K ⊂ H 1 (). We approximate the solution of the obstacle problem by searching the minimum on the discrete subset Kp,gD of the conforming p-FE space Vp which allows to control boundary and obstacle conditions at appropriate points of the domain. Namely, the points are given on the reference square [−1, 1]2 by the tensor product of Gauss–Lobatto points. The images of these points onto the quadrilaterals of the mesh define the control points. Existence and uniqueness of a minimum up ∈ Kp are obtained by Theorem 2. In Sect. 3 the application of theoretical results from Glowinsky [9] to the p-version yields the convergence of up towards

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the solution u on K with respect to  · H 1 () (see Theorem 3). Furthermore, the h-version convergence analysis from Hlaváˇcek et al. (cf. [11]) is extended to the p-version. This results in an a priori estimate for the error u − up H 1 () in Theorem 4 under the regularity assumption u ∈ H 2 (). In Sect. 4 we comment on the implementation issues. In Sect. 5 we present four numerical examples. We also compare the p-version with the h-version. 2 A p-version finite element method applied to an obstacle problem For a second order quasi-linear elliptic differential operator with mixed boundary conditions on a bounded Lipschitz domain  ⊂ R2 we consider the following obstacle problem (P) and derive an equivalent variational inequality (VI) (see Theorem 1). Let  := ∂ = D ∪ N where D = ∅ and N are simply connected and disjoint with outward directed unit normal n on . Problem (P) For given data f ∈ H −1 (), gD ∈ H 1/2 (D ), gN ∈ H −1/2 (N ), and ψ ∈ H 1 () with ψ ≤ gD a.e. in a neighborhood of D , σ ≥ 0, we look for u ∈ H 1 () satisfying P(u) := − div(ρ(|∇u|)∇u) + σ u − f ≥ 0

in ,

on D , ∂ PN (u) := ρ(|∇u|) u − gN ≥ 0 on N , ∂n (u − ψ)P(u) = 0 on , (u − ψ)PN (u) = 0 on N , and u ≥ ψ on 

(1)

u = gD

(2)

where ρ ∈ C1 (R≥0 ) satisfies for all t ∈ R≥0 ρ2 ≤ ρ(t) + tρ  (t) ≤ ρ3

ρ0 ≤ ρ(t) ≤ ρ1 , with constants ρi > 0, 0 ≤ i ≤ 3. Setting

(3)

t τρ(τ ) dτ ,

p(t) :=

(4)

0

we can define the functional A : H 1 () → R by 





 u2 dx −

p(|∇u|) dx + 12 σ

A(u) :=



 f u dx −



N

gN u| ds.

(5)

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Standard calculus gives for its Fréchet derivatives at u ∈ H 1 ()  DA(u; v) = 



D2 A(u; v1 , v2 ) =

  ρ(t)∇ T u ∇v + σ uv − fv dx −

 gN v| ds,

N

  ρ(t)∇ T v1 ∇v2 + tρ  (t)s1 s2 + σ v1 v2 dx

(6)



with t := |∇u| and si := si := 0.

(∇u)T t ∇vi

for all v1 , v2 ∈ H 1 (). Note, if t = 0, we set

Lemma 1 (i) Let σ = 0 in (5). Then A is strictly convex, DA is uniformly monotone, and D2 A is continuous with respect to the semi-norm |·|H 1 () , i.e., there exist real constants κl , κu with 0 < κl ≤ κu such that for all u, v ∈ H 1 () κl |u −v|2H 1 () ≤ DA(u; u − v) − DA(v; u −v) ≤ κu |u −v|2H 1 () , κl |v|2H 1 () ≤ D2 A(u; v, v) ≤ κu |v|2H 1 () .

(7) (8)

For σ > 0 in (5), the inequalities (7), (8) hold, if we replace the semi-norm by the norm in H 1 (). (ii) Suppose there exist constants 0 < ρ4 < ρ5 such that |ρ  (t)| ≤ ρ4 , |ρ  (t) + tρ  (t)| ≤ ρ5 for t ∈ R≥0 . Then, D3 A is continuous. Proof (i) We note that si ≤ |∇vi |, i = 1, 2, a.e. Using (3) and taking v1 = v2 = v in (6) we can write  D2 A(u; v, v) =

 ρ2 |∇v|2 dx +



  (ρ(t) − ρ2 )|∇v|2 + tρ  (t)s21 + σ v2 dx.



We estimate  0≤ 

  ρ(t) − ρ2 + tρ  (t) s21 dx ≤



  (ρ(t) − ρ2 )|∇v|2 + tρ  (t)s21 dx



and obtain (8) with κl := ρ2 . Equation (8) implies the uniform monotonicity of DA stated in the left inequality of (7). In case of σ > 0, the coercivity of DA2 and the uniform monotonicity of DA follow by taking κl := min{σ , ρ2 }. Using |tρ  (t)| ≤ |ρ(t)| + |ρ(t) + tρ  (t)| ≤ ρ1 + ρ3 [see (3)], the Cauchy Schwarz inequality, and again si ≤ |∇vi | (i, j = 1, 2), we obtain the continuity of D2 A in case of σ = 0 since for all u, v1 , v2 ∈ H 1 () D2 A(u; v1 , v2 ) ≤ (2ρ1 + ρ3 ) |v1 |H 1 () |v2 |H 1 () .

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In case of σ > 0, there follows for all u, v ∈ H 1 () D2 A(u; v, v) ≤ 2 max{2ρ1 + ρ3 , σ } v2H 1 () . Thus, the right inequalities of (7), (8) follow with κu := 2 max{2ρ1 + ρ3 , σ }. The strict convexity of A follows by standard convex functional analysis. (ii) The continuity of D3 A follows completely analogously showing that for all u, v1 , v2 , v3 ∈ H 1 () |D3 A(u; v1 , v2 , v3 )| ≤ (2ρ4 + ρ5 ) |v1 |H 1 () |v2 |H 1 () |v3 |H 1 () .

(9) 

The bound (9) will be needed to estimate the approximation error in the numerical experiments by Lemma 4. Remark 1 For 0 < ρ0 ≤ ρ(|∇u|) for all u ∈ H 1 (), there holds A(u) → +∞ Proof With t := |∇u| we have p(t) ≥

as |u|H 1 () → +∞. t 0

τρ0 dτ = 12 ρ0 t2 and

 p(|∇u|) dx ≥ 12 ρ0 |u|2H 1 () .





Next, we introduce the cone u ∈ K := {v ∈ Hg1D () | v ≥ ψ a.e. on }, where Hg1D () := {v | v ∈ H 1 (), v|D = gD a.e. on D }, and the coincidence set

:= {x ∈  | u(x) = ψ(x)}.

(10)

The following equivalence result is standard (see, e.g., [15]). Theorem 1 (i) There exists a unique u ∈ K which minimizes the functional A on K, i.e., A(u) ≤ A(v) ∀v ∈ K. (MP) (ii) Furthermore, u ∈ K solves (MP) if and only if u solves the variational inequality DA(u; v − u) ≥ 0 ∀v ∈ K. (VI)

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(iii) Additionally, the variational inequality (VI) and problem (P) are equivalent, i.e., the solution of (VI) is the weak solution of (P) and vice versa. Proof The existence of a minimum of A, its uniqueness, and the equivalence (ii) follow by means of convex functional analysis (cf. [15]) because there holds: (α) K is a convex, closed, and nonempty subset of the real reflexive Banach space H 1 () . (β) A is strictly convex on K, i.e., A((1 − t)u + tv) < (1 − t)A(u) + tA(v) for all u, v ∈ K, t ∈ (0, 1) (see Lemma 1). (γ) A(u) → ∞ as uH 1 () → ∞ (see Remark 1). Integration by parts of (VI) with appropriately chosen test functions v ∈ K yields the equivalence (iii) (cf. [11, p.90]). A detailed proof can be found in [13, Theorem 1.23]. 

Now, we apply the p-version of the finite element method to the varia˜ = (−1, 1)2 be the reference square; let T denote a tional inequality (VI). Let Q shape regular partition of  into a finite number of curvilinear quadrilaterals ˜ where FQ is a bounded diffeomorphism satisfying Q = FQ (Q) |FQ |1,∞,Q˜ ≤ C1 hQ ,

−1 |FQ |1,∞,Q ≤ C2 h−1 Q ,

|JFQ |0,∞,Q˜ ≤ C3 h2Q ,

|JF −1 |0,∞,Q ≤ C4 h−2 Q Q

with parameter hQ , proportional to the diameter of Q, and constants Ci , −1 are the Jacobians of 1 ≤ i ≤ 4. Here, JFQ = det DFQ and JF −1 = det DFQ Q

−1 , respectively, and FQ and FQ

|FQ |k,∞,Q˜ := sup |Dl FQ (˜x)|. ˜ x˜ ∈Q |l|=k

We define the FE space ˜ Q ∈ T }. Vp := {u ∈ H 1 () : u|Q ◦ FQ ∈ P2p (Q), ˜ p ≥ 1, denotes the space of tensor product polynomials on the where P2p (Q), reference square which are of degree at most p separately in x1 and x2 . Next, we introduce the affine subspace Vp,gD and the subset Kp,gD of Vp . Let Lp be the Legendre polynomial of degree p ≥ 1. The zeros of (1 − ξ 2 )Lp (ξ ) p+1

are distinct numbers ξi degree p. We set

∈ [−1, 1], 0 ≤ i ≤ p, called Gauss–Lobatto points of p+1

GQ,p ˜ := {(ξi

p+1

, ξj

) | 0 ≤ i, j ≤ p}.

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Using appropriate extensions of FQ we define for Q ∈ T GQ,p := {FQ (ξ ) | ξ ∈ GQ,p ˜ }

and

Gp :=



GQ,p .

Q∈T

We define Vp,gD := {w ∈ Vp | w(x) = gD (x), x ∈  D ∩ Gp } , and Kp,gD := {w ∈ Vp,gD | w(x) ≥ ψ(x), x ∈ Gp }. Here, it is worth to note that we do not have Kp,gD ⊂ K or Kp,gD ⊂ Kq,gD for p ≤ q in general. This inconsistency of the approximation subsets appears also in the analysis for the convergence rate of the h-version given by Falk in [7] and by Hlaváˇcek et al. in [11]. Nevertheless, these approaches can not be extended straightforwardly to the p-version since they both use the piecewise affine linearity of the approximating functions. Lemma 2 Vp,gD and Kp,gD are nonempty closed convex subsets of Vp . Proof From interpolation theory it is known that there exists an interpolating polynomial v ∈ Vp with v(x) ≥ ψ(x) for all x ∈ Gp and v(x) = gD (x) for all x ∈ D,p := Gp ∩  D . Thus Kp,gD is nonempty. The convexity of Kp,gD is trivial. Let vn → v converge strongly in H 1 (), where vn ∈ Kp,gD and v ∈ H 1 (). With vn (x) ≥ ψ(x) for all x ∈ Gp and vn (x) = g(x) for all x ∈ D,p , there follows v(x) ≥ ψ(x) for all x ∈ Gp and v(x) = g(x) for all x ∈ D,p . Therefore v ∈ Kp,gD .  Dropping the above claims ≥ ψ(x), the statement concerning Vp,gD follows. The unique p-version finite element approximation for the exact solution u of (VI) is obtained as follows. Theorem 2 (i) There exists a unique up ∈ Kp,gD which minimizes the functional A on Kp,gD , i.e., A(up ) ≤ A(v)

∀v ∈ Kp,gD .

(DMP)

(ii) Furthermore, up ∈ Kp,gD solves (DMP) if and only if up solves the variational inequality DA(up ; v − up ) ≥ 0

∀v ∈ Kp,gD .

(DVI)

Proof Analogously to the proof of Theorem 1 we obtain the existence of a minimum, its uniqueness, and the equivalence (ii) since Kp,gD is a convex, closed, and nonempty subset of the real reflexive Banach space Vp , A is strictly convex, 

and A(u) → ∞ for |u|H 1 () → ∞.

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3 Convergence results for the p-version The following approximation result is a straightforward extension of [4, Theorem 14.2] and will be used below in the proofs of Theorem 3 and 4. Let ip v the piecewise interpolating polynomial of v ∈ C() with respect to Gp and the mesh T . For any real numbers r and s satisfying s > 1 + 2r and 0 ≤ r ≤ 1, there exists a constant c > 0 depending only on s such that, for any function v ∈ H s (), the following estimate holds v − ip vH r () ≤ cpr−s vH s () .

(11)

The convergence of the minimizer up of Theorem 2 towards the minimizer u of Theorem 1 is stated in the following theorem: Theorem 3 Let ψ ∈ C0 () ∩ H 1 () and ψ ≤ gD in a neighborhood of D . Then, there holds lim up − uH 1 () = 0

p→∞

with the minimizers u of Theorem 1 and up of Theorem 2. ˆ ·, ·) is positive definite on Hg1D () and Vp for all uˆ ∈ Proof The form D2 A(u; H 1 () due to Lemma 1 (8) and the Poincaré inequality. Thus, it suffices with Theorem [9, Theorem I.5.2] to prove the following two hypotheses: H1 If (vp )p is such that vp ∈ Kp,gD for all p and vp converges weakly to v as p → ∞, then v ∈ K. H2 There exists a dense subset χ of K and a family of mappings rp : χ → Kp such that limp→∞ rp v = v strongly in H 1 () for all v ∈ χ . H1 will be shown in Lemma 3. H2 follows, if we take χ := C∞ () ∩ K and rp : H 1 () ∩ C0 () → Vp defined by rp v := ip v which yields rp v ∈ Kp,gD . The strong convergence in H 1 () is implied by the approximation result (11) 

because of v ∈ χ ⊂ H 2 (). Remark 2 Interpolation by polynomials of high degree can be unstable for functions f ∈ L∞ ((−1, 1)) (cf. Theorem of Faber). Here, the approximation result (11) and Glowinski’s framework used in the proof of Theorem 3 help to circumvent this difficulty because we are in H 1 (). Furthermore, Glowinski’s Theorem transforms the problem of approximating a function by a nonconforming sequence (vp )p with vp ∈ Kp,gD ⊂ K to the question if the cluster points of (vp )p with respect to the weak topology of V belong to the cone K (cf. H1). Lemma 3 If the sequence (vp )p with vp ∈ Kp,gD converges weakly to v for p → ∞, then v ∈ K.

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Proof Firstly, we introduce the Bernstein operator Bp for bivariate functions ˜ For a function φ ∈ C(Q), ˜ the formula on the closed reference square Q. Bp φ(x1 , x2 ) :=

p p  

φ



2k1 p

− 1, 2kp2 − 1 · qp,k1 (x1 ) · qp,k2 (x2 )

(12)

k1 =0 k2 =0

with qp,k (t) :=

  k p−k p t−1 1−t 2 2 k

˜ into P2p ([−1, 1]). It is known that we produces a linear map φ → Bp φ of C(Q) have the uniform convergence lim Bp φ − φ

˜ ∞,Q

p→∞

=0

(cf. [10, Chapter 4, p. 133]). Now, we consider φ ∈ C0 () and define its approximation φp by a combination of Bernstein polynomials on Q, Q ∈ T , i.e.,   −1 (x)) BQ,p φ(x) := Bp φ ◦ FQ (FQ

for x ∈ Q

and φp is given by φp |Q := BQ,p φ for all Q ∈ T . It follows that φp ∈ Vp

and

˜ ∀x ∈ Q.

lim φp (x) − φ(x)∞, = 0

p→∞

(13)

Further, if φ ≥ 0, we have φp ≥ 0 because the Bernstein operators BQ,p are monotone [cf. (12)]. Secondly, we define the piecewise interpolating polynomial ψp := ip ψ. From the approximation result (11) we know lim ψ − ψp L2 () = 0.

p→∞

Now, let the sequence (vp )p∈N , vp ∈ Kp,gD , converge weakly to v and let φ ≥ 0. Using the Gauss–Lobatto quadrature formula p+1

∃ρi

> 0,

0 ≤ i ≤ p, 1

∀f ∈ P2p−1 ([−1, 1]) :

f (ζ ) dζ = −1

p  i=0

p+1

f (ξi

p+1

)ρi

(14)

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and φ ∈ L∞ (Q) = (L1 (Q)) , we obtain that 

 (vp − ψp )φp−1 dx =

lim

p→∞ Q

(v − ψ)φ dx . Q

With Rellich’s embedding theorem (cf. [1, A 5.1]) there follows lim vp = v

p→∞

strongly in L2 () .

Thus, it suffices to show that v ≥ ψ almost everywhere. With (14) and the definition of ψp we get for all Q ∈ T  (vp − ψp )φp−1 dx =

p p     p+1 p+1  (vp − ψp )φp−1 FQ (ξi , ξj ) i=0 j=0

Q

p+1

·| det DFQ (ξi

p+1

, ξj

p+1 p+1 ρj

)| ρi

≥ 0.

(15)

The inequality follows since φp−1 (x) ≥ 0 for all x ∈ Q and (vp − ψp )(x) ≥ 0 for all x ∈ GQ,p due to the definition of Kp,gD . Combining (13) and (15) we obtain that for all φ ∈ C0 (Q) with φ ≥ 0  (v − ψ)φ dx ≥ 0

∀Q ∈ T ,

Q

hence v ≥ ψ almost everywhere on , i.e., v ∈ K.



With Theorem 3 the convergence of the p-version is proved. If we assume higher regularity of the solution u and of the obstacle ψ, i.e., u, ψ ∈ H 2 (Q), we obtain the following a priori error estimate which yields a convergence rate. This assumption of higher regularity of u, ψ is quite natural due to [12, Chapter IV]. Note, in general, Vp,gD ⊂ Hg1D () and Kp,gD ⊂ K. This nonconformity of the approximation subset requests an extension of the analysis for the h-version given in [11] and [7] for a Laplacian inequality. Particularly, we use the nonnegativity of P(u) on the coincidence set given in (10) and the partition of

into ∩ ϒp and \ ϒp where ϒp := {x ∈ |up (x) ≤ ip ψ(x)}. Theorem 4 Let u ∈ K and up ∈ Kp,gD be the minimizers of Theorem 1 and 2, respectively. Suppose u, ψ ∈ H 2 (), f ∈ L2 (), and gN ∈ H 1/2 (N ). Then there exist constants C1 , C2 > 0, independent of u, ψ, and p such that  1/2 −3/4 p u − up H 1 () ≤ C1 C3 uH 2 () + ψH 2 ()  1/2 −1/4 + C2 C3 uH 1 () + ψH 1 () p

(16)

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with 1/2  C3 := uH 2 () + P(u)L2 ( ) + PN (u)H 1/2 (N \∂(\ )) . For u > ψ on N , there holds  1/2 −1 p u − up H 1 () ≤ C1 C3 uH 2 () + ψH 2 ()  1/2 −1/2 +C2 C3 uH 1 () + ψH 1 () p .

(17)

Proof Due to the variational inequalities (VI) and (DVI) (see Theorem 1, 2) we have DA(u; u) ≤ DA(u; v) DA(up ; up ) ≤ DA(up ; vp )

∀v ∈ K;

(18)

∀vp ∈ Kp,gD .

(19)

Linearization gives for a θ ∈ [0, 1] DA(up ; v) = DA(u; v) + D2 A(u + θ (up − u); v, up − u). Therefore, using Lemma 1 (7) and (18), (19) we deduce with v ∈ K and vp ∈ Kp,gD κl |u − up |2H 1 () ≤ DA(u; u − up ) − DA(up ; u − up ) ≤ DA(u; v − up ) + DA(up ; vp − u) ≤ DA(u; v − up ) + DA(u; vp − u) + max {D2 A(u + θ (up − u); vp − u, up − u)} θ∈[0,1]

(20)

Next, we bound the first term in (20). We take v := max{up , ψ} and observe that v ∈ K [12, Chapter II, Theorem A.1]. Integrating DA(u; v − up ) by parts and using the notation P(u) from (1) and PN (u) from (2) gives DA(u; v − up ) = P(u), ψ − up L2 ( ) + PN (u), ψ − up H −1/2 ( − ) N

− with N := N \ ∂( \ ) due to Theorem 1(iii).

To estimate the right hand side of the last equation, we must cope with the consistency error Kp,gD ⊂ K, i.e., there exist x ∈ such that up (x) < ψ(x). Setting ϒp := {x ∈  | up (x) ≤ ip ψ(x)} and u¯ p (x) := max{ip ψ(x), up (x)} for x ∈ , we observe up ≥ ip ψ on  \ ϒp . Recalling P(u) ≥ 0, we can write P(u), ψ − up L2 ( ) ≤ P(u), ψ − ip ψL2 ( \ϒp ) +P(u), ψ − u¯ p L2 ( ∩ϒp ) + P(u), u¯ p − up L2 ( ∩ϒp ) .

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Noting that up is the piecewise interpolating polynomial of u¯ p ∈ H 1 (), i.e., up = ip u¯ p on , and u¯ p = ip ψ on ϒp , we obtain P(u), ψ −up L2 ( ) ≤ P(u), ψ −ip ψL2 ( ) + P(u), u¯ p − ip u¯ p L2 ( ∩ϒp )   ≤ P(u)L2 ( ) C4 p−2 ψH 2 () + C5 p−1 u¯ p H 1 () . where C4 , C5 > 0 are the constants from the approximation result (11). Making use of the corresponding estimate for PN (u), ψ − up H −1/2 ( − ) and of the N continuity of the trace mapping we can write   DA(u; v − up ) ≤ P(u)L2 ( ) C4 p−2 ψH 2 () + C5 p−1 u¯ p H 1 ()   3 1 +PN (u)H 1/2 ( − ) C6 p− 2 ψH 2 () + C7 p− 2 u¯ p H 1 () (21) N

where C6 , C7 > 0 are the constants yielded by combining (11) and the continuity of the trace mapping. Note, when estimating on N we only get p−3/2 , p−1/2 (and not p−2 , p−1 ) due to the restriction r ≥ 0 of (11). Now, we bound the second term in (20). Let vp := ip u ∈ Kp,gD be the interpolating polynomial of u. Again, we integrate DA by parts, use duality, and obtain with (11) DA(u; vp −u) = P(u), vp −uL2 ( ) +PN (u), vp −uH −1/2 ( − ) N

≤ P(u)L2 ( ) C4 p−2 uH 2 () +PN (u)H 1/2 ( − ) C6 p−3/2 uH 2 () . N

(22)

Using (8) we bound the third term in (20). For all μ > 0 and for all ˆ v1 , v2 ∈ H 1 () we have u, ˆ v1 , v2 ) ≤ κu ∇v1 , ∇v2 L2 () ≤ D2 A(u;

κu 2 2μ |v1 |H 1 ()

+

μκu 2 2 |v2 |H 1 () .

Setting μ := κl κu−1 , vp := ip u ∈ Kp,gD we obtain ˆ vp − u, up − u) ≤ D2 A(u;

κu2 2κl

C42 p−2 u2H 2 () +

κl 2 |up

− u|2H 1 () .

(23)

Inserting (21), (22), (23) into (20) gives κl 2 2 |u − up |H 1 ()

κ2

≤ C42 2κu p−2 u2H 2 () l   +P(u)L2 ( ) C4 p−2 (uH 2 () + ψH 2 () ) + C5 p−1 u¯ p H 1 ()    +PN (u)H 1/2 ( − ) C6 p−3/2 uH 2 () + ψH 2 () N +C7 p−1/2 u¯ p H 1 () . (24)

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469

Let CP > 0 be the constant of the Poincaré–Friedrich’s inequality vH 1 () ≤ CP |v|H 1 () which holds for all v ∈ Hg1D (), gD ≡ 0. As ip u = up on D , we can use the Poincaré–Friedrich’s inequality and the approximation result (11) to estimate u − up 2L2 () ≤ 2u − ip u2L2 () + 2ip u − up 2L2 () ≤ 2C52 p−2 u2H 2 () + 2CP2 |ip u − up |2H 1 () ≤ 2(1 + CP2 )C52 p−2 u2H 2 () + 2CP2 |u − up |2H 1 () . Due to [12, Chapter II, Theorem A.1] and the approximation result (11), there holds additionally u¯ p H 1 () ≤ 54 (uH 1 () + ψH 1 () ), if p is sufficiently large. Setting CQ := CP2 κ4 , C22 := 54 CQ max{C5 , C7 }, and l

κ2

C12 := max{2(1 + CP2 )C52 + CQ C42 2κu , CQ C4 , CQ C6 } l

we obtain (16). If u > ψ on N , the Neumann condition (2) of Problem (P) holds with PN (u) = 0 on N [see Theorem 1(iii)]. Thus PN (u)H 1/2 ( − ) = 0 in (24) N yields (17). 

Remark 3 Our numerical examples Ex. 1, Ex. 3, and Ex. 4 with a nonempty coincidence set and a Dirichlet boundary condition suggest convergence rates better than O(p−1 ). Here, the last term of the sum (24) becomes zero due to the Dirichlet boundary condition since N = ∅ implies PN (u)H 1/2 ( − ) = 0. N Thus, C5 u¯ p H 1 () in (24) seems to be very small for these examples. On the other hand, Example 2 which is the only example in Sect. 5 with a known solution u and that allows the exact computation of u − up H 1 () , indicates that Theorem 4 provides a sharp estimate O(p−1/2 ). Remark 4 Falk [7] and Hlaváˇcek et al. [11] both deduce a convergence rate of u − uh H 1 () = O(h) for the h-version for an obstacle problem with Dirichlet boundary conditions and the regularity assumptions of Theorem 4, i.e., the convergence order of the h-version for variational equalities is not diminished by the introduction of an obstacle. Due to the piecewise affine linearity of h-version FEM functions, they can use the property uh ≥ ih ψ. Here uh denotes the h-version solution and ih ψ the linear interpolant of the obstacle. Unfortunately, the corresponding p-version analog up ≥ ip ψ does not hold. Therefore, we had to introduce u¯ p (x) := max{ip ψ(x), up (x)} in the proof of Theorem 4. As we have only u¯ p ∈ H 1 (), this results in the reduction of the convergence order by the obstacle [see (21)].

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Remark 5 An inspection of the proof of Theorem 4 shows that we get the convergence O(p−1/2 ) in case of an obstacle free problem, i.e., = ∅. Furthermore, we get the convergence O(p−1 ) in case of an obstacle free problem without Neumann conditions, i.e., N = ∅. Thus, we get the optimal convergence order when only H 2 -regularity is assumed. When the solutions to variational equations are analytic even exponential convergence rates with respect to the number of unknowns can be shown (cf.[2, 3]). But for variational inequalities we do not know H k ()-regularity results for k > 2. 4 Implementation For the implementation of the p-version we take the following basis of Vp . With p+1 of degree p and the corresponding Lagrangian the Gauss–Lobatto points ξi interpolating polynomials

p λi (ξ )

⎧ ⎪ ⎨p :=

⎪ ⎩0

p+1

ξ −ξk k=0 ξ p+1 −ξ p+1 k k=i i

for ξ ∈ [−1, 1],

0 ≤ i ≤ p,

for ξ ∈ R\[−1, 1],

˜ p := (bij := λpi (ξ1 )λp (ξ2 ) | 0 ≤ i, j ≤ p). Using the transforwe define the basis B j mations FQ we get the local bases  BQ := (bQ,i,j | 0 ≤ i, j ≤ p) with bQ,i,j :=

−1 bij (FQ = (x)) for x ∈ Q, 0 for x ∈ \Q,

on the curvilinear quadrilaterals Q. Besides the local numbering 0 ≤ i, j ≤ p we introduce a global numbering k = 1, . . . , N := card Gp such that Xk := {(Q, i, j) | Q ∈ T , 0 ≤ i, j ≤ p, bQ,i,j (xk ) = 1} for xk ∈ Gp and define the global basis functions 

bk :=

bQ,i,j .

(Q,i,j)∈Xk

With the notation wT b = can rewrite

N

k=1 wk bk

Vp := {wT b | w ∈ RN },

for w := (wk )k=1,...,N , b := (bk )k=1,...,N we

Vp,gD := {wT b | w ∈ RN =g }, D

N and Kp,gD := {wT b | w ∈ RN =g ∩ R≥ψ }. D

A p-version finite element method

where and

471

N RN =g := {w ∈ R | wk = gD (xk ) D

RN ≥ψ

∀xk ∈ Gp ∩  D }

:= {w ∈ RN | wk ≥ ψ(xk )

∀xk ∈ Gp }.

Now, with A : H 1 () → R given in (5), we define the mapping A : RN =gD ∩ N R≥ψ → R by A(w) := A(wT b). Therefore, the discrete obstacle problem can be formulated equivalently as the minimization problem: N Find u which minimizes A on RN =g ∩ R≥ψ in the sense that D

A(u) ≤ A(v)

N ∀v ∈ RN =g ∩ R≥ψ . D

In case that A is of quadratic type, i.e., A(v) = 12 vT B v − vT f where B ∈ RN×N and f ∈ RN , it is a quadratic programming problem. It can be solved by relaxation methods (cf. [9, Chapter V]) or a generalized conjugate gradient algorithm (cf. [14]), known as Polyak algorithm. In Example 1 we use the matlab routine quadprog in large-scale mode which implements a subspace trust-region method based on the interior-reflective Newton method described by Coleman and Li in [5]. When A is not of quadratic type, one must solve a general large-scale nonlinear minimization problem with inequality constraints. Algorithms for this constrained minimization are proposed and implemented by Conn et al. [6] and Felkel in [8]. In [13, Section 4.3] the minimizer given by Felkel is specified to our situation of a strictly convex functional A. Roughly spoken, the minimum is computed by a combination of the projected gradient method with Newton’s method for an unconstrained minimization problem. 5 Numerical examples The following lemma provides useful estimates for the approximation error in the numerical experiments. Lemma 4 Let the functional A : H 1 () → R be given by (5) and let there exist positive constants ρi , i = 0, 1, . . . , 5, such that the assumptions of Lemma 1 and Theorem 1 are satisfied. Furthermore, let u ∈ K be the unique minimizer of A 2 , there holds according to Theorem 1. Then, for v ∈ K with |v − u|H 1 () ≤ 2ρ4ρ+ρ 5

|v − u|2H 1 () ≤

3 ρ2

|A(v) − A(u)|.

(25)

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Proof For v ∈ K there exists a θ ∈ [0, 1] such that A(v) = A(u) + DA(u; w) + 12 D2 A(u; w, w) + 16 θ D3 A(u; w, w, w) with w := v − u by the Taylor formula. Combining the three inequalities (VI) from Theorem 1(ii), (8), and (9), we get 2A(v) − 2A(u) ≥ (ρ2 − 13 (2ρ4 + ρ5 )|v − u|H 1 () )|v − u|2H 1 () . Assuming v ∈ K with |v − u|H 1 () ≤

ρ2 2ρ4 +ρ5

yields (25).



Example 1 (p-version) Consider the obstacle problem −u ≥ f , on Q := (−1, 1)2

with

(u − ψ)(u + f ) = 0, gD ≡ 0 on ∂Q

u ≥ ψ ≡ −1

(26)

and with f = −w,

w := −(x + 1)(y + 1)(e(x−1)(y−1) − 1).

(27)

The solution of (26) minimizes  ( 12 ∇ T v ∇v − fv) dx

A(v) :=

on K := {v ∈ H01 (Q)|v ≥ ψ}.



Furthermore, we know u ∈ H 2 () from [12, Theorem IV.2.3]. As approximation space we choose Vp and its discrete subsets Vp,gD and Kp,gD . The minimizer of the discrete minimization problem is called up . The integrals of the discrete problems are calculated by a Gauss–Lobatto quadrature with p + 4 points. The discrete minimization problem leads to a quadratic programming problem which is solved using the matlab routine quadprog. Each iteration involves the approximate solution of a large linear system using the method of diagonal preconditioned conjugate gradients (PCG). The interior PCG iteration is terminated when the relative residual is less than 10−14 . The algorithm stops when either the relative change of the solution vector up representing up in the Euclidean norm is ≤ 10−13 or the relative change of A(up ) is ≤ 10−14 . The obstacle problem on the square is visualized in Fig. 1 for the obstacle ψ ≡ −1 by a 3d and a contour plot. These plots give an insight on how much the obstacle condition is violated by the p-FE solution up for p = 10. The numerical results are listed in Tables 1 and 2. While u = w from (27) solves (26) and A(u) = −14.95831706718053 if ψ ≡ −∞, the exact solution u of (26) is not known if ψ ≡ −1. Here, we apply Lemma 4 and Poincaré’s inequality to estimate the error by u − up 2H1 (Q) ≤ C |A(up ) − A(u)|

with a constant C > 0,

A p-version finite element method

473

Fig. 1 The obstacle problem on the square (−1, 1)2 with ψ ≡ −1. The plots visualize the FE solution up ∈ K10,0 for p = 10 and gD ≡ 0 Table 1 Convergence results of Example 1 for an obstacle given by ψ ≡ −∞

p

Np

A(up )

|A(up ) − A∗ |

αp

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361

−6.56805e+00 −1.37450e+01 −1.49633e+01 −1.49643e+01 −1.49587e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01 −1.49583e+01

8.39e+00 1.21e+00 5.00e−03 5.99e−03 3.64e−04 1.20e−05 2.68e−07 4.39e−09 5.55e−11 6.52e−13 2.31e−14 2.84e−14 1.42e−13 4.44e−14 2.49e−14 1.21e−13 2.49e−14 1.78e−15 1.56e−13

– – –10.71 –10.40 –6.47 –18.46 –25.08 –31.49 –38.01 –43.93 –42.70 –18.75 – – – – – – –

despite up ∈ K. For ψ ≡ −1 the value A(u) = −12.109 is obtained by extrapolation assuming |A(up ) − A(u)| ≈ Cp−β with constants C, β > 0. The experimental convergence rates αp with respect to the polynomial degrees are computed from |A(up ) − A(u)| ≈ Cpα where C and α are constants independent of p. Note that in case of inhomogeneous boundary conditions and obstacle conditions the sequence A(up ) is not monotonously decreasing since Vp,gD ⊂ Vq,gD and Kp,gD ⊂ Kq,gD for p < q. Therefore, the experimental convergence rates are computed from  

  A(up ) − A(u)  p  log αp+2 = log  A(up+2 ) − A(u)  p+2

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Table 2 Convergence results of Example 1 for an obstacle given by ψ ≡ −1

p

Np

A(up )

|A(up ) − A∗ |

αp

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361

−6.10945e+00 −1.04487e+01 −1.27217e+01 −1.24264e+01 −1.20966e+01 −1.20735e+01 −1.21574e+01 −1.21550e+01 −1.21047e+01 −1.21054e+01 −1.21217e+01 −1.21230e+01 −1.21057e+01 −1.21051e+01 −1.21108e+01 −1.21143e+01 −1.21078e+01 −1.21068e+01 −1.21087e+01

6.00e+00 1.66e+00 6.13e−01 3.17e−01 1.24e−02 3.55e−02 4.84e−02 4.60e−02 4.28e−03 3.56e−03 1.27e−02 1.40e−02 3.27e−03 3.92e−03 1.84e−03 5.34e−03 1.25e−03 2.22e−03 3.05e−04

– – – – −5.63 −4.54 −3.66 −3.28 −2.08 −5.09 −3.30 −3.25 −0.80 0.32 −6.72 −3.58 −3.83 −2.41 −8.05

in case of ψ ≡ −∞ whereas for ψ ≡ −1 we take p + 4 instead of p + 2. The numerical results in Tables 1 and 2 show a faster convergence than |A(up ) − A(u)| ≤ O(p−2 ). The theoretical rate of the p-version due to the approximation estimate given by combining Theorem 4 with Lemma 4 requires only |A(up ) − A(u)| ≤ O(p−1 ). This maybe indicates that C2 in (16) of Theorem 4 is very small (cf. Remark 3) for this example. Example 2 (p-version) Again, we consider the obstacle problem given by (26). But now we choose f and ψ such that the problem has a known solution 43 1 3 2 − 121 u ∈ H 2 (). Using the polynomials p1 (r) := 108 108 r , p2 (r) := 4 + 7 r − 20 r 368 3 64 3 ˆ and ψˆ by + 27 r , and p3 (r) := 1 − 27 r , we define the functions w ⎧ ⎪ ⎨p1 (r) ˆ w(r) := p2 (r) ⎪ ⎩ 0

for 0 ≤ r < 23 , for 23 ≤ r < 34 , for 34 ≤ r,

⎧ ⎪ ⎨p3 (r) ˆ ψ(r) := p2 (r) ⎪ ⎩ p1 (r)

for 0 ≤ r < 14 , for 14 ≤ r < 23 , for 23 ≤ r.

  2 ˆ ◦ r and Writing r(x) := |x − xM |, xM := 0.2 0.1 for x ∈ [−1, 1] , we define w := w 2 ψ := ψˆ ◦ r on  := [−1, 1] (see Fig. 2). Now, substituting the obstacle ψ and the right hand side f in (26) by the newly defined ψ from this example and by  d2 ˆ + f := −w = − dr 2w



1 d ˆ r dr w

◦ r ∈ L2 (),

respectively, it follows that u := uˆ ◦ r = max{w, ψ} solves the obstacle problem. Straightforward analytical computations yield uL2 () = 0.746857, uH 1 () = 2.68143, and uH 2 () = 16.3473.

A p-version finite element method

475

ˆ ψˆ against r (left), and 3D plot of w (grayscaled surface), ψ (edge grid) on Fig. 2 2D plot of w,  := (−1, 1)2 (right). Max(w, ψ) is the solution of the obstacle problem given in Example 2

Fig. 3 The surface plots visualize up ∈ Kp,0 for p = 5, 12, 19, 26 from Example 2. The gray value of   the surface shows the deviation up − u (x) against the analytical solution. The obstacle condition up ≥ ψ is violated when the grid lines are visible. The gray values of the surface at these regions indicate that the violations are close to zero

We calculate the numerical approximations up of u as described in Example 1. The surface plots of Fig. 3 give an insight on how much the obstacle condition is violated by up for p ∈ {5, 12, 19, 26}. In order to check the quality of the used adaptive numerical quadrature, we compute uL2 () and uH 1 ()

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A. Krebs, E. P. Stephan

Fig. 4 Double logarithmic plot of the error err(p) := u − up H1 () against p, 2 ≤ p ≤ 35

numerically and compare these results with those obtained analytically; these numbers agree with eachother in the first six significant digits. As we know the exact solution u of the obstacle problem, we can compute the error u − up H 1 () much more precisely than in Example 1 using adaptive numerical quadrature. Figure 4 shows the double logarithmic plot of u − up H 1 () against p, 2 ≤ p ≤ 35. 2 Example 3 (h- and p-version) Let  := √ (−1, 1) and A be given by (5) with 2 σ = 0, f ≡ 0, N = ∅, and p(t) := 1 + t . Consider the minimal surface problem

∃!u ∈ K : A(u) ≤ A(v) ∀v ∈ K := {v ∈ Hg1D () | v ≥ ψ a.e. on } with gD (x) := (1 − x81 ) − (1 − x82 ) on ∂ and  ψ(x) :=

1 16

−∞

− x21 − 0.3

for |x1 | < 14 , for |x1 | ≥ 14 ,

on .

Unfortunately, the ρ(t) corresponding to p(t) [see (4)] does not fulfill the condition (3) demanded for existence and uniqueness of the minimum by Theorem 1. We can take ρ1 = ρ3 = 1 as upper bounds, but the lower bounds ρ0 > 0 and ρ2 > 0 do not exists because t can become arbitrarily large. But an extensive analysis of the contact minimal surface problem given by Kinderlehrer and Stampacchia [12, Theorem IV.4.3] shows that there exists a unique minimal surface u ∈ H 2 (), when we assume homogeneous boundary data, a convex domain  with smooth boundary ∂, and an obstacle ψ ∈ C2 (). This result can be extended to the specific conditions of this example. We compare the h- and the p-version as follows: For the h-version we start with a uniform square grid T0 on which we take the FE space V1 (T0 ). The squares of the grid have the length h0 . Then, each

A p-version finite element method Table 3 Convergence results of Example 3 for the h-version

Table 4 Convergence results of Example 3 for the p-version

477 ¯ |A(uk ) − A|

α

9.04436 9.44237 9.53080 9.54963 9.55350 9.55431

5.10e–01 1.12e−01 2.37e−02 4.87e−03 1.00e−03 1.91e−04

— −0.93 −1.04 −1.10 −1.12 −1.19

9.31889 9.50355 9.54362 9.55222 9.55405 9.55442

2.36e−01 5.09e−02 1.09e−02 2.28e−03 4.55e−04 7.77e−05

— −0.99 −1.06 −1.10 −1.15 −1.27

A(uk )

h0

k

Nk

2 5

1 2 3 4 5 6

16 81 361 1,521 6,241 25,281

2 7

1 2 3 4 5 6

36 169 729 3,025 12,321 49,729

h0

p

Np

2 5

1 2 3 4 5 6 7 8 9 10 11 12

16 81 196 361 576 841 1,156 1,521 1,936 2,401 2,916 3,481

2 7

1 2 3 4 5 6 7 8 9

36 169 400 729 1,156 1,681 2,304 3,025 3,844

¯ |A(up ) − A|

α

9.04436 9.58605 9.56674 9.55889 9.55635 9.55540 9.55497 9.55476 9.55466 9.55460 9.55457 9.55455

5.10e−01 3.16e−02 1.22e−02 4.39e−03 1.85e−03 9.01e−04 4.70e−04 2.58e−04 1.64e−04 1.01e−04 6.89e−05 4.66e−05

— — −1.49 −1.32 −1.75 −1.87 −1.97 −2.11 −2.04 −2.06 −2.12 −2.08

9.31889 9.57405 9.56025 9.55655 9.55539 9.55494 9.55473 9.55463 9.55457

2.36e−01 1.95e−02 5.75e−03 2.05e−03 8.89e−04 4.38e−04 2.30e−04 1.27e−04 7.28e−05

— — −1.54 −1.54 −1.76 −1.85 −1.96 −2.10 −2.25

A(up )

element Q ∈ T0 is divided into four elements by bisection of the edges yielding by recursive application the grids Tk and FE-spaces V1 (Tk ), k ≥ 1. The discrete minima A(uk ) on K1,gD (Tk ) for different initial mesh widths are listed in Table 3. For the p-version, again, we start with a uniform square grid T0 with the initial mesh parameter h0 on which we take the FE space Vp (T0 ), p = 1. Then, we increase the polynomial degree p by 1 and compute the discrete minima A(up ) on Kp,gD (T0 ) (see Table 4). The constrained discrete minimum problems are computed using [13, Algorithm 4.3] which combines projected gradient iterations and Newton’s methods. The iterations are performed until the Euclidean norm of the projected gradient

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Fig. 5 Surface plots of u from Example 3: The approximation is computed according to the mesh parameters h = 18 and p = 4. 435 of the 3, 969 degrees of freedom are active. The wire frame model of the cylinder in the plots shows the coincidence set . The right plot visualizes the area where the obstacle condition is violated, i.e., ψ(x) > up (x)



(∇ψ A(u))i =

max{0, (∇A(u))i }, (∇A(u))i ,

if ui ≤ ψi , if ui > ψi ,

i ∈ {1, . . . , N}

fulfills ∇ψ A(u)2 ≤  := 10−10 . The right plot of Fig. 5 shows that for those x ∈  where the obstacle condition is violated, we have |ψ(x) − up (x)| ≤ 2 · 10−3 for h = 18 and p = 4. The p-version on the mesh with mesh size h0 and the h-version with initial mesh size h0 as well as the h-version with the tested mesh widths all give the ¯ = 9.5545. Denoting the number of unknowns by same extrapolated value A ¯ = O(N −1 ) for the h-version. Assuming (25) of Nk Table 3 shows |Ak − A| k Lemma 4 to hold, then gives uh − uH 1 () = O(h) since N = O(h−2 ). Table 4 ¯ = O(Np−3/2 ) for the p-versions and assuming (25) to hold indicates |Ap − A|

gives up − uH 1 () = O(p−3/2 ) since N = O(p2 ). The accuracy of the best h-version solution calculated from 49, 729 unknowns is already achieved by the p-version with less than 3, 000 unknowns. For the Newton steps of [13, Algorithm 4.3] it suffices to compute the matrixvector multiplication of the Hessian of A with the coefficient vector. An efficient implementation for this product with costs of O(p4 ), i.e., O(N 2 ), is described in [13, Appendix B]. The costs for a respective matrix-vector multiplication for the h-version are O(card T ), i.e., O(N). In spite of this p-version drawback the constrained minimization problems given by the p-version were solved much faster than those given by the h-version since the set of constrained variables could be identified faster due to the smaller number of unknowns. Example 4 (h- and p-version) We consider the same minimization problem as in Example 3 but with gD ≡ 0 and

A p-version finite element method Table 5 Convergence results of Example 4 for the h-version

Table 6 Convergence results of Example 4 for the p-version

 ψ(x) :=

479 ¯ |A(uk ) − A|

α

4.18788 4.28220 4.30337 4.30949 4.31111 4.31160

1.24e−01 2.96e−02 8.40e−03 2.28e−03 6.61e−04 1.71e−04

— −0.88 −0.84 −0.91 −0.88 −0.97

4.24838 4.29280 4.30692 4.31046 4.31143 4.31168

6.34e−02 1.90e−02 4.85e−03 1.31e−03 3.39e−04 9.09e−05

— −0.78 −0.93 −0.92 −0.96 −0.94

¯ |A(up ) − A|

α

A(uk )

h0

k

Nk

2 5

1 2 3 4 5 6

16 81 361 1,521 6,241 25,281

2 7

1 2 3 4 5 6

36 169 729 3,025 12,321 49,729

A(up )

h0

p

2 5

1 2 3 4 5 6 7 8 9 10 11 12

16 81 196 361 576 841 1,156 1,521 1,936 2,401 2,916 3,481

4.18788 4.30886 4.30934 4.31169 4.31183 4.31184 4.31196 4.31167 4.31202 4.31191 4.31170 4.31190

1.24e−01 2.91e−03 2.43e−03 8.08e−05 6.11e−05 6.61e−05 1.93e−04 9.75e−05 2.54e−04 1.38e−04 6.95e−05 1.29e−04

— — −1.57 −2.40 −3.41 −0.24 1.65 0.66 0.53 0.75 −3.17 −0.17

2 7

1 2 3 4 5 6 7 8 9 10

36 169 400 729 1,156 1,681 2,304 3,025 3,844 4,761

4.24838 4.30706 4.31097 4.31243 4.31151 4.31202 4.31174 4.31187 4.31187 4.31175

6.34e−02 4.71e−03 7.99e−04 6.56e−04 2.59e−04 2.55e−04 2.90e−05 9.76e−05 1.05e−04 2.49e−05

— — −1.82 −1.35 −1.06 −1.13 −3.17 −1.63 2.51 −3.01

1 16

−∞

Np

− |x − xm |2 +

1 4

for |x − xM | < 14 , for |x − xM | ≥ 14 ,

  2 where xM = 0.3 0.1 . Again, we know u ∈ H () from [12, Theorem IV.4.3]. Here, the p-version on the mesh with mesh size h0 and the h-version with ¯ = 4.3118. Table 5 shows initial mesh size h0 give the same extrapolated value A −3/4 ¯ |Ak − A| = O(Nk ) for the h-version which corresponds to uh − uH 1 () = O(h3/4 ). Table 6 confirms convergence and numerical stability of the p-version.

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A. Krebs, E. P. Stephan

For the mesh size h0 = 25 the experimental convergence rates for p ≥ 6 do not indicate further convergence. We explain this by the exactness of the solution for p = 5 and p = 6. As the approximation can hardly be improved, further convergence cannot be shown. In addition, we note that the experimental con¯ Nevertheless, the vergence rates depend highly on the extrapolated value A. p-version performs stable and achieves the best approximation of the h-version calculated from 49, 729 unknowns already with less than 3, 000 unknowns for the initial mesh sizes h0 = 25 , 27 . References 1. Alt, H.W.: Lineare Funktionalanalysis. Eine anwendungsorientierte Einführung. (Linear functional analysis. An application oriented introduction). 2. verbesserte Auflage. Springer, Berlin Heidelberg New York (1991) 2. Babuška, I., Guo, B.Q.: Regularity of the solution of elliptic problems with piecewise analytic data. I. Boundary value problems for linear elliptic equation of second order. SIAM J. Math. Anal. 19(1), 172–203 (1988) 3. Babuška, I., Guo, B.Q.: Regularity of the solution of elliptic problems with piecewise analytic data. II. The trace spaces and application to the boundary value problems with nonhomogeneous boundary conditions. SIAM J. Math. Anal. 20(4), 763–781 (1989) 4. Bernardi, C., Maday, Y.: Spectral methods. In: Handbook of Numerical Analysis, vol. V. Handb. Numer. Anal., V, North-Holland, Amsterdam pp. 209–485 (1997) 5. Coleman, T.F., Li, Y.: A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables. SIAM J. Optim 6(4), 1040–1058 (1996) 6. Conn, A.R. Gould, N.I.M., Toint, P.L.: LANCELOT. A Fortran package for large-scale nonlinear optimization (Release A). Springer Series in Computational Mathematics, vol. 17. Springer, Berlin Heidelberg New York (1992) 7. Falk, R.S.: Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28, 963–971 (1974) 8. Felkel, R.: On solving large scale nonlinear programming problems using iterative methods. Ph.D. thesis. Shaker Verlag, Aachen TU Darmstadt Darmstadt, Fachbereich Mathematik (1999) 9. Glowinski, R.: Numerical methods for nonlinear variational problems. Springer Series in Computational Physics. New York etc. Springer, Berlin Heidelberg New York (1984) 10. Hämmerlin, G., Hoffmann, K.H.: Numerical mathematics. Transl. from the German by Larry Schumaker. Undergraduate Texts in Mathematics; Readings in Mathematics, vol. xi, Springer, Berlin Heidelberg New York p. 422 (1991) 11. Hlaváˇcek, I., Haslinger, J., Neˇcas, J., Lovíšek, J.: Solution of variational inequalities in mechanics. Applied Mathematical Sciences, vol. 66. Springer, Berlin Heidelberg New York (1988) 12. Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. Pure and Applied Mathematics, vol. 88. Academic, New York (A Subsidiary of Harcourt Brace Jovanovich, Publishers) (1980) 13. Krebs, A.: On solving nonlinear variational inequalities by p-version finite elements. Ph.D thesis, Universität Hannover, Hannover Institut für Angewandte Mathematik (2004) 14. O’Leary, D.P.: A generalized conjugate gradient algorithm for solving a class of quadratic programming problems. Linear Algebra Appl. 34, 371–399 (1980) 15. Zeidler, E.: Nonlinear functional analysis and its applications. III. Variational methods and optimization, Translated from the German by Leo F. Boron. Springer, Berlin Heidelberg New York (1985)