# A Posteriori Error Estimates for the Generalized Overlapping Domain Decomposition Methods

Authors: Hakima Benlarbi; Ahmed-Salah Chibi
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

A posteriori error estimates for the generalized overlapping domain decomposition method (GODDM) (i.e., with Robin boundary conditions on the interfaces), for second order boundary value problems,
are derived. We show that the error estimate in the continuous case depends on the differences of the traces of the subdomain solutions on the interfaces. After discretization of the domain by finite elements we use the techniques of the residuala posteriori error analysis to get an a posteriori error estimate for the discrete solutions on subdomains. The results of some numerical experiments are presented to support the theory.


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## Body

## 1. Introduction

We consider the generalized overlapping domain decomposition method that is, with Robin transmission conditions on the interfaces [1, 2] for second order boundary value problems on a bounded domain Ω with Dirichlet boundary conditions. The a priori estimate of the error is given in several papers, see for instance Lions [3] in which a variational formulation of the classical Schwarz method is derived. In Chan et al. [4] a geometry related convergence results are obtained. Douglas and Huang [1] studied the accelerated version of the GODDM, Engquist and Zhao [2] studied the convergence for simple (rectangular or circular) geometries; however, these authors did not give a criterion to stop the iterative process. All these results can also be found in the recent books on domain decomposition methods of Quarteroni and Valli [5], Toselli and Widlund [6]. Recently Maday and Magoulès [7, 8] presented an improved version of the Schwarz method for highly heterogeneous media. This method uses new optimized interface conditions specially designed to take into account the heterogeneity between the subdomains on the interfaces. A recent overview of the current state of the art on domain decomposition methods can be found in two special issues of the computer methods in applied mechanics and engineering journal, edited by Farhat and Le Tallec [9], Magoulès and Rixen [10] and in Nataf [11].In general, thea priori estimate is not suitable for assessing the quality of the approximate solution on subdomains since it depends mainly on the exact solution itself which is unknown. The alternative approach is to use the approximate solution itself in order to find such an estimate. This approach, known as a posteriori estimate, became very popular in the nineties of the last century with finite element methods, see the monographs [12, 13] and the references therein. In their paper Otto and Lube [14] gave an a posteriori estimate for a nonoverlapping domain decomposition algorithm that said that “the better the local solutions fit together at the interface the better the errors of the subdomain solutions will be.” This error estimate enables us to know with certainty when one must stop the iterative process as soon as the required global precision is reached. A posteriori error analysis was also used by Bernardi et al. [15] to determine an optimal value of the penalty parameter for penalty domain decomposition methods to construct fast solvers.In various situations it is better to use overlapping decompositions for faster convergence rate, for example, the case of decomposition into simple subdomains where uniform discretizations are possible. So the aim of this paper is to show that a similar result to that in [14], in the case of the GODDM, holds. It is obtained via the introduction of two auxiliary problems defined each over two nonoverlapping subdomains. These auxiliary problems are needed for the analysis and not for the computation.The paper is organized as follows. In Section1, we introduce some necessary notations, then we give a variational formulation of our model problem. We establish, in Section 2, a stopping criterion for the iterative process in the continuous case. In Section 3, an a posteriori error estimate is proposed for the convergence of the discretized solution using the finite element method on subdomains. We conclude this section by an adaptation of the techniques of the residual a posteriori error analysis to give an a posteriori estimate in the discrete case.


### 1.1. Problem Formulation and Domain Decomposition Method

LetΩ be a bounded domain in ℝ2 with a piecewise C1,1 boundary ∂Ω. We denote (·,·)Ω and ∥·∥0,Ω the usual inner product and norm of L2(Ω). Let H1(Ω) be the usual Sobolev space with norm ∥·∥1,Ω and seminorm |·|1,Ω; H01(Ω) is the subspace of functions of H1(Ω) vanishing on ∂Ω.LetΩ1 and Ω2 be two subdomains of Ω (to be defined later), when Γi⊂∂Ωi we need the space Wi=H001/2(Γi)={v|Γi:v∈H1(Ωi),v=0on∂Ωi∖Γi} which is a subspace of H1/2(Γi); i=1,2 equipped with the norm
(1.1)∥φ∥Wi=infv∈H1(Ωi),v=φonΓi|v|1,Ωi.
On the subdomain Ωi we use the space Vi=H01(Ω)∩H1(Ωi) which has as a trace space Wi=Vi|Γi=H001/2(Γi), i=1,2. By 〈·,·〉Γi we denote the inner product of L2(Γi).Let us consider the following model problem:(1.2)-Δu=f∈L2(Ω),u=0on∂Ω,
The variational formulation of this problem is given by
(1.3)Findu∈Vsolutionofa(u,v)=(f,v)Ω,∀v∈V,
where a(u,v)=∫Ω‍∇u∇vdx and(f,v)Ω=∫Ω‍fvdx for all v∈V=H01(Ω). We split the domain Ω into two overlapping subdomains Ω1 and Ω2 such that
(1.4)Ω1∩Ω2=Ω12,∂Ωi∩Ωj=Γi,i≠j(i,j=1,2).The generalized overlapping domain decomposition method is written as follows: setui0 in Ωi and construct the sequence uin+1, i=1,2 in parallel for n=0,1,2,…(1.5)-Δu1n+1=finΩ1,u1n+1=0on∂Ω1∩∂Ω,∂u1n+1∂η1+α1u1n+1=∂u2n∂η1+α1u2nonΓ1,(1.6)-Δu2n+1=finΩ2,u2n+1=0on∂Ω2∩∂Ω,∂u2n+1∂η2+α2u2n+1=∂u1n∂η2+α2u1nonΓ2,
where ηi is the exterior normal to Ωi and αi is a real parameter, i=1,2.The convergence of the GODDM (1.5) and (1.6) is proved in [2, 5]. Our main interest is to obtain an a posteriori error estimate we need for stopping the iterative process as soon as the required global precision is reached.The weak formulations of problems (1.5), (1.6) are given, respectively, by
(1.7)u1n+1∈V1:a1(u1n+1,v1)+〈α1u1n+1,v1〉Γ1=(f,v1)+〈∂u2n∂η1+α1u2n,v1〉Γ1,∀v1∈V1,(1.8)u2n+1∈V2:a2(u2n+1,v1)+〈α2u2n+1,v1〉Γ2=(f,v2)+〈∂u1n∂η2+α2u1n,v1〉Γ2,∀v2∈V2,
where Vi is defined above. It can also be written as
(1.9)Vi={v∈H1(Ωi):v∣∂Ωi∩∂Ω=0},ai(u,v)=(∇u,∇v)Ωi.


## 1.1. Problem Formulation and Domain Decomposition Method

LetΩ be a bounded domain in ℝ2 with a piecewise C1,1 boundary ∂Ω. We denote (·,·)Ω and ∥·∥0,Ω the usual inner product and norm of L2(Ω). Let H1(Ω) be the usual Sobolev space with norm ∥·∥1,Ω and seminorm |·|1,Ω; H01(Ω) is the subspace of functions of H1(Ω) vanishing on ∂Ω.LetΩ1 and Ω2 be two subdomains of Ω (to be defined later), when Γi⊂∂Ωi we need the space Wi=H001/2(Γi)={v|Γi:v∈H1(Ωi),v=0on∂Ωi∖Γi} which is a subspace of H1/2(Γi); i=1,2 equipped with the norm
(1.1)∥φ∥Wi=infv∈H1(Ωi),v=φonΓi|v|1,Ωi.
On the subdomain Ωi we use the space Vi=H01(Ω)∩H1(Ωi) which has as a trace space Wi=Vi|Γi=H001/2(Γi), i=1,2. By 〈·,·〉Γi we denote the inner product of L2(Γi).Let us consider the following model problem:(1.2)-Δu=f∈L2(Ω),u=0on∂Ω,
The variational formulation of this problem is given by
(1.3)Findu∈Vsolutionofa(u,v)=(f,v)Ω,∀v∈V,
where a(u,v)=∫Ω‍∇u∇vdx and(f,v)Ω=∫Ω‍fvdx for all v∈V=H01(Ω). We split the domain Ω into two overlapping subdomains Ω1 and Ω2 such that
(1.4)Ω1∩Ω2=Ω12,∂Ωi∩Ωj=Γi,i≠j(i,j=1,2).The generalized overlapping domain decomposition method is written as follows: setui0 in Ωi and construct the sequence uin+1, i=1,2 in parallel for n=0,1,2,…(1.5)-Δu1n+1=finΩ1,u1n+1=0on∂Ω1∩∂Ω,∂u1n+1∂η1+α1u1n+1=∂u2n∂η1+α1u2nonΓ1,(1.6)-Δu2n+1=finΩ2,u2n+1=0on∂Ω2∩∂Ω,∂u2n+1∂η2+α2u2n+1=∂u1n∂η2+α2u1nonΓ2,
where ηi is the exterior normal to Ωi and αi is a real parameter, i=1,2.The convergence of the GODDM (1.5) and (1.6) is proved in [2, 5]. Our main interest is to obtain an a posteriori error estimate we need for stopping the iterative process as soon as the required global precision is reached.The weak formulations of problems (1.5), (1.6) are given, respectively, by
(1.7)u1n+1∈V1:a1(u1n+1,v1)+〈α1u1n+1,v1〉Γ1=(f,v1)+〈∂u2n∂η1+α1u2n,v1〉Γ1,∀v1∈V1,(1.8)u2n+1∈V2:a2(u2n+1,v1)+〈α2u2n+1,v1〉Γ2=(f,v2)+〈∂u1n∂η2+α2u1n,v1〉Γ2,∀v2∈V2,
where Vi is defined above. It can also be written as
(1.9)Vi={v∈H1(Ωi):v∣∂Ωi∩∂Ω=0},ai(u,v)=(∇u,∇v)Ωi.


## 2. A Posteriori Error Estimate in the Continuous Case

Since it is numerically easier to compare the subdomain solutions on the interfacesΓ1 and Γ2 rather than on the overlap Ω12, thus we need to introduce two auxiliary problems defined on nonoverlapping subdomains of Ω. This idea allows us to obtain the a posteriori error estimate by following the steps of Otto and Lube [14]. We get these auxiliary problems by coupling each one of the problems (1.5) and (1.6) with another problem in a nonoverlapping way over Ω. These auxiliary problems are needed for the analysis and not for the computation, to get the estimate. We see that we can go from the GODDM to a couple of nonoverlapping domain decomposition methods with Robin boundary conditions on the interfaces Γ1 and Γ2. To define these auxiliary problems we need to split the domain Ω into two sets of disjoint subdomains: (Ω1,Ω3) and (Ω2,Ω4), see Figure 1, such that
(2.1)Ω=Ω1∪Ω3,withΩ1∩Ω3=∅,Ω=Ω2∪Ω4withΩ2∩Ω4=∅.Two examples of domain decompositions (Schur and Schwarz).
(a)
(b)Let(u1n+1,u2n+1) be the solutions of problems (1.5), (1.6). We define the couple (u1n+1,u3n+1) over (Ω1,Ω3) to be the solution of the following nonoverlapping problems:
(2.2)-Δu1n+1=finΩ1,u1n+1=0on∂Ω1∩∂Ω,∂u1n+1∂η1+α1u1n+1=∂u2n∂η1+α1u2nonΓ1,(2.3)-Δu3n+1=finΩ3,u3n+1=0in∂Ω3∩∂Ω,∂u3n+1∂η3+α3u3n+1=∂u1n∂η3+α3u1nonΓ1.
We can write u2n=u3n+ϵ1n on Γ1 (in the 3rd equation in (2.2)) that is, ϵ1n is the difference between the overlapping and the nonoverlapping solutions u2n and u3n (in problems (1.5)-(1.6) and (2.2)-(2.3), resp.) in Ω3. Because both overlapping and nonoverlapping problems converge, see [2, 5] that is, u2n and u3n tend to u2=u|Ω2, ϵ1n should tend to naught an n tends to infinity in V2.By multiplying the first equation in (2.2) by v1∈V1 and integrating by parts we obtain
(2.4)∫Ω1‍∇u1n+1∇v1dx+∫Γ1‍α1u1n+1v1dx=∫Ω1‍fv1dx+∫Γ1‍Λ3nv1dσ,
where Λ3n is given by
(2.5)Λ3n=∂u3n∂η1+α1u3n+∂ϵ1n∂η1+α1ϵ1n.
On the other hand, using the last equation in (2.3) and putting
(2.6)θ1n=∂ϵ1n∂η1+α1ϵ1n,
we get
(2.7)Λ3n+1=∂u3n+1∂η1+α1u3n+1+θ1n+1=-∂u3n+1∂η3+α1u3n+1+θ1n+1=α3u3n+1-∂u1n∂η3-α3u1n+α1u3n+1+θ1n+1=(α1+α3)u3n+1-Λ1n+ζn,
where ζn=θ1n+1+θ3n. From this result we can write the following algorithm which is equivalent to the auxiliary nonoverlapping problem (2.2)-(2.3). We need this algorithm and two lemmas for obtaining an a posteriori error estimate for this problem.Algorithm 2.1.
The sequences(u1n+1,u3n+1)n∈ℕ solutions of problems (2.2), (2.3) satisfy the following domain decomposition algorithm denoted ALG.G.D.D.M.(1)
LetΛi0∈W1* be an initial value, i=1,3(W1* is the dual of W1).(2)
GivenΛjn∈W1* solve for i=1,3, i≠j: Find uin+1∈Vi solutions of
(2.8)ai(uin+1,vi)+〈αiuin+1,vi〉Γ1=F(vi)+〈Λjn,vi〉Γ1,∀vi∈Vi.(3)
Compute new dataΛin+1∈W1*, i=1,3 from
(2.9)〈Λin+1,φ〉Γ1=〈(αi+αj)uin+1,φ〉Γ1-〈Λjn,φ〉Γ1+〈ζn,φ〉Γ1,∀φ∈W1,j≠i.(4)
Setn=n+1, go to Step 2.Lemma 2.2.
Letui=u|Ωi, ein+1=uin+1-ui,ηin=Λin-Λi and ξ1n=ζn-ζ. Then for i=1,3,i≠j, the following relations hold
(2.10)ai(ein+1,vi)+〈αiein+1,vi〉Γ1=〈ηjn,vi〉Γ1,∀vi∈Vi,(2.11)〈ηin+1,φ〉Γ1=〈(αi+αj)ein+1,φ〉Γ1-〈ηjn,φ〉Γ1+〈ξ1n,φ〉Γ1,∀φ∈W1.The proof follows directly from steps 2 and 3 of ALG.G.D.D.M.Lemma 2.3.
By lettingC be a generic constant which has different values at different places one get for i=1,3,i≠j(2.12)〈ηjn-1-αiein,w〉Γ1≤C∥ein∥1,Ωi∥w∥W1,〈αiw+ξ1n,ein+1〉Γ1≤C∥ein+1∥1,Ωi∥w∥W1.Proof.
Using (2.10) and the fact that the trace mapping Tri:Vi→Wi and its inverse are continuous, we obtain
(2.13)〈ηjn-1-αiein,w〉Γ1=ai(ein,Tri-1w)=(∇ein,∇Tri-1w)≤|ein|1,Ωi|Tri-1w|1,Ωi≤CTri-1∥ein∥1,Ωi∥w∥W1.
On the otherhand we have
(2.14)〈αiw+ξ1nein+1〉Γ1=∫Γ1‍(αiw+ξ1n)ein+1ds≤[|αi|∥w∥0,Γ1+|ξ1n|0,Γ1]∥ein+1∥0,Γ1≤max(|αi|,|ξ1n|0,Γ1)∥w∥0,Γ1∥ein+1∥0,Γ1≤C∥ein+1∥1,Ωi∥w∥W1.
This ends the proof.Based on the previous two lemmas, we can obtain the followinga posteriorierror estimate for the nonoverlapping domain decomposition problem (2.2)-(2.3).Proposition 2.4.
For the sequences(u1n+1,u3n+1)n∈ℕ solutions of problems (2.2), (2.3) one has the following a posteriori error estimation
(2.15)∥e1n+1∥1,Ω1+∥e3n∥1,Ω3≤C∥u1n+1-u3n∥H001/2(Γ1).Proof.
From (2.10) and (2.11) we have
(2.16)a1(e1n+1,v1)+a3(e3n,v3)=〈η3n-α1e1n+1,v1〉Γ1+〈η1n-1-α3e3n,v3〉Γ1=〈α1(e3n-e1n+1)+ξ1n,v1〉Γ1+〈η1n-1-α3e3n,v3-v1〉Γ1.
Taking v1=e1n+1 and v2=e2n, then using Lemma 2.3 we get
(2.17)12(∥e1n+1∥1,Ω1+∥e3n∥1,Ω3)2≤∥e1n+1∥1,Ω12+∥e3n∥1,Ω32=a1(e1n+1,e1n+1)+a3(e3n,e3n)≤〈α1(e3n-e1n+1)+ξ1n,e1n+1〉Γ1+〈η1n-1-α3e3n,e3n-e1n+1〉Γ1≤C1∥e1n+1∥1,Ω1∥e3n-e1n+1∥H001/2(Γ1)+C2∥e3n∥1,Ω3∥e3n-e1n+1∥H001/2(Γ1)≤C{∥e1n+1∥1,Ω1+∥e3n∥1,Ω3}∥u1n+1-u3n∥H001/2(Γ1),
which is the required result.In the similar way, we define another nonoverlapping auxiliary problem over(Ω2,Ω4). The sequences (u2n+1,u4n+1)n∈ℕ are solutions of
(2.18)-Δu2n+1=finΩ2,u2n+1=0on∂Ω2∩∂Ω,∂u2n+1∂η2+α2u2n+1=∂u1n∂η2+α2u1nonΓ2,(2.19)-Δu4n+1=finΩ4,u4n+1=0on∂Ω4∩∂Ω,∂u4n+1∂η4+α4u4n+1=∂u2n∂η4+α4u2nonΓ2,
where we can write u1n=u4n+ϵ2n on Γ2 (in the 3rd equation in (2.18)) that is, ϵ2n is the difference between the overlapping and the nonoverlapping solutions u1n and u4n (in problems (1.5)-(1.6) and (2.18)-(2.19) resp.) in Ω4. As both sequences u1n and u4n converge towards u1, ϵ2n should tend to naught as n tends to infinity. Using Lemmas 2.2 and 2.3 over (Ω2,Ω4) we obtain the a posteriori error estimate for this auxiliary problem.Proposition 2.5.
For the sequences(u2n+1,u4n+1)n∈ℕ solutions of problems (2.18), (2.19) one has the following a posteriori error estimate
(2.20)∥u2n+1-u2∥1,Ω2+∥u4n-u4∥1,Ω4≤C∥u2n+1-u4n∥H001/2(Γ2).Remark 2.6.
We remark from Propositions2.4 and 2.5 that we can go from sequences of solutions to problems (1.5) and (1.6) to sequences of solutions to problems generated by nonoverlapping domain decomposition problems of Robin type transmission conditions across the interfaces Γ1 and Γ2. This result enable us to compare the subdomain iterations of the GODDM on the interfaces Γ1 and Γ2 rather than on the overlap region Ω12.
In the following theorem we are going to use this fact to obtain ana posteriori error estimate for the continuous generalized overlapping domain decomposition method (1.5)-(1.6).Theorem 2.7.
Letui=u|Ωi where u is the solution of problem (1.2). For the sequences (u1n+1,u2n+1)n∈ℕ solutions of problems (1.5) and (1.6), one has the following a posteriori error estimate
(2.21)∥u1n+1-u1∥1,Ω1+∥u2n-u2∥1,Ω2≤C{∥u1n+1-u2n∥H001/2(Γ1)+∥u2n-u1n-1∥H001/2(Γ2)+∥ϵ1n∥H001/2(Γ1)+∥ϵ2n-1∥H001/2(Γ2)}.Proof.
We use two auxiliary problems defined each over two nonoverlapping subdomains ofΩ. These two problems are defined over (Ω1,Ω3) and (Ω2,Ω4), respectively. From the previous two propositions we have
(2.22)∥u1n+1-u1∥1,Ω1+∥u2n-u2∥2,Ω2≤∥u1n+1-u1∥1,Ω1+∥u3n-u3∥1,Ω3+||u2n-u2||1,Ω2+||u4n-1-u4||1,Ω4≤C∥u1n+1-u3n∥H001/2(Γ1)+C∥u2n-u4n-1∥H001/2(Γ2).
From the definitions of ϵ1n and ϵ2n, we have
(2.23)∥u1n+1-u1∥1,Ω1+∥u2n-u2∥1,Ω2≤C∥u2n-u1n-1-ϵ2n-1∥H001/2(Γ2)+C∥u1n+1-u2n-ϵ1n∥H001/2(Γ1)≤C{∥u1n+1-u2n∥H001/2(Γ1)+∥u2n-u1n-1∥H001/2(Γ2)+∥ϵ1n∥H001/2(Γ1)+∥ϵ2n-1∥H001/2(Γ2)},
which is the required result.


## 3. A Posteriori Error Estimate in the Discrete Case

In this section, we consider the discretization of the variational problems (1.7) and (1.8). Let τh be a triangulation of Ω compatible with the decomposition and Vh⊂V is the subspace of continuous functions (polynomials of degree k, k≥1) which vanish over ∂Ω. We have
(3.1)Vi,h=Vh∣Ωi,Wi,h=Vh∣Γi.Wi,h is a subspace of H001/2(Γi) which consists of continuous piecewise polynomial functions on Γi which vanish at the end points of Γi(i=1,2).Letuh∈Vh be the solution of the discrete problem associated with (1.3), ui,h=uh∣Ωi. We construct the sequences (ui,hn+1)i=1,2,ui,hn+1∈Vi,h solutions of the discrete versions of (1.5) and (1.6) at the (n+1)thiteration, that is,
(3.2)-Δu1,hn+1=fonΩ1,u1,hn+1=0on∂Ω1∩∂Ω,∂u1,hn+1∂η1+α1u1,hn+1=∂u2,hn∂η1+α1u2,hnonΓ1,(3.3)-Δu2,hn+1=fonΩ2,u2,hn+1=0on∂Ω2∩∂Ω,∂u2,hn+1∂η2+α2u2,hn+1=∂u1,hn∂η2+α2u1,hnonΓ2.
In a similar manner to that of Section 2, we introduce two discrete auxiliary problems, one over (Ω1, Ω3), the other over (Ω2, Ω4). We let
(3.4)-Δu1,hn+1=finΩ1,u1,hn+1=0on∂Ω1∩∂Ω,∂u1,hn+1∂η1+α1u1,hn+1=∂u2,hn∂η1+α1u2,hnonΓ1,(3.5)-Δu3,hn+1=finΩ3,u3,hn+1=0on∂Ω3∩∂Ω,∂u3,hn+1∂η3+α3u1,hn+1=∂u1,hn∂η3+α3u1,hnonΓ1.
We can write u2,hn=u3,hn+ϵ1,hn on Γ1, (in the 3rd equation in (3.4)), that is, ϵ1,hn is the difference between the discrete overlapping and the nonoverlapping solutions u2,hn and u3,hn (in problems (3.2)-(3.3) and (3.4)-(3.5) resp.,) in Ω3. Because both u2,hn and u3,hn converges to u2,ϵ1,hn should tend to naught as n tends to infinity.Similarly, over(Ω2, Ω4) we consider the sequences (u2,hn+1,u4,hn+1)n∈ℕ solutions of the following discrete problems:
(3.6)-Δu2,hn+1=finΩ2,u2,hn+1=0on∂Ω2∩∂Ω,∂u2,hn+1∂η2+α2u2,hn+1=∂u1,hn∂η2+α2u1,hnonΓ2,(3.7)-Δu4,hn+1=finΩ4,u4,hn+1=0on∂Ω4∩∂Ω,∂u4,hn+1∂η4+α4u4,hn+1=∂u2,hn∂η4+α4u2,hnonΓ2,
where we can also write u1,hn=u4,hn+ϵ2,hn on Γ2,, ϵ2,hn tends to naught as n tends to infinity as explained for the previous problems. We can obtain the discrete counterparts of Propositions 2.4 and 2.5 by doing almost the same analysis as in Section 2 (i.e., passing from continuous spaces to discrete subspaces and from continuous sequences to discrete ones). Therefore
(3.8)∥u1,hn+1-u1,h∥1,Ω1+∥u3,hn-u3,h∥1,Ω3≤C∥u1,hn+1-u3,hn∥W1,h,∥u2,hn+1-u2,h∥1,Ω2+∥u4,hn-u4,h∥1,Ω4≤C∥u2,hn+1-u4,hn∥W2,h,
where the discrete trace subspace norm ∥·∥Wi,h defined over Γi(i=1,2) is taken to be
(3.9)∥wh∥Wi,h2=∫Γi‍wh2dx+∬Γi(wh(x)-wh(y))2|x-y|2dxdy.
Hence, using the previous two inequalities, we have
(3.10)∥u1,hn+1-u1,h∥1,Ω1+∥u2,hn-u2,h∥1,Ω2⩽C{∥u2,hn-u1,hn-1∥W2,h+∥u1,hn+1-u2,hn∥W1,h+∥ϵ1,hn∥W1,h+∥ϵ2,hn-1∥W2,h}.
We remark that (3.10) is the discrete version of (2.21).Remark 3.1.
Let us note that (3.10) is an estimate of the error between the approximate solution ui,h and the approximate solution of the discretized overlapping domain decomposition algorithm ui,hn+1,i=1,2.
Next we will obtain an estimate of the error between the approximate solutionui,hn+1 and the exact solution u. We introduce some necessary notations. We denote by
(3.11)εh={Easideofτh;E∉∂Ω}.
For every T∈τh and E∈εh, we define
(3.12)ωT=∪{T′∈τh;T′∩T≠ϕ},ωE=∪{T′∈τh;T′∩E≠ϕ}.
For a function f which is not necessarily continuous across two neighboring elements of τh having E as a common side, [f] denotes the jump of f across E and ηE the normal vector of E.
We have the following theorem which gives thea posteriori error estimate for the discrete GODDM.Theorem 3.2.
Letui=u∣Ωi where u is the solution of problem (1.2), the sequences (u1,hn+1,u2,hn+1)n∈ℕ are solutions of problems (3.2)-(3.3). Then there exists a constant C independent of h such that
(3.13)∥u1,hn+1-u1∥1,Ω1+∥u2,hn-u2∥1,Ω2≤C{∑i=12‍(∑T∈τh‍(ηiT)2+hT2∥f-fh∥0,T2)1/2+∑i=12‍ηΓi},
where
(3.14)ηΓi=∥ui,h*-uj,h*-1∥Wi,h+∥ϵi,h*∥Wi,h,i,j=1,2,i≠j,ηiT=hT∥fh+Δui,h*∥0,T+∑E∈εh‍hE1/2∥[∂ui,h*∂ηE]∥0,E.
The symbol * corresponds to n+1 when i=1 and to n when i=2.Proof.
The proof is based on the technique of the residuala posteriori estimation (see [13]) and Theorem 2.7. We give the main steps. By the triangle inequality we have
(3.15)∑i=12‍∥ui-ui,h*∥1,Ωi≤∑i=12‍∥ui-ui,h∥1,Ωi+∑i=12‍∥ui,h-ui,h*∥1,Ωi.
The second term on the right-hand side of (3.15) is bounded as in (3.10) by
(3.16)∑i=12‍∥ui,h-ui,h*∥1,Ωi≤C∑i=12‍(∥ui*-uj*-1∥Wi,h+∥ϵi,h*∥Wi,h).
To bound the first term on the r.h.s of (3.15) we use the residual equation and apply the technique of the residual a posteriori error estimation, see [13], to get for vh∈Vh and vi,h=vh|Ωi(3.17)ai(ui-ui,h,v)=ai(ui-ui,h,v-vh)=∑T⊂Ωi‍∫‍(fh+Δui,h)(vi-vi,h)-∑E⊂Ωi‍∫E‍[∂ui,h∂ηE](vi-vi,h)ds=-∑E⊂Γi‍∫E‍∂ui,h∂ηE(vi-vi,h)ds+∑T⊂Ωi‍(f-fh,vi-vi,h)Ωi+〈∂ui∂ηi,u-vh〉Γi,
where fh is any approximation of f by polynomials of degree at most k, therefore
(3.18)∑i=12‍ai(ui-ui,h,v)≤∑i=12‍{∑T⊂Ωi‍||f+Δui,h||0,T||vi-vi,h||0,T+∑E⊂Ωi‍||[∂ui,h∂ηE]||0,E||vi-vi,h||0,E+∑E⊂Γi‍||∂ui,h∂ηE||0,E||vi-vi,h||0,E+∑T⊂Ωi‍||f-fh||0,T||vi-vi,h||0,T+||∂ui∂ηi||0,Γi||u-vh||0,Γi}≤∑i=12‍(∑T⊂Ωi‍(ηiT)2+hT2∥f-fh∥0,T2)1/2∑i=12‍|vi|1,Ωi.
We use the fact that
(3.19)∥ui-ui,h∥1,Ωi≤supv∈Viai(ui-ui,h,v)∥v∥1,Ωi.
Finally, by combining (3.15), (3.18), and (3.19) the required result follows.


### 3.1. Numerical Examples

As an illustration of the theoretical results obtained in this article and to be able to use thea posteriori error estimate (3.10), we consider the following problem
(3.20)-Δu=f∈L2(Ω),u=0on∂Ω,
with Ω=]0,1[2 for the first and second examples and Ω=]0,1[×]0,1/2[∪]0,1/2[×]0,1[ for the third example (i.e., an L-shaped region). The exact solution is taken to be u(x,y)=xy(x-1)(y-1)exy for the first example, u(x,y)=sin8πx·sin8πy for the second example, and finally u(x,y) = xy(x-1)(y-1)(x-1/2)(y-1/2) for the third example.We first compute the bilinear Galerkin solutionuh over Ω and then apply the generalized overlapping domain decomposition method (1.7)-(1.8) to compute the bilinear sequences ui,hn+1(i=1,2) to be able to look at the behavior of the constant C within (3.10) as the mesh size becomes small (h-1=8,16,32,64). We take Ω1=]0,3/4[×]0,1[ and Ω2=]1/4,1[×]0,1[ for the first and second examples. As for the third example we have Ω1=]0,1[×]0,1/2[ and Ω2=]0,1/2[×]0,1[. We denote by
(3.21)Eri=∥ui,h-ui,hn∥1,Ωi,Trer1=∥u1,hn+1-u2,hn∥W1,h,Trer2=∥u2,hn-u1,hn-1∥W2,h,Quotient=(Er1+Er2)(Trer1+Trer2).
The generalized overlapping domain decomposition method, with α1=α2=0.5, converges. The iterations have been stopped when the relative error between two subsequent iterates is less than ϵ=10-6.The Quotient in Tables1, 2, 3, 4, 5, and 6 represents an estimate of the constant C in (3.10). It happens to tend to a constant value around 1 which what we expect (i.e., we have asymptotic exactness). We present, for each example, results with and without the a posteriori estimate as a stopping criteria. We see that we obtain the same results but with less iterations when using the a posteriori estimate as a stopping criteria for a moderately small mesh size for the first and third examples. But for the second example, the difference in iteration count is clear between the two approaches.Table 1
Results on the square withouta posteriori estimates as a stopping criteria for the first example.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
1.7957693
(
-
02
)
8.1821001
(
-
03
)
4.5210766
(
-
03
)
3.1683034
(
-
03
)
E
r
2
1.0956427
(
-
02
)
9.6854433
(
-
03
)
8.7326244
(
-
03
)
6.9200024
(
-
03
)
Trer
1
2.2513498
(
-
02
)
0.1504199
(
-
02
)
8.1897348
(
-
03
)
3.4817880
(
-
02
)
Trer
2
1.7198801
(
-
02
)
2.1677755
(
-
02
)
9.3967836
(
-
03
)
6.9538120
(
-
03
)
Iterations
12
10
7
5
Quotient
0.7280898
0.7507522
0.7536285
0.9667203Table 2
Results on the square witha posteriori estimates as a stopping criteria for the first example.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
1.3457623
(
-
02
)
4.1221001
(
-
03
)
8.5210766
(
-
03
)
3.1683034
(
-
03
)
E
r
2
1.0556422
(
-
02
)
5.2454431
(
-
03
)
8.4322244
(
-
03
)
6.2200024
(
-
03
)
Trer
1
2.2513498
(
-
02
)
5.1504199
(
-
03
)
2.1897348
(
-
02
)
3.4817880
(
-
03
)
Trer
2
2.4423401
(
-
02
)
6.1677755
(
-
03
)
1.3967836
(
-
03
)
6.9538120
(
-
03
)
Iterations
3
5
6
7
Quotient
0.5116240
0.7276534
0.7277927
0.8996421Table 3
Results on the square withouta posteriori estimates as a stopping criteria for the second example.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
1.7957693
(
-
02
)
2.5598288
(
-
03
)
4.7627655
(
-
03
)
0.8936322
(
-
04
)
E
r
2
1.0956427
(
-
02
)
4.5353208
(
-
02
)
8.1970235
(
-
03
)
6.3020810
(
-
03
)
Trer
1
2.2513498
(
-
02
)
5.2993342
(
-
02
)
1.1897348
(
-
02
)
3.4817880
(
-
03
)
Trer
2
1.7198801
(
-
08
)
6.0068501
(
-
09
)
1.3967836
(
-
08
)
6.9538120
(
-
08
)
Iterations
10
9
7
5
Quotient
0.7280898
0.9041330
1.089299
1.560316Table 4
Results on the square witha posteriori estimates as a stopping criteria for the second example.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
4.7345582
(
-
02
)
8.1821001
(
-
04
)
4.8510591
(
-
03
)
1.4817880
(
-
02
)
E
r
2
1.3897220
(
-
02
)
9.6854433
(
-
02
)
8.1924358
(
-
03
)
6.9200024
(
-
03
)
Trer
1
9.6471474
(
-
02
)
0.1504199
(
-
02
)
1.1897348
(
-
02
)
1.4817880
(
-
02
)
Trer
2
9.1859622
(
-
09
)
0.0000000
(
+
00
)
0.3967836
(
-
09
)
2.9538120
(
-
08
)
Iterations
8
6
5
4
Quotient
0.6348280
0.6493334
1.096336
1.018248Table 5
Results on theL-shaped region without a posteriori estimates as a stopping criteria.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
3.5915386
(
-
02
)
16.3642002
(
-
03
)
9.0421132
(
-
03
)
6.3366068
(
-
03
)
E
r
2
2.19128497
(
-
02
)
19.3708866
(
-
03
)
12.4732488
(
-
03
)
9.8440048
(
-
03
)
Trer
1
4.50217996
(
-
02
)
0.2008398
(
-
02
)
4.3794696
(
-
02
)
2.9635760
(
-
02
)
Trer
2
3.4397602
(
-
02
)
4.3355510
(
-
02
)
2.7935672
(
-
03
)
5.9076240
(
-
03
)
Iterations
10
8
6
6
Quotient
0.5443332
0.6562642
0.8380865
1.008344Table 6
Results on theL-shaped region with a posteriori estimates as a stopping criteria.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
2.5815386
(
-
02
)
14.3642002
(
-
03
)
9.0421132
(
-
03
)
4.3366122
(
-
03
)
E
r
2
2.19128487
(
-
02
)
12.3708866
(
-
03
)
12.4732488
(
-
03
)
6.8440223
(
-
03
)
Trer
1
4.60217446
(
-
02
)
0.4008398
(
-
02
)
4.3794696
(
-
02
)
4.8635660
(
-
03
)
Trer
2
3.3397402
(
-
02
)
2.3355510
(
-
02
)
2.7935672
(
-
03
)
6.8076220
(
-
03
)
Iterations
4
5
6
7
Quotient
0.6009663
0.9170200
0.8380865
0.9579689


### 3.2. Concluding Remarks

(1)
From Theorem3.2 we see that the subdomain errors depend on the differences of the latter on the interface, on the data f, and on the approximate solution uh. The first two terms of (3.13) are the contribution to the a posteriori error from the finite element discretization whereas the last term is the domain decomposition error. Numerical results indicate that the finite element part of the error estimate dominates the domain decomposition part.
(2)
As far as the asymptotic exactness of the estimator based on (3.13) is concerned, Ainsworth and Oden in [12, Theorem  3.5] gave a necessary condition for this property to hold. This condition says that the local residuals in (3.13) should be computed exactly.
(3)
The a posteriori error analysis for a second order boundary value problem or Stokes problem is possible. Also, similar results can be obtained in the general case of more than two subdomains.


## 3.1. Numerical Examples

As an illustration of the theoretical results obtained in this article and to be able to use thea posteriori error estimate (3.10), we consider the following problem
(3.20)-Δu=f∈L2(Ω),u=0on∂Ω,
with Ω=]0,1[2 for the first and second examples and Ω=]0,1[×]0,1/2[∪]0,1/2[×]0,1[ for the third example (i.e., an L-shaped region). The exact solution is taken to be u(x,y)=xy(x-1)(y-1)exy for the first example, u(x,y)=sin8πx·sin8πy for the second example, and finally u(x,y) = xy(x-1)(y-1)(x-1/2)(y-1/2) for the third example.We first compute the bilinear Galerkin solutionuh over Ω and then apply the generalized overlapping domain decomposition method (1.7)-(1.8) to compute the bilinear sequences ui,hn+1(i=1,2) to be able to look at the behavior of the constant C within (3.10) as the mesh size becomes small (h-1=8,16,32,64). We take Ω1=]0,3/4[×]0,1[ and Ω2=]1/4,1[×]0,1[ for the first and second examples. As for the third example we have Ω1=]0,1[×]0,1/2[ and Ω2=]0,1/2[×]0,1[. We denote by
(3.21)Eri=∥ui,h-ui,hn∥1,Ωi,Trer1=∥u1,hn+1-u2,hn∥W1,h,Trer2=∥u2,hn-u1,hn-1∥W2,h,Quotient=(Er1+Er2)(Trer1+Trer2).
The generalized overlapping domain decomposition method, with α1=α2=0.5, converges. The iterations have been stopped when the relative error between two subsequent iterates is less than ϵ=10-6.The Quotient in Tables1, 2, 3, 4, 5, and 6 represents an estimate of the constant C in (3.10). It happens to tend to a constant value around 1 which what we expect (i.e., we have asymptotic exactness). We present, for each example, results with and without the a posteriori estimate as a stopping criteria. We see that we obtain the same results but with less iterations when using the a posteriori estimate as a stopping criteria for a moderately small mesh size for the first and third examples. But for the second example, the difference in iteration count is clear between the two approaches.Table 1
Results on the square withouta posteriori estimates as a stopping criteria for the first example.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
1.7957693
(
-
02
)
8.1821001
(
-
03
)
4.5210766
(
-
03
)
3.1683034
(
-
03
)
E
r
2
1.0956427
(
-
02
)
9.6854433
(
-
03
)
8.7326244
(
-
03
)
6.9200024
(
-
03
)
Trer
1
2.2513498
(
-
02
)
0.1504199
(
-
02
)
8.1897348
(
-
03
)
3.4817880
(
-
02
)
Trer
2
1.7198801
(
-
02
)
2.1677755
(
-
02
)
9.3967836
(
-
03
)
6.9538120
(
-
03
)
Iterations
12
10
7
5
Quotient
0.7280898
0.7507522
0.7536285
0.9667203Table 2
Results on the square witha posteriori estimates as a stopping criteria for the first example.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
1.3457623
(
-
02
)
4.1221001
(
-
03
)
8.5210766
(
-
03
)
3.1683034
(
-
03
)
E
r
2
1.0556422
(
-
02
)
5.2454431
(
-
03
)
8.4322244
(
-
03
)
6.2200024
(
-
03
)
Trer
1
2.2513498
(
-
02
)
5.1504199
(
-
03
)
2.1897348
(
-
02
)
3.4817880
(
-
03
)
Trer
2
2.4423401
(
-
02
)
6.1677755
(
-
03
)
1.3967836
(
-
03
)
6.9538120
(
-
03
)
Iterations
3
5
6
7
Quotient
0.5116240
0.7276534
0.7277927
0.8996421Table 3
Results on the square withouta posteriori estimates as a stopping criteria for the second example.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
1.7957693
(
-
02
)
2.5598288
(
-
03
)
4.7627655
(
-
03
)
0.8936322
(
-
04
)
E
r
2
1.0956427
(
-
02
)
4.5353208
(
-
02
)
8.1970235
(
-
03
)
6.3020810
(
-
03
)
Trer
1
2.2513498
(
-
02
)
5.2993342
(
-
02
)
1.1897348
(
-
02
)
3.4817880
(
-
03
)
Trer
2
1.7198801
(
-
08
)
6.0068501
(
-
09
)
1.3967836
(
-
08
)
6.9538120
(
-
08
)
Iterations
10
9
7
5
Quotient
0.7280898
0.9041330
1.089299
1.560316Table 4
Results on the square witha posteriori estimates as a stopping criteria for the second example.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
4.7345582
(
-
02
)
8.1821001
(
-
04
)
4.8510591
(
-
03
)
1.4817880
(
-
02
)
E
r
2
1.3897220
(
-
02
)
9.6854433
(
-
02
)
8.1924358
(
-
03
)
6.9200024
(
-
03
)
Trer
1
9.6471474
(
-
02
)
0.1504199
(
-
02
)
1.1897348
(
-
02
)
1.4817880
(
-
02
)
Trer
2
9.1859622
(
-
09
)
0.0000000
(
+
00
)
0.3967836
(
-
09
)
2.9538120
(
-
08
)
Iterations
8
6
5
4
Quotient
0.6348280
0.6493334
1.096336
1.018248Table 5
Results on theL-shaped region without a posteriori estimates as a stopping criteria.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
3.5915386
(
-
02
)
16.3642002
(
-
03
)
9.0421132
(
-
03
)
6.3366068
(
-
03
)
E
r
2
2.19128497
(
-
02
)
19.3708866
(
-
03
)
12.4732488
(
-
03
)
9.8440048
(
-
03
)
Trer
1
4.50217996
(
-
02
)
0.2008398
(
-
02
)
4.3794696
(
-
02
)
2.9635760
(
-
02
)
Trer
2
3.4397602
(
-
02
)
4.3355510
(
-
02
)
2.7935672
(
-
03
)
5.9076240
(
-
03
)
Iterations
10
8
6
6
Quotient
0.5443332
0.6562642
0.8380865
1.008344Table 6
Results on theL-shaped region with a posteriori estimates as a stopping criteria.
h
1
/
8
1
/
16
1
/
32
1
/
64
E
r
1
2.5815386
(
-
02
)
14.3642002
(
-
03
)
9.0421132
(
-
03
)
4.3366122
(
-
03
)
E
r
2
2.19128487
(
-
02
)
12.3708866
(
-
03
)
12.4732488
(
-
03
)
6.8440223
(
-
03
)
Trer
1
4.60217446
(
-
02
)
0.4008398
(
-
02
)
4.3794696
(
-
02
)
4.8635660
(
-
03
)
Trer
2
3.3397402
(
-
02
)
2.3355510
(
-
02
)
2.7935672
(
-
03
)
6.8076220
(
-
03
)
Iterations
4
5
6
7
Quotient
0.6009663
0.9170200
0.8380865
0.9579689


## 3.2. Concluding Remarks

(1)
From Theorem3.2 we see that the subdomain errors depend on the differences of the latter on the interface, on the data f, and on the approximate solution uh. The first two terms of (3.13) are the contribution to the a posteriori error from the finite element discretization whereas the last term is the domain decomposition error. Numerical results indicate that the finite element part of the error estimate dominates the domain decomposition part.
(2)
As far as the asymptotic exactness of the estimator based on (3.13) is concerned, Ainsworth and Oden in [12, Theorem  3.5] gave a necessary condition for this property to hold. This condition says that the local residuals in (3.13) should be computed exactly.
(3)
The a posteriori error analysis for a second order boundary value problem or Stokes problem is possible. Also, similar results can be obtained in the general case of more than two subdomains.



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*Source: 947085-2012-12-17.xml*