European Congress on Computational Methods in Applied Sciences and EngineeringECCOMAS 2004

P. Neittaanmaki, T. Rossi, S. Korotov, E. Onate, J. Periaux, and D. Knorzer (eds.)Jyvaskyla, 24–28 July 2004

ADAPTIVE HP-VERSIONS OF BOUNDARY ELEMENTMETHODS FOR ELASTIC CONTACT PROBLEMS

M. Maischak?, E. P. Stephan?

?Institut fur Angewandte Mathematik,Universitat Hannover, Welfengarten 1, 30167 Hannover, Germany

e-mail: maischak,stephan@ifam.uni-hannover.de,

web-page: http://www.ifam.uni-hannover.de

Key words: Signorini problem, boundary elements, hp-version, a posteriori error esti-mator, adaptive algorithm.

Abstract. The variational formulation of Signorini-problems leads to a variational in-equality on a convex subset. Applying the symmetric formulation of the boundary elementmethod to this variational inequality introduces the Poincare-Steklov operator, which canbe represented in its discretized form by the Schur-complement of the dense Galerkin-matrices for the single layer potential, the double layer potential and the hypersingularintegral operator. Considering the difficulties in discretizing the convex subset involved,traditionally only the h-version is used for the discretization. Only recently, the investiga-tion of p- and hp-versions for Signorini problems started. Here, convergence results for thequasi-uniform hp-version of BEM Signorini problems are shown, as well as a-posteriorierror estimates for the hp-version, which can be used as error indicators for adaptive hp-algorithms. Numerical experiments are presented which underline the theoretical results.

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M. Maischak, E. P. Stephan

1 INTRODUCTION

In this paper we present a convergence analysis for the boundary element approxima-tion, obtained by the hp-version, of a class of unilateral problems, namely for the Signorinicontact problem of the Lame-operator. The problem under consideration is the deforma-tion u of a linear elastic body occupying a bounded domain Ω with Lipschitz boundaryΓ which is partly in unilateral contact with a supporting, frictionless rigid foundation.This paper can be seen as the generalization of [15] to the Lame-equation. Making useof boundary integral operators as in Han [11] the Signorini problem is reformulated as avariational inequality on Γ for u and its traction ϕ; here the admissible virtual displace-ment u belongs to the convex cone KΓ of H1/2(Γ) functions v satisfying vn ≤ gn on theSignorini boundary ΓS for a given contact function gn. We introduce the hp-version ofthe corresponding Galerkin boundary element scheme by enforcing the contact conditionin the affinely transformed Gauss-Lobatto points on the single elements. This yields aconvex closed subset KΓ

hp of the boundary element space σhp of continuous, piecewisepolynomials of degree p on a regular, quasi-uniform partition ωh of Γ. For a non-concavecontact function gn or polynomial degree p ≥ 2 of the Galerkin approximation the setKΓ

hp is not a subset of KΓ. The Galerkin approximation for the traction T (u) on Γ iscomputed with discontinuous piecewise polynomials of degree q forming the space τhp.Adapting and extending the discretization theory of Glowinski [9] we prove convergenceof the Galerkin solution in the energy norm without any regularity assumptions (The-orem 2). In the proofs in [15], where the Gauss-Lobatto quadrature places a key role,we extend the arguments in Gwinner, Stephan [10]; their approach uses Newton-Cotesformulas and is restricted to the pure h-version with low polynomial degrees. Undermild regularity assumptions we derive with Theorem 3 an a priori error estimate for theGalerkin solution. Here we apply the results of [15] for the Laplacian, where we extendedthe results of Falk [8] (dealing with the h-version of the finite element method) to thehp-version BEM for a variational inequality with non-local boundary integral operators.The proof is based on an equivalent formulation of the Signorini contact problem via thePoincare-Steklov operator and uses heavily the discrete version of this operator.

For the Galerkin solution of the variational inequality we derive an efficient and reli-able a posteriori error estimate. The hierarchical error estimators are computed by theenrichment of the boundary element spaces σhp and τhp with bubble functions on eachelement. This enrichment defines two-level subspace decompositions with correspondingadditive Schwarz operators where the latter have condition numbers which depend onlylogarithmically on the polynomial degrees. In our analysis for the a posteriori error esti-mate we assume a saturation assumption to hold for the Galerkin error, as it is commonfor hierarchical bases in the finite element method [2]. We state the results for d = 2dimensions, but the arguments can be carried over to higher dimensions. We present athree-step adaptive algorithm for the hp-version which is steered by the hierarchical errorestimators. As our numerical experiments show this three-step algorithm leads to appro-

2

M. Maischak, E. P. Stephan

priate mesh refinement and a reasonable polynomial degree distribution. We believe thatour analysis can also be extended to more general problems than the model problem con-sidered here, namely to, e.g. elasticity problems with non-linear behavior in Ω where onehas, of course, to apply a fem-bem coupling as in [5] and also to problems with friction,see [16] for the h-version.

In the following we will denote by Hs(Γ) = [Hs(Γ)]d the Sobolev spaces involved forvector-valued functions.

2 THE HP-VERSION OF THE LAME PROBLEM WITH SIGNORINI

BOUNDARY CONDITION

2.1 Variational formulation

Let Ω ⊂ IRd, d ≥ 2, be a bounded domain with Lipschitz boundary Γ = ∂Ω which is adisjoint union of ΓD, ΓN and ΓS 6= ∅. We consider the following problem.

Given h ∈ H−1/2(ΓN ∪ ΓS), g ∈ H1/2(ΓD ∪ ΓS) ∩ [C0(ΓD ∪ ΓS)]d, find u ∈ H1(Ω) s.t.

∆∗u := µ∆u + (λ+ µ) grad div u = 0 in Ω,u = g on ΓD,

ϕ := T (u) = h on ΓN ,un ≤ gn, ϕn ≤ hn, (un − gn)(ϕn − hn) = 0, ϕt = ht on ΓS

(1)

Here, normal and tangential components are given by un = u ·n and ut = u− un ·n, etc.The traction operator is defined by (T u)i = λni div u + µ ∂

∂nui + µ〈 ∂u

∂xi,n〉.

It is well known that this problem can be formulated equivalently as a variationalinequality over Ω [9] with the help of the convex subset K := v ∈ H1(Ω) : v|ΓD

=

g|ΓD, v · n|ΓS

≤ gn|ΓS, i.e. find u ∈ K, such that

β(u, v − u) ≥ l(v − u) ∀v ∈ K (2)

with

β(u,v) = µ

d∑

i=1

∫

Ω

∇ui∇vi dx+ (λ+ µ)

∫

Ω

div u div v dx,

l(v) =

∫

ΓN∪ΓS

h · v ds.

Let R be the space of rigid body movements, e.g. for d = 2 we have R = r ∈ H1(Ω) :r = (ω1 + ω3x2, ω2 − ω3x1);ω1, ω2, ω3 ∈ IR. Then there holds the following theorem.

Theorem 1 [13, Theorem 6.1] The variational inequality (2) has a unique solution ifand only if ΓD 6= ∅ or l(v) < 0 ∀v ∈ R ∩ K\0.

3

M. Maischak, E. P. Stephan

In the following we will reformulate (1) as a variational inequality involving only bound-ary integral operators on Γ.

First, we introduce the relevant boundary integral operators and review their mappingproperties. With the fundamental solution for the Laplacian,

G(x, y) =

− 1ω2

log |x− y| if d = 2,1

ωd|x− y|2−d if d ≥ 3

,

with ω2 = 2π, ωd = 4πd/2

Γ( d−2

2), and the fundamental solution for the Lame-equation

S(x, y) =λ+ 3µ

µ(λ+ 2µ)

G(x, y)I +λ+ µ

ωd(λ+ 3µ)

(x− y)(x− y)t

|x− y|d

we define the operators of the single layer potential V , the double layer potential K, itsformal adjoint K ′, and the hypersingular integral operator W for x ∈ Γ, ϕ,u ∈ [C∞(Γ)]d

as follows:

Vϕ(x) :=

∫

Γ

S(x, y)ϕ(y) dsy, Ku(x) :=

∫

Γ

TyS(x, y)u(y) dsy,

K ′ϕ(x) := Tx

∫

Γ

S(x, y)ϕ(y) dsy, Wu(x) := −Tx

∫

Γ

TyS(x, y)u(y) dsy.

For a distribution w on Γ we define (if possible) Vw, Kw, etc. by approximating w withsmooth functions. For |σ| < 1/2 the mappings

V : H−1/2+σ(Γ) → H1/2+σ(Γ), K : H1/2+σ(Γ) → H1/2+σ(Γ),

K ′ : H−1/2+σ(Γ) → H−1/2+σ(Γ), W : H1/2+σ(Γ) → H−1/2+σ(Γ),

are linear and continuous [6, 7]. W : H1/2(Γ)/R → H−1/2(Γ) is strongly positive definiteand V : H−1/2(Γ) → H1/2(Γ) is strongly positive definite, if diam(Ω) is small enough.

Then, an equivalent formulation to (1) is given by problem (L):Find (u,ϕ) ∈ KΓ × H−1/2(Γ) such that

〈u,W (v − u)〉 + 〈ϕ, (I +K)(v − u)〉 ≥ 2l(v − u)〈ψ, Vϕ〉 − 〈ψ, (I +K)u〉 = 0 ∀(v,ψ) ∈ KΓ × H−1/2(Γ) (3)

whereKΓ := v ∈ H1/2(Γ) : v|ΓD

= g|ΓD,v · n|ΓS

≤ gn|ΓS. (4)

We can write equivalently the system (L) with the coercive and non-symmetric bilinearform

B(u,ϕ;v,ψ) := 〈Wu,v〉 + 〈(I +K)tϕ,v〉 + 〈Vϕ,ψ〉 − 〈(I +K)u,ψ〉

4

M. Maischak, E. P. Stephan

as: Find (u,ϕ) ∈ KΓ × H−1/2(Γ) such that

B(u,ϕ;v − u,ψ) ≥ 2l(v − u) ∀(v,ψ) ∈ KΓ × H−1/2(Γ) (5)

On the other hand, eliminating ϕ in (L) leads to the equivalent problem (S):Find u ∈ KΓ such that

〈Su,v − u〉 ≥ l(v − u) ∀v ∈ KΓ. (6)

with the symmetric and positive definite (on H1/2(Γ)/R) Poincare-Steklov operator forthe interior problem

S :=1

2(W + (K ′ + I)V −1(K + I)) : H1/2(Γ) → H−1/2(Γ). (7)

For existence and uniqueness of the solution of problems (S) and (L), respectively, see [10].

2.2 Numerical approximation

Let ωh, γh denote two not necessarily identical regular partitions of Γ, such that allcorners of Γ and all “end points” ΓS ∩ ΓN , ΓN ∩ ΓD, ΓD ∩ ΓS are nodes of ωh, γh. Oftenwe use ωh = γh, but this is only necessary when we are approximating the boundary.

Let ~p = (pe)e∈ωh, ~q = (qe)e∈γh

be degree vectors which associate each element of ωh orγh with a polynomial degree pe ≥ 1 or qe ≥ 0.

On the interval [−1, 1] we choose N + 1 Gauss-Lobatto points, i.e. the points ξN+1j ,

0 ≤ j ≤ N , that are the zeros of (1−ξ2)L′N(ξ), where LN denotes the Legendre polynomial

of degree N . It is known (cf. [3, Prop. 2.2, (2.3)]) that there exist positive weight factors

%N+1j := 1

N(N+1)L2N (ξN+1

j )such that ∀φ ∈ P2N−1([−1, 1]) :

∑Nj=0 φ(ξj)%

N+1j =

∫ 1

−1φ(ξ) dξ.

N = 1 • •−1 1

N = 2 • • •

N = 3 • • • •

N = 4 • • • • •

N = 5 • • • • • •

Figure 1: Gauss-Lobatto points

By an affine transformation we define the set of Gauss-Lobatto points Ge,hp on eachelement e of the partition ωh of Γ, corresponding to the polynomial degree pe and setGhp :=

⋃

e∈ωhGe,hp.

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M. Maischak, E. P. Stephan

Figure 2: Lagrange polynomials, N = 5

On the partition ωh of Γ we introduce σhp as the space of continuous polynomials withσhp|e ⊆ [Ppe(e)]

d, ∀e ∈ ωh. A suitable basis of σhp is given by the Lagrange interpolationpolynomials on the set of Gauss-Lobatto points of each element, see Figure 2, becauselater we must evaluate the values of the solution on Ghp. On the partition γh of Γ weintroduce τhp as the space of piecewise polynomials with τhp|e ⊆ [Pqe(e)]

d, ∀e ∈ γh; here asuitable basis is given by the Legendre polynomials.

Thus, our test and trial spaces are defined as

σhp := vhp ∈ [C0(Γ; IR)]d : vhp|e ∈ [Ppe(e)]d,∀e ∈ ωh, (8)

τhp := ψhp ∈ [L2(Γ; IR)]d : ψhp|e ∈ [Pqe(e)]d,∀e ∈ γh. (9)

Now, we chose

KΓhp := v ∈ σhp | ∀x ∈ Ghp ∩ ΓS : v(x) · n ≤ gn(x) and

∀x ∈ Ghp ∩ ΓD : v(x) = g(x). (10)

which is a convex, closed subset of σhp. Note that KΓhp 6⊆ KΓ for p ≥ 2 (as Figure 2

illustrates) or for non-concave contact function gn.Within the above, the hp-version of (L) is given by the discrete problem (Lhp):Find (uhp,ϕhp) ∈ KΓ

hp × τhp such that

B(uhp,ϕhp;v − uhp,ψ) ≥ 2l(v − uhp) ∀(v,ψ) ∈ KΓhp × τhp. (11)

Via the canonical imbeddings

jhp : σhp → H1/2(Γ), khp : τhp → H−1/2(Γ)

and their duals j∗hp, and k∗hp the discrete Poincare-Steklov operator Shp : σhp → σ∗

hp

Shp :=1

2(j∗hpWjhp + j∗hp(I +K ′)khp(k

∗

hpV khp)−1k∗hp(I +K)jhp) (12)

6

M. Maischak, E. P. Stephan

is well defined (see [4]). With (12) the discrete problem (Shp) for the general hp-version(6) reads: Find uhp ∈ KΓ

hp such that

〈Shpuhp,vhp − uhp〉 ≥ l(vhp − uhp) ∀vhp ∈ KΓhp. (13)

Theorem 2 Let (ωh, γh)h∈I be a family of quasi-uniform meshes, such that h := max|e|, e ∈ωh or e ∈ γh, where I ⊆ (0,∞) with 0 ∈ I. Let ~p = (pe)e∈ωh

, such that pe = p for alle ∈ ωh, and let ~q = (qe)e∈γh

, such that qe = p− 1 for all e ∈ γh.Let the solutions (u,ϕ) of (5) and (uhp,ϕhp) of (11) exist uniquely. Suppose that for

the polygonal domain Ω, there are only a finite number of ”end points” ΓS ∩ ΓD, ΓD ∩ ΓN ,ΓN ∩ ΓS. Then there holds

limp→∞

‖(u − uhp,ϕ−ϕhp)‖H1/2(Γ)×H−1/2(Γ) = 0, if h fixed,

andlimh→0

‖(u − uhp,ϕ−ϕhp)‖H1/2(Γ)×H−1/2(Γ) = 0, if p fixed.

Proof. Analogously to the proof of Theorem 1 in [15] for the Laplacian. ut

Remark 1 For a proof in case of the h-version see [10], where Newton-Cotes formulaswere used as numerical quadrature rules; here Gauss-Lobatto quadrature is used instead.

With Theorem 2 the convergence of the quasi-uniform hp-version is proved. If we assumehigher regularity of the solution (u,ϕ) of (L) and of the contact function g in (1), i.e.u,g ∈ H3/2(Γ), ϕ ∈ H1/2(Γ) we obtain the following a priori error estimate which proposesa convergence rate of O(h1/4p−1/4). Note, that this assumption is quite natural since asmooth contact function g implies a smooth solution (cf. [14]).

Theorem 3 Let (ωh, γh)h be a family of quasi-uniform meshes, such that h := max|e|, e ∈ωh or e ∈ γh and let ~p = (pe)e∈ωh

, pe = p, ~q = (qe)e∈γh, qe = p− 1.

Let (u,ϕ) ∈ KΓ×H−1/2(Γ) be the solution of problem (L) and let (uhp,ϕhp) ∈ KΓhp×τhp

be the solution of problem (Lhp) and assume u,g ∈ H3/2(Γ) and h∗ −Su ∈ [L2(Γ)]d, withh∗|ΓN∪ΓS

= h and h∗|ΓD= 0.

Then there holds

‖(u − uhp,ϕ−ϕhp)‖H1/2(Γ)×H−1/2(Γ) ≤ C h1/4p−1/4‖u‖H3/2(Γ)

for h→ 0 or p→ ∞.

Proof. Analogously to the proof of Theorem 3 in [15] for the Laplacian. ut

7

M. Maischak, E. P. Stephan

3 A POSTERIORI ERROR ESTIMATES

3.1 The adaptive hp-version

Let the finite dimensional spaces σhp and τhp be defined as in (8) and (9), respectively.We extend the spaces σhp and τhp by bubble functions given on each element in ωh andγh.

Each element e ∈ ωh is associated with a polynomial degree pe and the affine mappingFe with Fe(ξ) = x(ξ) ∈ e for ξ ∈ [−1, 1].

With the Legendre polynomial Lj of degree j we set

ψ0(ξ) :=1 − ξ

2, ψ1(ξ) :=

1 − ξ

2, ψj(ξ) :=

√

2j − 1

2

∫ ξ

−1

Lj−1(t) dt, 2 ≤ j,

and take the space σe to be

σe = span[ψe,pe+1]2, ψe,j := ψj(F

−1e (x)). (14)

On the other hand, each element e ∈ γh is associated with a polynomial degree qe. Setting

φ0(t) :=1

2, φj(t) :=

√

2j + 1

2Lj(t) 1 ≤ j,

we take the space τe to be

τe = span[φe,qe+1]2, φe,j(x) := φj(F

−1e (x)). (15)

Hence we obtain the subspace decompositions

σh,p+1 := σhp ⊕ Lp, Lp :=⊕

e∈ωh

σe, (16)

τh,p+1 := τhp ⊕ λp, λp :=⊕

e∈γh

τe. (17)

The polynomial degree vector associated with σh,p+1 is (pe + 1)e∈ωhand the polynomial

degree vector associated with τh,p+1 is (qe + 1)e∈γh.

Let Php : σh,p+1 → σhp, Php,e : σh,p+1 → σe, php : τh,p+1 → τhp, php,e : τh,p+1 → τebe the Galerkin projections with respect to the bilinear forms 〈W ·, ·〉 and 〈V ·, ·〉. For allu ∈ σh,p+1 we define Php and Php,e by

〈WPhpu,v〉 = 〈Wu,v〉 ∀v ∈ σhp (18)

〈WPhp,eu,v〉 = 〈Wu,v〉 ∀v ∈ σe, e ∈ ωh (19)

and for all ϕ ∈ τh,p+1 we define php and php,e by

〈V phpϕ,φ〉 = 〈Vϕ,φ〉 ∀φ ∈ τhp (20)

〈V php,eϕ,φ〉 = 〈Vϕ,φ〉 ∀φ ∈ τe, e ∈ γh. (21)

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M. Maischak, E. P. Stephan

Finally, we define the two-level Schwarz operators

Pσ := Php +∑

e∈ωh

Php,e and pτ := php +∑

e∈γh

php,e. (22)

The following lemma states that the operators Pσ and pτ have condition numbersdepending only logarithmically on pmax and are independent of h.

Lemma 1 There exist constants c1, c2 > 0 and C1, C2 > 0 independent of h and p =(pe)e∈ωh

, q = (qe)e∈γhsuch that

C1(1 + log pmax)−2‖v‖2

W ≤ ‖Phpv‖2W +

∑

e∈ωh

‖Php,ev‖2W ≤ C2‖v‖

2W ∀v ∈ σh,p+1,

c1(1 + log qmax)−2‖ψ‖2

V ≤ ‖phpψ‖2V +

∑

e∈γh

‖php,eψ‖2V ≤ c2‖ψ‖

2V ∀ψ ∈ τh,p+1

with pmax := maxpe, e ∈ ωh and qmax := maxqe, e ∈ γh

Proof. Analogously to the proof of Lemma 4 in [15], based on [12, Lemma 3.2 andTheorem 4.1]. utFor all (u,ϕ) ∈ H1/2(Γ)/R×H−1/2(Γ) we define the norm ‖(u,ϕ)‖H := (‖u‖2

W +‖ϕ‖2V )1/2,

which is equivalent to (‖u‖2H1/2(Γ)/IR

+ ‖ϕ‖2H−1/2(Γ)

)1/2. The norm ‖(u,ϕ)‖H is generated

by the bilinear forma(u,ϕ;v,ψ) := 〈Wu,v〉 + 〈Vϕ,ψ〉.

Let (u,ϕ) be the solution of the variational inequality (5) and let (uhp,ϕhp) ∈ σhp×τhp,(uh,p+1,ϕh,p+1) ∈ σh,p+1 × τh,p+1 be the solutions of the corresponding discrete problems.

As for finite element problems (see, e.g. [2], [1]) we make the saturation assumption:There exists a parameter 0 ≤ κ < 1 such that for all discrete spaces holds:

‖(u − uh,p+1,ϕ−ϕh,p+1)‖H ≤ κ‖(u − uhp,ϕ−ϕhp)‖H. (23)

From the saturation condition and the triangle inequality we obtain immediately:

(1 − κ)‖(u − uhp,ϕ−ϕhp)‖H ≤ ‖(uh,p+1 − uhp,ϕh,p+1 −ϕhp‖H

≤ (1 + κ)‖(u − uhp,ϕ−ϕhp)‖H.

Remark 2 We have u ∈ KΓ ⊂ H1/2(Γ) and uh,p+1 ∈ KΓh,p+1 and uhp ∈ KΓ

hp. Therehold the continuous Dirichlet condition uΓD

= g and the discrete Dirichlet conditionuhp(x) = g(x)∀x ∈ Ghp, etc. We also have that the ”endpoints” ΓN ∩ ΓD and ΓS ∩ ΓD arenodes of the partition ωh. Due to the use of Gauss-Lobatto nodes in the construction ofGhp we have that the endpoints are in Ghp, therefore there are several points where u anduhp are identical, i.e. the difference vanishes. The same argument applies to the differenceuh,p+1 −uhp due to using the same mesh ωh. Therefore the free constants in ‖ · ‖W is fixedand ‖ · ‖W is indeed a norm.

9

M. Maischak, E. P. Stephan

Theorem 4 We assume that (23) holds. Then, there are constants ζ1, ζ2 > 0 such that

ζ1 · ηhp ≤ ‖(u − uhp,ϕ−ϕhp)‖H ≤ ζ2(1 + log maxpmax, qmax) · ηhp (24)

where

ηhp :=(

Θ2hp + η2

u,hp + η2ϕ,hp

)1/2, Θhp := ‖Phpeh,p+1‖W , (25)

ηu,hp :=(

∑

e∈ωh

Θ2hp,e

)1/2

, Θhp,e := ‖Peeh,p+1‖W , (26)

ηϕ,hp :=(

∑

e∈γh

θ2hp,e

)1/2

, θ2hp,e := Rt

e,qe+1V−1ee Re,qe+1, (27)

Re,qe+1 := B(uhp,ϕhp; 0, [ψe,qe+1]2), Vee := 〈V [ψe,qe+1]

2, [ψe,qe+1]2〉, (28)

with the auxiliary problem:eh,p+1 ∈ Kuhp

:= v − uhp |v ∈ KΓh,p+1 = KΓ

h,p+1 − uhp solves

W (eh,p+1,v − eh,p+1) ≥ 2l(v − eh,p+1) −B(uhp,ϕhp;v − eh,p+1, 0) ∀v ∈ Kuhp. (29)

Proof. Analogously to the proof of Theorem 6 in [15]. utNow, we formulate an hp-adaptive three-step algorithm which uses the error indicatorsfrom Theorem 4 to generate a sequence of locally refined meshes. We neglect the factor(1 + log maxpmax, qmax)

2 and estimate the global error by

ηhp :=

(

Θ2hp +

∑

e∈ωh

Θ2hp,e +

∑

e∈γh

θ2hp,e

)1/2

. (30)

Algorithm 1 Let the parameters ε > 0, 0 < δ1 < δ2 < 1 and initial subdivisions ωh0, γh0

of Γ and initial polynomial degree vectors p0 and q0 be given. With σ(hp)0 and τ(hp)0 wedenote the initial test and trial spaces. For k = 0, 1, 2, . . .

1. Compute the solution (u(hp)k,ϕ(hp)k

) ∈ KΓ(hp)k

× τhp of (Lhp).

2. Solve the defect problem (29) for eh,p+1.

3. Compute the error indicator Θhp.

4. For each e ∈ ωhkcompute the local error indicator Θhp,e.

5. For each e ∈ γhkcompute the local error indicator θhp,e.

6. Compute the global error estimate ηhp. Stop if ηhp < ε.

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M. Maischak, E. P. Stephan

7. Determine Θhp,e′isuch that card(e ∈ ωhk

: Θhp,e < Θhp,e′i) = δi card(ωhk

) fori = 1, 2 and determine θhp,e′′i

such that card(e ∈ γhk: θhp,e < θhp,e′′i

) = δi card(γhk)

for i = 1, 2.

8. Adaption step.

h–adaptive algorithm: If Θhp,e ≥ Θhp,e′1, split the element e ∈ ωhk

into two el-ements of the same size and inherit the polynomial degree. If θhp,e ≥ θhp,e′′

1,

split the element e ∈ γhkinto two elements of the same size and inherit the

polynomial degree.

p–adaptive algorithm: If Θhp,e ≥ Θhp,e′1, increase the polynomial degree on e ∈

ωhkby 1. If θhp,e ≥ θhp,e′′

1, increase the polynomial degree on e ∈ γhk

by 1.

hp–adaptive algorithm: If Θhp,e′2≥ Θhp,e ≥ Θhp,e′

1, increase the polynomial de-

gree on e ∈ ωhkby 1. If Θhp,e > Θhp,e′

2split the element e ∈ ωhk

into twoelements of the same size and inherit the polynomial degree. If Θhp,e < Θhp,e′

1

do nothing.

If θhp,e′′2≥ θhp,e ≥ θhp,e′′

1, increase the polynomial degree on e ∈ γhk

by 1. Ifθhp,e > θhp,e′′

2split the element e ∈ γhk

into two elements of the same size andinherit the polynomial degree. If θhp,e < θhp,e′′

1do nothing.

This defines new meshes ωhk+1, γhk+1

and enlarged spaces σ(hp)k+1⊃ σ(hp)k

, τ(hp)k+1⊃

τ(hp)k. Goto 1.

The above refinement strategy ensures that at least 100 · δ1 percent of the elements arerefined.

Remark 3 The algorithm 1 can be simplified, if we demand that there always holds ωh =γh. We still compute both sets of error indicators, but both refinement criteria will beapplied simultaneously to ωh.

3.2 The adaptive h-version

For the completeness of presentation we also consider the pure h-version where a dif-ferent subspace enrichment is used.

Let the finite dimensional spaces σh and τh be defined as σhp in (8) with p = 1 andτhp in (9) with q = 0, respectively. Defining nh := dimσh and nh/2 := dimσh/2 thenn := nh/2 − nh is the number of new nodes in the refined mesh ωh/2, which is given bybisecting all elements in ωh. The new nodes are denoted by νe, e ∈ ωh. For each of thesenew nodes νe we define the piecewise linear basis functions be by

be(νe) = 1, be(∂e) = 0, e ∈ ωh. (31)

With σe = span[be]2 we denote the two-dimensional space spanned by be.

11

M. Maischak, E. P. Stephan

¡¡¡¡

¡¡¡¡

¡¡¡¡

¡¡¡¡

¡¡¡¡

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

¢¢¢¢

¢¢¢¢

¢¢¢¢

¢¢¢¢

¢¢¢¢

¢¢¢¢

AA

AA

AA

AA

AA

AA

AA

AA

AA

AA

AA

AA

be

Figure 3: 2-level decomposition of ωh/2

Next, we consider bisections of all elements e ∈ γh, i.e. e = e1 ∪ e2, |e1| = |e2|, e ∈ γh.With each element e ∈ γh we associate the Haar basis function βe defined as

βe(x) :=

1 if x ∈ e1

−1 if x ∈ e2

0 if x ∈ Γ\e(32)

and the two-dimensional space τe = span[βe]2. Hence we obtain the following subspace

βe

Figure 4: 2-level decomposition of τh/2

decompositions

σh/2 = σh ⊕ Lh, Lh :=⊕

e∈ωh

σe, (33)

τh/2 = τh ⊕ λh, λh :=⊕

e∈γh

τe. (34)

Let Ph : σh/2 → σh, Pe : σh/2 → σe, ph : τh/2 → τh and pe : τh/2 → τe be the Galerkinprojections with respect to the bilinear forms 〈W ·, ·〉 and 〈V ·, ·〉.

For all u ∈ σh/2 we define Ph and Pe by

〈WPhu,v〉 = 〈Wu,v〉 ∀v ∈ σh, (35)

〈WPeu,v〉 = 〈Wu,v〉 ∀v ∈ σe (36)

and for all ϕ ∈ τh/2 we define ph and pe by

〈V phϕ,φ〉 = 〈Vϕ,φ〉 ∀φ ∈ τh,

〈V peϕ,φ〉 = 〈Vϕ,φ〉 ∀φ ∈ τe.

12

M. Maischak, E. P. Stephan

Finally, we introduce the two-level Schwarz operators

P := Ph +∑

e∈ωh

Pe and p := ph +∑

e∈γh

pe. (37)

The following lemma states that the operators P and p have bounded condition numbers.

Lemma 2 [17, 18] There are constants C1, C2 > 0 and c1, c2 > 0 independent of h suchthat

C1‖v‖2W ≤ ‖Phv‖

2W +

∑

e∈ωh

‖Pev‖2W ≤ C2‖v‖

2W ∀v ∈ σh/2

c1‖ψ‖2V ≤ ‖phψ‖

2V +

∑

e∈γh

‖peψ‖2V ≤ c2‖ψ‖

2W ∀ψ ∈ τh/2.

As for finite elements (see, e.g. [1],[2]) we make the saturation assumption:Let (u,ϕ), (uh,ϕh) be the solutions of (5) and (11) and let (uh/2,ϕh/2) be the solution

of (11), where every mesh element has been split into two equal parts. Let there exist aconstant 0 ≤ κ < 1 and a parameter h0 > 0 such that for all meshes with mesh parameterh ≤ h0 (largest element size) there holds

‖(u − uh/2,ϕ−ϕh/2)‖H ≤ κ‖(u − uh,ϕ−ϕh)‖H. (38)

Remark 4 The arguments from Remark 2 apply here, too. Therefore we can use ‖ · ‖W

to describe the error.

From the saturation condition and the triangle inequality we obtain immediately that forall h ≤ h0:

(1 − κ)‖(u − uh,ϕ−ϕh)‖H ≤ ‖(uh/2 − uh,ϕh/2 −ϕh)‖H

≤ (1 + κ)‖(u − uh,ϕ−ϕh)‖H.

Next, we call σh/2 (τh/2) the refinement of all elements in σh (τh) and KΓh/2 the correspond-

ing convex subset of σh/2. Then the above Theorem 4 immediately gives the followingresult. Its proof follows analogously to the proof of Theorem 4 by replacing the subscriptsh, p+ 1 with h/2 and hp with h.

Proposition 1 We assume (38). Let (uh,ϕh) be the solution of (11) on KΓh × τh. Then

there are constants ζ1, ζ2 > 0 such that

ζ1 · ηh ≤ ‖(u − uh,ϕ−ϕh)‖H ≤ ζ2 · ηh (39)

13

M. Maischak, E. P. Stephan

where

ηh :=(

Θ2h + η2

D,h + η2N,h

)1/2, Θh := ‖Pheh/2‖W , (40)

ηD,h :=(

∑

e∈ωh

Θ2e,h

)1/2

, Θe,h := ‖Peeh/2‖W , (41)

ηN,h :=(

∑

e∈γh

θ2e,h

)1/2

, θ2e,h := Rt

e,h/2V−1ee Re,h/2, (42)

Re,h/2 := B(uh,ϕh; 0, [βe,h/2]2), Vee := 〈V [βe,h/2]

2, [βe,h/2]2〉, (43)

with the auxiliary problem:eh/2 ∈ Kuh

:= v − uh |v ∈ KΓh/2 = KΓ

h/2 − uh solves

W (eh/2,v − eh/2) ≥ 2l(v − eh/2) −B(uh,ϕh;v − eh/2, 0) ∀v ∈ Kuh. (44)

4 NUMERICAL EXAMPLES

Here we investigate the deformation of a elastic bar with an elasticity module E = 2000and a Poisson number ν = 0.3. The rigid obstacle partly supports the bar, see Figure 5,i.e. we have the obstacle function g ≡ 0. We apply a constant load T (u) = (0,−160)from above on ΓN,1. The remaining part of the boundary is load free. The exact value

????????????

ΓS

ΓN,1

ΓN,2 ΓN,2

¡¡¡¡¡¡¡¡¡¡¡¡¡

Figure 5: Model geometry

of the potential J(u) := 12〈Su,u〉 − l(u) = −35.413 is obtained by extrapolation of the

potential values of the uniform h-version.In the adaptive computations we use the merged version of the adaptive algorithm 1, i.e.

we choose ωh = γh. We compute the Dirichlet and Neumann indicators independently andmerge the refinement decisions. Here we have used the steering parameters δ = δ1 = 0.80and δ2 = 0.90, i.e. 20% of all elements will be refined and in case of a hp-version, thepolynomial degrees of 10% of all elements will be increased, whereas the the 10% of allelements with the largest indicator values will be bisected.

In Figure 6 we present the error in the energy norm (estimated by the error in thepotential value) for the uniform h-version, for the adaptive h-version steered by the hat-indicator, the adaptive h-version steered by the bubble-indicator and the adaptive hp-version steered by the bubble-indicator The errors, convergence rates α and values of

14

M. Maischak, E. P. Stephan

0.01

0.1

1

100 1000

erro

r in

ene

rgy

norm

degrees of freedom

h-version (uniform)h-version (hat-adaptive)

h-version (bubble-adaptive)hp-version (bubble-adaptive)

Figure 6: Convergence of the Lame Contact Problem

dimσk JS(uN) ‖u − uk‖S Θ ηu,h ηϕ,h α32 -34.3631304348 1.0280416 0.23266 2.4493 2.200264 -34.7135180401 0.8405248 0.15341 1.7724 1.5229 0.291

128 -35.0095916450 0.6406312 0.09086 1.2825 1.0848 0.392256 -35.1958530996 0.4734415 0.04936 0.9206 0.7683 0.436512 -35.3002571848 0.3460387 0.02571 0.6560 0.5436 0.452

1024 -35.3554291310 0.2541080 0.01311 0.4656 0.3845 0.445

Table 1: Error and hat-Haar-indicators for the uniform h-version (Lame)

the h-indicators ηu,h, ηϕ,h for the uniform h-version are presented in Table 1. Errors,convergence rates and values of the bubble-indicators ηu,hp, ηϕ,hp for the adaptive h-version are presented in Table 2, the corresponding meshes are shown in Figure 7. Themeshes and polynomial degrees for the adaptive hp-version are shown in Figure 8. It canbe observed that the convergence rates for the adaptive versions are significantly higherthan for the uniform h-version. The error indicators Θ, ηu,hp and ηϕ,hp tend to zero for theuniform and the adaptive version. It has to be emphasized that the coarse grid indicatorΘ, which can not be localized, is always an order of magnitude smaller than the otherindicators which are used to steer the adaptive refinement. Figure 8 shows that our three-step algorithm 1 creates not only appropriate meshes but also increases the polynomialdegrees as expected due to the varying smoothness of the exact solution.

15

M. Maischak, E. P. Stephan

k dim σk JS(uk) ‖u − uk‖S Θ ηu,hp ηϕ,hp α0 32 -34.3631304348 1.0246314 2.3785 1.3758 2.42791 44 -34.5922098253 0.9059747 1.2402 1.0186 1.7048 0.3862 56 -34.8731082156 0.7347733 0.6692 0.7577 1.2359 0.8693 72 -35.1141090700 0.5467092 0.3990 0.5723 0.8908 1.1764 92 -35.2792451745 0.3657251 0.1803 0.3823 0.6138 1.6405 116 -35.3390467499 0.2719435 0.0940 0.2716 0.4368 1.2786 148 -35.3733112175 0.1992204 0.0521 0.1964 0.3131 1.2777 184 -35.3938810245 0.1382714 0.0275 0.1375 0.2200 1.6778 238 -35.4033732813 0.0981158 0.0108 0.0968 0.1538 1.3339 316 -35.4083096430 0.0684862 0.0052 0.0676 0.1083 1.268

10 410 -35.4104836903 0.0501628 0.0026 0.0479 0.0764 1.196

Table 2: Error and bubble-indicators for the adaptive h-version (Lame)

Figure 7: h-adaptive generated meshes for Lame-BEM (bubble-indicator)

16

M. Maischak, E. P. Stephan

1 1 1 1 1 1

1

1

1 1 1 1 1 1

1

1

2 1 1 2 2 1 1 2

1

1

1 1 1 1 1 1

1

1

2 2 1 1 3 3 1 1 2 2

1

1

1 1 1 1 1 1

1

1

2 2 2 1 1 3 3 1 1 2 2 2

1

1

1 1 1 1 1 1

1

1

2 2 2 1 1 1 4 4 1 1 2 2 2 2

1

1

1 1 1 1 1 1

1

1

2 2 2 1 1 1 1 5 4 4 1 1 2 2 2 2 2

1

1

1 1 2 1 1 1

1

1

2 2 2 1 1 2 1 1 5 5 4 5 1 1 1 2 2 2 2 2

1

1

1 2 3 2 2 1

1

1

Figure 8: hp-adaptive generated meshes for Lame-BEM (bubble-indicator)

REFERENCES

[1] R. Bank and R. Smith, A posteriori error estimates based on hierarchical bases,SIAM J. Numer. Anal., 30 (1993), pp. 921–935.

[2] R. E. Bank, Hierarchical bases and the finite element method, Acta Numerica,(1996), pp. 1–43.

[3] C. Bernardi and Y. Maday, Polynomial interpolation results in Sobolev spaces,J. Comput. Appl. Math., 43 (1992), pp. 53–80.

[4] C. Carstensen, Interface problem in holonomic elastoplasticity, Math. Methods.Appl. Sci., 16 (1993), pp. 819–835.

[5] C. Carstensen and J. Gwinner, FEM and BEM coupling for a nonlinear trans-mission problem with Signorini contact, SIAM J. Numer. Anal., 34 (1997), pp. 1845–1864.

[6] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary re-sults, SIAM J. Math. Anal., 19(3) (1988), pp. 613–626.

[7] M. Costabel and E. P. Stephan, A direct boundary integral equation method fortransmission problems, J. Math. Anal. Appl., 106 (1985), pp. 367–413.

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M. Maischak, E. P. Stephan

[8] R. S. Falk, Error estimates for the approximation of a class of variational inequal-ities, Math. Comp., 28 (1974), pp. 963–971.

[9] R. Glowinski, Numerical methods for nonlinear variational problems, Springer Se-ries in Computational Physics, Springer-Verlag, Berlin, 1984.

[10] J. Gwinner and E. P. Stephan, A boundary element procedure for contact prob-lems in linear elastostatics, RAIRO Math.Mod.Num.Anal., 27 (1993), pp. 457–480.

[11] H. Han, A direct boundary element method for Signorini problems, Math. Comp.,55 (1990), pp. 115–128.

[12] N. Heuer, M. Mellado, and E. P. Stephan, hp-adaptive two-level methods forboundary integral equations on curves, Computing, 67 (2001), pp. 305–334.

[13] N. Kikuchi and J. Oden, Contact Problems in Elasticity: a Study of VariationalInequalities and Finite Element Methods, SIAM, Philadelphia, 1988.

[14] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequal-ities and their applications, Academic Press, 1980.

[15] M. Maischak and E. P. Stephan, Adaptive hp-versions of bem for Signoriniproblems. to appear in Applied Numerical Mathematics, 2004.

[16] , A fem-bem coupling method for a nonlinear transmission problem modellingCoulomb Friction Contact. to appear in Computer Methods in Applied Mechanicsand Engineering, 2004.

[17] P. Mund and E. P. Stephan, An adaptive two-level method for the coupling ofnonlinear fem-bem equations, SIAM J.Numer.Analysis, 36 (1999), pp. 1001–1021.

[18] T. Tran and E. P. Stephan, Additive Schwarz method for the h-version boundaryelement method, Applicable Analysis, 60 (1996), pp. 63–84.

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