AN ASYMPTOTIC BEHAVIOR AND A POSTERIORI ERROR ESTIMATES FOR THE GENERALIZED SCHWARTZ METHOD OF ADVECTION-DIFFUSION EQUATION?

Salah BOULAARAS

Department of Mathematics,College of Sciences and Arts,Al-Ras,Qassim University,Kingdom of Saudi Arabia

Laboratory of Fundamental and Applied Mathematics of Oran(LMFAO),University of Oran 1,Ahmed Benbella,Oran,Algeria

E-mail:saleh boulaares@yahoo.fr;S.Boularas@qu.edu.sa

Mohammed Said TOUATI BRAHIM Smail BOUZENADA Abderrahmane ZARAI

Department of Mathematics and Computer Science,Larbi Tebessi University,12002 Tebessa,Algeria

E-mail:touatibrahimsaid39@yahoo.com;bouzenadas@gmail.com;zaraiabdoo@yahoo.fr

Abstract In this paper,a posteriori error estimates for the generalized Schwartz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are proved by using the Euler time scheme combined with Galerkin spatial method.Furthermore,an asymptotic behavior in Sobolev norm is deduced using Benssoussan-Lions’algorithm.Finally,the results of some numerical experiments are presented to support the theory.

Key words a posteriori error estimates;GODDM;advection-diffusion;Galerkin method;Benssoussan-Lions’algorithm

1 Introduction

We consider the following advection diffusion equation.

where D is a diffusion coefficient satisfies

Σ is a set in RN×R defined as Σ = ? ×[0,T]with T<+∞ ,and ? is a smooth bounded domain of RNwith boundary Γ and the right hand side f is a regular function satisfies

For equation(1.1)one has the following weak formulation: find u ∈

where

The symbol(·,·)?signifies the inner product in L2(?)and(·,·)Γindicates the inner product of L2(Γ).

The alternative method of Schwarz can be applied to resolve elliptic boundary value problems on domains that consist of two or more overlapping subdomains.Herman Amandus Schwarz created it in 1890.Scientists used this method to resolve stationary or evolutionary boundary value problems on domains which consist of two or more overlapping subdomains(see[1–9,18–27]).The resolution of these problems is approximated by an infinite sequence of functions derived from the solution of a sequence of elliptic or parabolic boundary value problems in each of the subdomains.A wide analysis of Schwarz method for nonlinear boundary value problems was presented in(see[14–16,22]).Moreover,the effectiveness of Schwarz methods for these problems,mainly in fluid mechanics,was proved in lot of papers,see[17,23–27]and in[24],a priori errors estimates in the elliptic case were given,this results in a weak formulation of Schwarz’s classical method is resulted.Then,in[23],geometry related convergence results were given and a fast version of the generalized overlapping domain decomposition method(GODDM)is presented.Furthermore,in[16],the simple rectangular or circular geometries convergence was analysed.

Lately in[18,19],the Schwarz method of the elliptic equations,for extremely heterogeneous media,was applied.The method uses current optimized boundary conditions particularly designed to take in consideration the heterogeneity between the subdomains on the boundaries.In general,the a priori estimate for elliptic equations is not desirable to evaluate the quality of the approximate solutions on subdomains,since it basically relied on the exact solution itself,which is unknown.

Error analysis in uniform norm for overlapping nonmatching grids methods for stationary problems are studied(see[15–17]).To define the goal of this work,we proceed as in[4].More precisely,we adopt an approach that links geometrical convergence results.Then,we proved the overlapping domain decomposition method combined with a finite element approximation for elliptic free boundary problems related to impulse control problem with respect to the mixed boundary conditions for Laplace operator,where a uniform norm of an overlapping Schwarz method on nonmatching grids has been used.Moreover,[5],we extended the last results to the parabolic case with respect to the boundary conditions using the finite difference scheme combined with a Galerkin methods,we prove that the discretization on every subdomain converges in uniform norm.Then,a result of asymptotic behavior in Sobolev norm is given.The same study was conducted in the evolutionary free boundary problems in[1,6,7].

In order to complete the previous works,we apply in this paper the same study for the hyperbolic equations,which is a very active area of research.The main distinguishing feature of the initial boundary value problems is the fact that perturbations propagate with finite speed.Another characterizing aspect is that the boundary treatment is not as simple as that for elliptic or parabolic equations.According to the sign of the equation coefficients,the in flow and out flow boundary regions become determined,from case to case,where boundary conditions have to be prescribed.The situation becomes more complex for systems of hyperbolic equations,where the boundary treatment must undergo a local characteristic analysis.If not implemented conveniently,the numerical realization of hyperbolic boundary conditions is a potential source of spurious instabilities.Hyperbolic problems also feature the presence of discontinuous solutions,arising in nonlinear equations,as well as in linear problems with discontinuous initial data.In order to account for non smooth solutions,the problem is not set in differential form but rather in a weak form in which spatial derivatives are no longer acting on the solution but only on smooth test functions.Simply put,both theta scheme with Galerkin methods combined with GODDM are derived directly from the differential form of the equation.

In this paper,we prove an a posteriori error estimates for the generalized overlapping domain decomposition method with Dirichlet boundary conditions on the boundaries for the discrete solutions on subdomains for a class of advection-diffusion equations with linear source terms using Euler time scheme combined with a finite element spatial approximation,similar to that in[5],which investigated Laplace equation and parabolic free boundary problems which are mentioned above.Moreover,an asymptotic behavior in Sobolev norm is deduced using Benssoussan–Lions’algorithms(see[6]).

The outline of the paper is as follows:in Section 2,we introduce some necessary notations,definitions then we give the weak formulation of a class of advection-diffusion equations.In Sections 3 and 4,a posteriori error estimate for both continuous and discrete cases are proposed for the convergence of the discrete solution using the theta time scheme combined with a Galerkin method on subdomains.

2 The Generalized Overlapping Domain Decomposition of Advection-Diffusion Equations

Let ? be a bounded domain in R2with a piecewise C1,1boundary ??.We consider a simple decomposition of ? into two overlapping subdomaine ?1and ?2such that

We need the spaces

and

which is a subspace of

equipped with the norm

and set

as the part of ??iinside denote bythe outward normal vector on Γij.

We discretize problem(1.3)with respect to time using the Euler time scheme,then we have

Problem(2.7)can be reformulated as the following coercive system of elliptic variational equation

We define the continuous counterparts of Schwarz sequences for problem(1.3),respectivelysolution of

The weak formulation of problems(2.10)is to find

2.1 The Space-Continuous for Generalized Overlapping Domain Decomposition

According to(2.8),(2.10)and(2.11),we can write the following problem,respectively byfor m=0,1,2,···such that

where niis the exterior normal to ?iand zi∈ L∞(??i??),zi>0 is a real parameter,i=1,2 to accelerate the convergence,this is accomplished by

3 A Posteriori Error Estimate in the Continuous Case

We need to introduce two auxiliary problems defined on nonoverlapping subdomains of ?nonoverlapping subdomains of ?.This idea allows us to obtain the a posteriori error estimate by following the steps of Otto and Lube[24].We get these auxiliary problems by coupling each one of problems(2.12)and(2.13)with a different problem in a nonoverlapping way over ?.

To define these auxiliary problems we need to split the domain ? into two sets of disjoint subdomains(?1,?3)and(?2,?4)such that

Lube and Otto[24]proved there exists a constant C>0 such that for the errorn∈N and i=1,2 holds

Applying Green formula with the new boundary conditions of generalized Schwarz alternating method defined in(2.12),we obtain

thus problem(2.12)is equivalent to: findsuch that

Multiply the first equation by v1∈V1and integration by part and by putting

then(2.12)can be reformulated as the following system of elliptic variational equations,using the Green formula

and(3.3),

On the other hand by setting

Using(3.11),we have

We can write the following algorithm which is equivalent to the auxiliary nonoverlapping problem(3.9)and(3.10).We need this algorithm to get an a posteriori error estimate for the presented problem.

3.1 Algorithm

decomposition algorithm.

Step 1 k=0.

Step 4 Compute

Step 6 Set m=m+1 go to Step 3.

Step 7 Set k=k+1 go to Step 2.

Proof First,we have

Since b(·,·)is a coercive bilinear form,then

Therefore,

Lemma 3.2 By letting C be a generic constant which has different values at different places,we get for i,j=1,3,i 6=j

Proof By using Lemma 3.1 and the fact that the trace mapping Tri:Vi?→ Wiand its is inverse are continuous,we obtain i,j=1,3,i 6=j

For the second estimate,we have

Proposition 3.3 For the sequencesolutions of(3.9)and(3.10),we have the following a posteriori error estimation

Proof From(3.13)and(3.15),we have

Similarly,we define another nonoverlapping auxiliary problems over(?2,?4).We get the same result

Proposition 3.4 For the sequencesWe get the the similar following a posteriori error estimation

Proof The proof is very similar to the proof of Proposition 3.3. ?

Proof By using two nonoverlapping auxiliary problems over(?1,?3)and(?2,?4)resp.From the previous two propositions we have

Thus,it can be deduced

4 A Posteriori Error Estimate in the Discrete Case

In this section,we consider the discretization of problem(2.9).Let τhbe a decomposition of ? into open triangles,compatible with the discretization.A triangle is denote by K wich its diameter by hK,an edge by E,and the length of the edge by hEand Vh?H10is the subspace of continuous functions which vanish over??.We have

i,hon Γiwhich vanish at the end points of Γi(i=1,2).

4.1 The Space Discretization

Let ? be decomposed into triangles and τhdenote the set of all those elements h>0 is the mesh size.We assume that the family τhis regular and quasi-uniform.We consider the usual basis of affine functions ?ii={1,···,m(h)}defined by ?i(Mj)= δijwhere Mjis a summit of the considered triangulation.

We discretize in space,i.e.,that we approach the spaceby a space discretization of finite dimensional Vh?In a second step,we discretize the problem with respect to time using the Euler scheme.Therefore,we search a sequence of elements∈Vhwhich approaches un(tn),tn=n?t,with initial data=u0h.Now,we apply Euler scheme on the following to the semi-discrete approximation for vh∈Vh.

In similar manner to that of the previous section,we introduce two auxiliary problems,we define for(?1,?3),

and for(?2,?4),

Proposition 4.1 We can obtain the discrete counterparts of Propositions 3.3 and 3.4 by doing almost the same analysis as in section above(i.e.,passing from continuous spaces to discrete subspaces and from continuous sequences to discrete ones).Therefore,

and so we get the discrete case of Theorem 3.5,(4.6)and(4.7),

5 An Asymptotic Behavior for the Problem

5.1 A Fixed Point Mapping Associated with Discrete Problem

We define for i=1,2,3,4 the following mapping

5.2 An Iterative Discrete Algorithm

with gi,0is a linear and a regular function.

Now,we give the following discrete algorithm

Proof We note that

Setting

On the other hand,we have

It is very clear that if Fi(wi)≧Fi(wi)thenThus

But the role of wiandwiare symmetrical,thus we have a similar prof

Proposition 5.2 Under the previous hypotheses and notations,we have the following estimate of convergent

where u∞,m+1is an asymptotic continuous solution and ui,h0is a solution of(5.3).

ProofWe have

and also we have

Theorem 5.3 Under the previous hypotheses,notations,results,we have for i=1,···,4,k=1,···,n,m=1,2,···,

Proof Using Proposition 4.1 and 5.2,it can be easily deduced(5.6)using the triangulation inequality.

6 Numerical Example

In this section,we give a simple numerical example.Consider the following advection diffusion equation

where ?=]0.1[,u(0,x)=0,T=1 and

The exact solution of the problem is

For the finite element approximation,we take uniform partition and linear conforming element.For the domain decomposition,we use the following decompositions ?1=]0,0.55[,?2=]0.45,1[.

We compute the bilinear semi-implicit scheme combined with Galerkin solution in ? and and we apply the generalized overlappingdomain decomposition method to compute the bilinear sequences(s=1,2)to be able to look at the behavior of the constant C,where the space stepsa nd the time steps of discetization

The generalized overlapping domain decomposition method,with α1= α2=0.55,converges.The iterations have been stopped when the relative error between two subsequent iterates is less than 10?6,we get the following results

Finally,we can deduce the asymptotic behavior

as the following result

In the tables above,we also see that the iteration number is roughly related to h and?t,and the order of convergence is in a good agreement with our estimates(5.6).

7 Conclusion

In this paper,a posteriori error estimates for the generalized Shwarz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are derived using Euler time scheme combined with Galerkin spatial method.Furthermore,a result of asymptotic behavior in uniform norm is deduced by using Benssoussan-Lions’algorithm.In the next work.The geometrical convergence of both the continuous and discrete corresponding Schwarz algorithms error estimate of a new class of non linear elliptic PDEs will be proved and the results of some numerical experiments will be presented to support the theory.

AcknowledgementsThe first author gratefully acknowledge Qassim University in Kingdom of Saudi Arabia and this presented work is in memory of his father(1910–1999)Mr.Mahmoud ben Mouha Boulaaras.