Iteratively solving a kind of Signorini transmission problem in a unbounded domain

Vol. 39, N 4, 2005, pp. 715–726 DOI: 10.1051/m2an:2005031

ITERATIVELY SOLVING A KIND OF SIGNORINI TRANSMISSION PROBLEM IN A UNBOUNDED DOMAIN

Qiya Hu

and Dehao Yu

Abstract. In this paper, we are concerned with a kind of Signorini transmission problem in a unbounded domain. A variational inequality is derived when discretizing this problem by coupled FEM-BEM. To solve such variational inequality, an iterative method, which can be viewed as a vari- ant of the D-N alternative method, will be introduced. In the iterative method, the finite element part and the boundary element part can be solved independently. It will be shown that the convergence speed of this iteration is independent of the mesh size. Besides, a combination between this method and the steepest descent method is also discussed.

Mathematics Subject Classification. 65N30, 65R20, 73C50.

Received: April 15, 2004. Revised: April 3, 2005.

1. Introduction

In this paper we analyze the following transmission problem inR². Let Ω be a bounded domain with Lipschitz boundary Γ. To describe mixed boundary conditions, let Γ = Γ_t∪Γ_swhere Γ_tand Γ_sare nonempty, disjoint, and open in Γ. Let n denote the unit normal on Γ defined almost everywhere pointing from Ω into Ω_c:=R²\Ω. Assume that¯ f ∈L²(Ω), u₀∈H¹²(Γ) andt₀∈L²(Γ).

As the interior part, we consider the nonlinear partial differential equation

div(p(|∇u|)· ∇u) +f = 0 in Ω, (1.1)

where p : [0,∞) → [0,∞) is a continuous function with t·p(t) being monotonously increasing with t. In the exterior part, we consider the Laplace equation

u= 0 in Ω_c, (1.2)

with the radiation condition (note Lem. 3.6 of [2])

u(x) =a+o(1) (|x| → ∞), (1.3)

Keywords and phrases. Signorini contact, FEM-BEM coupling, variational inequality, D-N alternation, convergence rate.

1Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics, Chinese Academy of Sciences, Beijing 100080, China. [email protected];[email protected]

The work of Q. Hu was supported by the National Natural Science Foundation No. 10371129.

The work of D. Yu was supported by the State Major Key Project for Basic Researches of China G19990328.

c EDP Sciences, SMAI 2005

Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2005031

where a is a real constant. Writing u₁ := u|Ω and u₂ := u|Ωc, the traction on Γ are given by the traces of p(|∇u|)^∂u_∂n¹ and−^∂u_∂n². We consider transmission conditions on Γ_t,

u₁|_Γt =u₂|_Γt+u₀|_Γt and p(|∇u|)∂u₁

∂n|_Γt = ∂u₂

∂n|_Γt+t₀|_Γt, (1.4) and Signorini conditions on Γ_s,

u₁|Γs≤u2|Γs+u₀|Γs, (1.5)

p(|∇u|)∂u₁

∂n|Γs = ∂u₂

∂n|Γs+t₀|Γs ≤0 (1.6)

0 =p(|∇u|)∂u₁

∂n|Γs·(u₂+u₀−u₁)|Γs. (1.7) This problem can be considered as a scalar model of the two-body contact problem between a linear elastic unbounded medium and a deformable body allowing some nonlinear monotone stress strain relationship (refer to [4, 12]). Some similar problems have been considered in [5, 8, 9, 14]. The paper [2] introduced a coupled FEM- BEM variational inequality of the problem (1)–(5), and considered the corresponding approximation problem.

Moreover, an asymptotic error estimate has been derived in that paper.

However, there is no literature to study the numerical method for solving this kind of coupled FEM-BEM variational inequality. In the present paper we introduce an iterative method for solving (1)–(5). This method can be viewed as a variant of the D-N alternative method, which has been applied to solving the Poisson equations in bounded domain and unbounded domain (see [7,19,20]). It will be shown that the solution sequence generated by this iteration converges to the solution of the coupled system given in [2], and the convergence speed is independent of the mesh size. The new method has obvious advantages: (1) the finite element problem and boundary element problem are solved respectively; (2) the memory requirement was decreased.

This paper is organized as follows. In Section 2, we describe the coupled variational problem in suitable way. In Section 3, a basic iteration method is described, and its convergence results are given. In Section 4, we analyze the convergence rate. In Section 5, we discuss a combination between this iterative method and the steepest descent method.

2. A coupled system derived by FEM and BEM

In this section, we describe the variational problems considered by [2]. For convenience, we would like to use the equivalent forms to (24) and (28) in [2].

LetH¹(Ω), H¹²(Γ), H₀¹(Ω) andH⁻¹²(Γ) denote the usual Sobolev spaces (refer to [10]). We define the (non- linear) functionalA:H¹(Ω)×H¹(Ω)→Rand the linear functionalF :H¹(Ω)×H¹²(Γ)→Rby

A(u, v) =

Ωp(|∇u|)(∇u)^t· ∇vdx, u, v∈H¹(Ω) and

F(u, ϕ) =

Ωf·udx+

Γt₀·ϕds, u∈H¹(Ω), ϕ∈H¹²(Γ).

To define a symmetric and positive definite boundary operator, letγ(x, y) be the fundamental solution for the Laplacian,i.e.

γ(x, y) =− 1

2πlog|x−y|, x, y∈R².

Set

V φ(z) = 2

Γφ(x)·γ(z, x)ds_x (V :H⁻¹²(Γ)→H¹²(Γ)), Kφ(z) = 2

Γφ(x)· ∂

∂n_xγ(z, x)ds_x (K:H¹²(Γ)→H¹²(Γ)), Kφ(z) = 2

Γφ(x)· ∂

∂n_zγ(z, x)ds_x (K :H⁻¹²(Γ)→H⁻¹²(Γ)), W φ(z) = ∂

∂n_zKφ(z) (W :H¹²(Γ)→H⁻¹²(Γ)).

It is well known thatV :H⁻¹²(Γ)→H¹²(Γ) is symmetric and positive definite, andW :H¹²(Γ)→H⁻¹²(Γ) is sym- metric and positive semi-definite under suitable assumption (see [2,15]). Thus, the operatorS:H¹²(Γ)→H⁻¹²(Γ) defined by

S=1

2(W + (K−I)V⁻¹(K−I))

is also symmetric and positive definite. Note that the operator S is just the Dirichlet−N eumann map or Steklov−P oincareoperator (refer to [2, 18]).

Let·,· be theL²inner product on Γ. Set

L(u, ϕ) =F(u, ϕ) +Sa, ϕ , u∈H¹(Ω), ϕ∈H¹²(Γ).

For the functionp(t), we make the natural assumption: p₁≤p(t)≤p₂ andα≤p(t) +tp(t)≤β for constants p₁, p₂, α, β >0 (refer to [2, 11]). Below we fix the constantain some way to force uniqueness (see [2]).

LetE=H¹(Ω)×H¹²(Γ). Set

D={(u, ϕ)∈E:u=ϕ+u₀on Γ_tandu≤ϕ+u₀ a.e.on Γ_s}. The variational form of the problem (1)–(5) is: to find (u, ϕ)∈D such that

A(u, v−u) +Sϕ, ϕ−ψ ≥L(v−u, ϕ−ψ),

∀(v, ψ)∈D. (2.1)

If (u, ϕ) is the solution of (2.1), then we can obtain the solution of (1)–(5) byu₁=uand u₂(z) = 1

2(Kϕ+V S(ϕ−a))(z) +a, z∈Ω_c.

Without loss of generality, we assume that the domain Ω is a polygon. Let Ω be divided into some regular quasi-uniform triangles with diameter h. Then let H_h¹ denote the finite-dimensional space of continuous and piecewise linear function relating to this partition. The mesh in Ω leads to a mesh of the boundary, so that we set

H_h¹² ={u|_Γ:u∈H_h¹}.

Moreover, we may consider H_h⁻¹² as the piecewise constant trial functions. For simplicity of exposition, we assume thatu₀∈H_h¹². LetE_h=H_h¹×H_h¹², and set

D_h={(uh, ϕ_h)∈E_h:u_h=ϕ_h+u₀on Γ_t andu_h≤ϕ_h+u₀ a.e.on Γ_s}.

Let

i_h:H_h¹² →H¹²(Γ)

and

j_h:H_h⁻¹² →H⁻¹²(Γ) denote the canonical imbedding with its duali^∗_h andj_h^∗

i^∗_h:H⁻¹²(Γ)→(H_h¹²)^∗ and

j^∗_h:H¹²(Γ) →(H_h⁻¹²)^∗. Set

S_h=1

2(i^∗_hW i_h+i^∗_h(K−I)j_h(j_h^∗V j_h)⁻¹j^∗_h(K−I)i_h).

The discrete problem associated with (2.1) is: to find (u_h, ϕ_h)∈D_hsuch that A(u_h, v_h−u_h) +S_hϕ_h, ψ_h−ϕ_h ≥L_h(v_h−u_h, ψ_h−ϕ_h),

∀(v_h, ψ_h)∈D_h (2.2)

where

L_h(v, ψ) =F(v, ψ) +S_ha, ψ .

It has been shown in [2] that the discrete problem (2.2) (and (2.1)) has a unique solution (by transforming it to an equivalent form). Moreover, an asymptotic estimate of the errorsu_h−uandϕ_h−ϕwas also derived. If (u_h, ϕ_h) is the solution of (2.2), then we can obtain the approximate solution of (1)–(5) by u_1h=u_h and

u_2h(z) =1

2(Kϕ_h+V S(ϕ_h−a))(z) +a, z∈Ω_c.

Remark 2.1. Since the stiffness matrix of the boundary integral operatorS_h is dense, it is very expensive to solve the variational inequality (2.2) by the existing methods (refer to [6,19]). For this reason, we will introduce an iterative method to solve (2.2) in the next section.

3. A basic iterative method

At first, we describe the algorithm.

For a givenλ∈H¹²(Γ), we define

Hˆ_λ¹={v∈H_h¹:v=λ+u₀ on Γ_tandv≤λ+u₀ a.e. on Γ_s}.

For a givenw∈H_h¹², letw∈H_h¹ denote the zero extension ofw(which equals won Γ, and equals zero at the internal nodes of Ω).

Algorithm 3.1Letϕ⁰_h∈H_h¹² be given. Whenϕⁿ_h ∈H_h¹² have been gotten,ϕⁿ⁺¹_h ∈H_h¹² will be obtained by step 1^◦: to finduⁿ⁺¹_h ∈Hˆ_ϕ¹n

h such that

A(uⁿ⁺¹_h , v_h−uⁿ⁺¹_h )≥(f, v_h−uⁿ⁺¹_h ),∀v_h∈Hˆ_ϕ¹n

h; (3.1)

step 2^◦: to findpⁿ⁺¹_h ∈H_h¹² such that

S_hpⁿ⁺¹_h , ψ_h = (f, ψ_h)−A(uⁿ⁺¹_h , ψ_h) +t₀+Sa, ψ_h , ∀ψ_h∈H_h¹²; (3.2) step 3^◦: set

ϕⁿ⁺¹_h =θpⁿ⁺¹_h + (1−θ)ϕⁿ_h, 0< θ <1. (3.3)

Remark 3.1. In essence, the step 1^◦ and the step 2^◦ can be regarded respectively as solving a Dirichlet (inequality) problem on Ω (by FEM) and as solving a Neumann problem on Ω_c(by BEM). In the implementation of the step 2^◦, it is unnecessary to consider the operatori^∗_h or j_h^∗. Instead, only the Galerkin method is used (refer to [1]).

Remark 3.2. The main merits of this algorithm are: (1) the finite element problem and the boundary element problem are solved independently, the variational inequality (3.1) and the boundary integral equation (3.2) can be solved by the existing methods (see [3, 6]); (2) it has smaller memory requirement than the algorithms to solve globally the variational inequality (2.2).

Now, we give the convergence result of this iteration.

Since S_h : H_h¹² → (H_h¹²)^∗ is a symmetric and positive definite operator, we can define the norm · _Sh = (S_h·,· )¹². Letϕ_h∈H¹² be determined by the variational inequality (2.2), and seteⁿ_h=ϕ_h−ϕⁿ_h. Theorem 3.1. There exists a constant c₀ independent of the mesh size h, such that when0 < θ < _c¹

0+1, the D-N alternative algorithm is convergent. Moreover, we have

eⁿ⁺¹_h ²_Sh ≤(1−2θ+ 2(c₀+ 1)θ²)eⁿ_h²_Sh. (3.4) In particular, when θ= _2(c¹

0+1), the upper bound of (3.4)is minimal and there holds eⁿ⁺¹_h ²_Sh ≤(1− 1

2(c₀+ 1))eⁿ_h²_Sh. (3.5)

Remark 3.3. Setuⁿ_1h=uⁿ_h and

uⁿ_2h(z) =1

2(Kϕⁿ_h+V S(ϕⁿ_h−a))(z) +a, z∈Ω_c.

Theorem 3.1 proves thatuⁿ_1handuⁿ_2hconverge to u_1h andu_2hrespectively. In fact, we have uⁿ_1h−u_1h1,Ω≤C

1− 1

2(c₀+ 1) ⁿ₂

e⁰_hSh,

withC being a constant independent ofhandn. The erroruⁿ_2h−u_2hhas a similar estimate, but it involves a particular norm on the unbounded domain Ω_c.

4. Analysis of the convergence rate

Since the nonlinear variational inequality is involved, Theorem 3.1 can not be derived by the standard eigenvalue method, which has been used to analyze the convergence rate of the usual iterative method (com- pare [7, 19]). The proof of Theorem 3.1 is based on four lemmas.

Lemma 4.1. The variational inequality (2.2)can be decomposed into the following sub-problems

A(u_h, v_h−u_h)≥(f, vh−u_h), ∀vh∈Hˆ_ϕ¹_h (4.1) and

S_hϕ_h, ψ_h = (f, ψ_h)−A(u_h, ψ_h) +t₀+Sa, ψ_h , ∀ψ_h∈H_h¹². (4.2)

Proof. The variational inequality (4.1) follows by settingψ_h=ϕ_h in (2.2).

For any w_h∈H_h¹², we set v_h =u_h+w_h (or u_h−w_h) andψ_h =ϕ_h+w_h (or ϕ_h−w_h) in (2.2). Then, the functionϕ_h∈H_h¹² satisfies

A(u_h, w_h) +Shϕ_h, w_h ≥(f, w_h) +t0+Sa, w_h and

A(u_h,−wh) +Shϕ_h,−wh ≥(f,−wh) +t0+Sa,−wh .

These two inequalities imply the equation (4.2).

Set

µⁿ_h=pⁿ⁺¹_h −ϕ_h. Lemma 4.2. The following inequality is valid:

A(u_h, u_h−uⁿ⁺¹_h )−A(uⁿ⁺¹_h , u_h−uⁿ⁺¹_h )≤ Shµⁿ_h, eⁿ_h . (4.3) Proof. It can be verified directly that

Shµⁿ_h, eⁿ_h = Shµⁿ_h, u_h−uⁿ⁺¹_h +Shµⁿ_h, ϕ_h+u₀−u_h +Shµⁿ_h, uⁿ⁺¹_h −(ϕⁿ_h+u₀)

= Shµⁿ_h, u_h−uⁿ⁺¹_h +Shpⁿ⁺¹_h , ϕ_h+u₀−u_h − Shϕ_h, ϕ_h+u₀−u_h

+S_hpⁿ⁺¹_h , uⁿ⁺¹_h −(ϕⁿ_h+u₀) − S_hϕ_h, uⁿ⁺¹_h −(ϕⁿ_h+u₀) . (4.4) Setw_h=ϕ_h+u₀−u_h|_Γ. From the definition ofpⁿ⁺¹_h (see step 2^◦), we have

S_hpⁿ⁺¹_h , ϕ_h+u₀−u_h =S_hpⁿ⁺¹_h , w_h = (f, w_h)−A(uⁿ⁺¹_h , w_h) +t₀+Sa, w_h

= t₀+Sa, w_h −[(f,−w_h)−A(uⁿ⁺¹_h ,−w_h)]

= t₀+Sa, w_h −[(f,(uⁿ⁺¹_h −w_h)−uⁿ⁺¹_h )

−A(uⁿ⁺¹_h ,(uⁿ⁺¹_h −w_h)−uⁿ⁺¹_h )]. (4.5) Sinceu_h∈Hˆ_ϕ¹_h, we know that

w_h|_Γt = 0 and w_h|_Γs ≥0 (a.e.on Γ_s). (4.6) Note thatuⁿ⁺¹_h ∈Hˆ_ϕ¹n

h, we have

uⁿ⁺¹_h |Γt =ϕⁿ_h+u₀ and uⁿ⁺¹_h |Γt ≤ϕⁿ_h+u₀(a.e.on Γ_s), by (4.6), this leads to

(uⁿ⁺¹_h −w_h)|_Γt =ϕⁿ_h+u₀ and (uⁿ⁺¹_h −w_h)|_Γt≤ϕⁿ_h+u₀ (a.e. on Γ_s).

Namely,uⁿ⁺¹_h −w_h∈Hˆ_ϕ¹n

h. Thus, by (4.5) and the definition ofuⁿ⁺¹_h (see step 1^◦) we obtain

S_hpⁿ⁺¹_h , ϕ_h+u₀−u_h ≥ t₀+Sa, w_h =t₀+Sa, ϕ_h+u₀−u_h . (4.7) Similarly, we can prove

−S_hϕ_h, uⁿ⁺¹_h −(ϕⁿ_h+u₀) = S_hϕ_h, ϕⁿ_h+u₀−uⁿ⁺¹_h

≥ t₀+Sa, ϕⁿ_h+u₀−uⁿ⁺¹_h . (4.8) On the other hand, let ˆv_h∈H_h¹denote a suitable extension ofϕ_h+u₀such that ˆv_h−u_his just the zero extension ofϕ_h+u₀−u_h|Γ. From (4.2), we have

S_hϕ_h, ϕ_h+u₀−u_h = (f,vˆ_h−u_h)−A(u_h,vˆ_h−u_h) +t₀+Sa, ϕ_h+u₀−u_h). (4.9)

It is clear that ˆv_h∈Hˆ_ϕ¹_h. Thus, by (4.1) and (4.9) we obtain

Shϕ_h, ϕ_h+u₀−u_h ≤ t0+Sa, ϕ_h+u₀−u_h . Namely,

−Shϕ_h, ϕ_h+u₀−u_h ≥ −t0+Sa, ϕ_h+u₀−u_h . (4.10) In an analogous way, we can prove

Shpⁿ⁺¹_h , uⁿ⁺¹_h −(ϕⁿ_h+u₀) = −Shpⁿ⁺¹_h , ϕⁿ_h+u₀−uⁿ⁺¹_h

≥ −t0+Sa, ϕⁿ_h+u₀−uⁿ⁺¹_h . (4.11) Using (4.4), together with (4.7)–(4.8) and (4.10)–(4.11), yields

S_hµⁿ_h, eⁿ_h ≥ S_hµⁿ_h, u_h−uⁿ⁺¹_h = S_hpⁿ⁺¹_h , u_h−uⁿ⁺¹_h − S_hϕ_h, u_h−uⁿ⁺¹_h . (4.12) Subtracting (4.2) from (3.2), and choosingψ_h= (u_h−uⁿ⁺¹_h )|_Γ, we obtain

S_hpⁿ⁺¹_h , u_h−uⁿ⁺¹_h − S_hϕ_h, u_h−uⁿ⁺¹_h =A(u_h, e_hΓ)−A(uⁿ⁺¹_h , e_hΓ) (4.13) withe_hΓ= (u_h−uⁿ⁺¹_h )|_Γ. It follows by (3.1) and (4.1) that

A(uⁿ⁺¹_h , w_h) = (f, w_h), ∀w_h ∈H_h¹∩H₀¹(Ω) and

A(u_h, w_h) = (f, w_h), ∀w_h∈H_h¹∩H₀¹(Ω).

Hence,

A(uⁿ⁺¹_h , e_hΓ−(u_h−uⁿ⁺¹_h )) = (f, e_hΓ−(u_h−uⁿ⁺¹_h )) =A(u_h, e_hΓ−(u_h−uⁿ⁺¹_h )).

Namely,

A(u_h, e_hΓ)−A(uⁿ⁺¹_h , e_hΓ) =A(u_h, u_h−uⁿ⁺¹_h )−A(uⁿ⁺¹_h , u_h−uⁿ⁺¹_h ). (4.14)

This, together with (4.12) and (4.13), gives the desired result.

By (4.3) and the monotonicity of the (nonlinear) functionalA(·,·) (see [2]), we obtain Corollary 4.1. We have the inequality

S_hµⁿ_h, eⁿ_h ≥0. (4.15)

For a givenλ∈H_h¹², letλ∈H_h¹ denote the discrete harmonic extension ofλ.

Lemma 4.3. There exists a positive constant csuch that

|µⁿ_h|²_1,Ω≤cS_hµⁿ_h, µⁿ_h . (4.16) Proof. Let H(µⁿ_h) ∈ H¹(Ω) denote the (continuous) harmonic extension. Since µⁿ_h is just the orthogonal projection ofH(µⁿ_h), we have

|µⁿ_h|²_1,Ω≤ |H(µⁿ_h)|²_1,Ω. (4.17) It follows by the well-known property ofH(µⁿ_h) that there is a positive constantc₁independent ofh, such that

|H(µⁿ_h)|²_1,Ω≤c₁µⁿ_h²1 2,Γ. Thus, by (4.17) and Lemma 5.1 in [2], we obtain

|µⁿ_h|²_1,Ω≤c₁µⁿ_h²1

2,Γ ≤cShµⁿ_h, µⁿ_h .

Lemma 4.4. There exists a positive constant c₀ such that

S_hµⁿ_h, µⁿ_h ≤c₀S_heⁿ_h, eⁿ_h . (4.18) Proof. Subtracting (4.2) from (3.2), and choosingψ_h=µⁿ_h, yields

S_hµⁿ_h, µⁿ_h = S_hpⁿ⁺¹_h , µⁿ_h − S_hϕ_h, µⁿ_h

= A(u_h, µⁿ_h)−A(uⁿ⁺¹_h , µⁿ_h). (4.19) Since (µⁿ_h−µⁿ_h)|Γ = 0, we have (refer to (4.14))

A(u_h, µⁿ_h)−A(uⁿ⁺¹_h , µⁿ_h) =A(u_h,µⁿ_h)−A(uⁿ⁺¹_h ,µⁿ_h).

Substituting the above equality into (4.19), yields

S_hµⁿ_h, µⁿ_h =A(u_h,µⁿ_h)−A(uⁿ⁺¹_h ,µⁿ_h). (4.20) On the other hand, by the assumptions to the functionp(t) (see Sect. 2), we can prove (refer to [2, 11])

A(u_h,µⁿ_h)−A(uⁿ⁺¹_h ,µⁿ_h)≤β|uh−uⁿ⁺¹_h |1,Ω· |µⁿ_h|1,Ω (4.21) and

α|uh−uⁿ⁺¹_h |²_1,Ω≤A(u_h, u_h−uⁿ⁺¹_h )−A(uⁿ⁺¹_h , u_h−uⁿ⁺¹_h ). (4.22) Hence, by (4.20)–(4.22), we obtain

Shµⁿ_h, µⁿ_h ≤α⁻¹²β[A(u_h, u_h−uⁿ⁺¹_h )−A(uⁿ⁺¹_h , u_h−uⁿ⁺¹_h )]¹² · |µⁿ_h|1,Ω. Furthermore, it follows by (4.3) that

S_hµⁿ_h, µⁿ_h ≤α⁻¹²βS_hµⁿ_h, eⁿ_h ¹² · |µⁿ_h|_1,Ω, (4.23) which, together with (4.16), yields

S_hµⁿ_h, µⁿ_h ≤α⁻¹β²cS_hµⁿ_h, eⁿ_h .

SinceS_h:H_h¹² →(H_h¹²)^∗ is a symmetric and positive definite, by the Cauchy-Schwarz inequality, we obtain Shµⁿ_h, µⁿ_h ≤α⁻¹β²cShµⁿ_h, µⁿ_h ¹² · Sheⁿ_h, eⁿ_h ¹².

Namely,

Shµⁿ_h, µⁿ_h ≤α⁻²β⁴c²Sheⁿ_h, eⁿ_h .

This means that the inequality (4.18) is valid withc₀=α⁻²β²c². Now, we can prove our main result easily.

Proof of Theorem 3.1. From (3.3), we have

eⁿ⁺¹_h =eⁿ_h−θg_hⁿ. (4.24)

wheregⁿ_h=pⁿ⁺¹_h −ϕⁿ_h. It follows by (4.24) that

eⁿ⁺¹_h ²_Sh =eⁿ_h²_Sh−2θS_heⁿ_h, g_hⁿ +θ²gⁿ_h²_Sh. (4.25)

Namely,

eⁿ⁺¹_h ²_Sh =

1−2θS_heⁿ_h, g_hⁿ

eⁿ_h²_Sh +θ²g_hⁿ²_Sh eⁿ_h²_Sh

eⁿ_h²_Sh (eⁿ_h = 0). (4.26) By Corollary 4.1, yields (note thatgⁿ_h=eⁿ_h+µⁿ_h)

S_heⁿ_h, g_hⁿ =eⁿ_h²_Sh+S_heⁿ_h, µⁿ_h ≥ eⁿ_h²_Sh. (4.27) Moreover, it follows by Lemma 4.4 that

g_hⁿ²_Sh = µⁿ_h+eⁿ_h²_Sh

≤ 2(µⁿ_h²_Sh+eⁿ_h²_Sh)

≤ 2(c₀+ 1)eⁿ_h²_Sh). (4.28)

Using (4.26), together with (4.27) and (4.28), we deduce to (3.4).

It is clear that when 0< θ < _c¹

0+1, we have

eⁿ⁺¹_h ²_Sh<eⁿ_h²_Sh. It can be verified directly that whenθ=_2(c¹

0+1), the function g(θ) = 1−2θ+ 2(c₀+ 1)θ²

reaches the minimum value 1−_2(c0¹₊₁₎·

Remark 4.1. Some iterative methods for nonlinear minimization problems have been developed by [16,17]. It is easy to see that the methods as well as the convergence results can be extended to the case of nonlinear equation.

On the other hand, the unknownϕ_h satisfies a nonlinear equation, and Algorithm 3.1 can be interpreted as a preconditioned Richardson iteration (refer to the next section). Thus, Theorem 3.1 may be derived by using the extended results.

5. A combination between the Algorithm 3.1 and the steepest descent method

In most applications, it is difficult to estimate exactly the constantcin (4.16) (orc₀ in Theorem 3.1). To solve this problem, we consider the steepest descent method, by which we can determine a (variable) relaxation parameterθ_n in then−thiteration (refer to [6, 13]).

For a givenλ∈H_h¹², let Ψ(λ)∈Hˆ_λ¹ denote the solution of the variational inequality

A(Ψ(λ), v−Ψ(λ))≥(f, v−Ψ(λ)), ∀v∈Hˆ_λ¹. (5.1) We define the (nonlinear) operatorS_Ω:H_h¹² →(H_h¹²)^∗ by

S_Ωλ, µ =A(Ψ(λ),µ)¯ −(f,µ),¯ ∀µ∈H_h¹². Since Ψ(ϕ_h) =u_h, it follows by Lemma 4.1 thatϕ_hsatisfies the interface equation

Sϕˆ _h=g, (5.2)

where

Sˆ=S_Ω+S_h and g=t₀+S_ha.

It is clear that pⁿ⁺¹_h = S_h⁻¹(g−S_Ωϕⁿ_h). Thus, the algorithm 3.1 given in Section 3 can be regarded as the preconditioned Richardson iteration

ϕⁿ⁺¹_h =ϕⁿ_h+θS_h⁻¹(g−Sϕˆ ⁿ_h). (5.3) Now, the (variable) parameterθ (=θ_n) in (5.3) is chosen such that the function

G(θ) =Sϕˆ _h−Sϕˆ ⁿ⁺¹_h ²_S−1 h

reaches its minimum value whenθ=θ_n.

Some algorithms of determining the parameterθ_n are given in [13]. For the combined method, the main cost still results from the Algorithm 3.1 described in Section 3.

Let{ϕⁿ_h} denote the solution sequence generated by this method.

Theorem 5.1. For the combined method, we have the following convergence result:

ϕ_h−ϕⁿ_h²_Sh ≤

1− 1

2(c₀+ 1) _n

Sϕˆ _h−Sϕˆ ⁰_h²_S−1

h , (5.4)

wherec₀ is the constant given in the last section.

To prove this theorem, we need two lemmas.

The following result can be proved like (4.3) (refer to Cor. 4.1).

Lemma 5.1. The following inequality is valid for anyλ, µ∈H_h¹²

λ−µ, S_Ωλ−S_Ωµ ≥A(Ψ(λ),Ψ(λ)−Ψ(µ))−A(Ψ(µ),Ψ(λ)−Ψ(µ))≥0. (5.5) Lemma 5.2. For any λ, µ∈H_h¹², we have

S_Ωλ−S_Ωµ²_S−1

h ≤c₀λ−µ²_Sh. (5.6)

Proof. From the definition of the operatorS_Ω, we have S_Ωλ−S_Ωµ²_S−1

h =A(Ψ(λ),w) −A(Ψ(µ),w), where

w=S_h⁻¹(S_Ωλ−S_Ωµ).

Thus, we can prove, in an analogous way with (4.18), that S_Ωλ−S_Ωµ²_S−1

h ≤α⁻¹β²cA(Ψ(λ),Ψ(λ)−Ψ(µ))−A(Ψ(µ),Ψ(λ)−Ψ(µ)). (5.7) On the other hand, it follows by (5.5) that

A(Ψ(λ),Ψ(λ)−Ψ(µ))−A(Ψ(µ),Ψ(λ)−Ψ(µ)) ≤ λ−µ, S_Ωλ−S_Ωµ

≤ S_h¹²(λ−µ) · S_h⁻¹²(S_Ωλ−S_Ωµ). (5.8)

Using (5.7) and (5.8), we deduce to (5.6).

Proof of Theorem 5.1. It follows, by a well-known identity, that Sϕˆ _h−Sϕˆ ⁿ⁺¹_h ²_S−1

h − Sϕˆ _h−Sϕˆ ⁿ_h²_S−1

h = 2Sϕˆ _h−Sϕˆ ⁿ_h,Sϕˆ ⁿ_h−Sϕˆ ⁿ⁺¹_{h S}−1

h +Sϕˆ ⁿ_h−Sϕˆ ⁿ⁺¹_h ²_S−1

h . (5.9)