16667827

Domain decomposition techniques for

reaction±di€usion equations in

two-dimensional regions with re-entrant corners

J.I. Ramos

_{, E. Soler}

a_{Departamento de Lenguajes y Ciencias de la Computacion, E.T.S. Ingenieros Industriales,} Universidad de Malaga, Plaza El Ejido, s/n 29013, Malaga, Spain

b_{Departamento de Lenguajes y Ciencias de la Computacion, E.T.S. Ingenieros Informatica,} Universidad de Malaga, Campus Universitario de Teatinos, 29071 Malaga, Spain

Abstract

A system of two non-linear reaction±di€usion equations is solved numerically by means of linearized h-methods and both overlapping and non-overlapping domain decomposition techniques in two-dimensional regions with re-entrant corners. Two numerical methods based on either approximate factorization (AF) or the bi-conjugate-gradient-stabilized (BiCGstab) technique are employed. A study of the e€ects of the number of overlapping grid lines on both the accuracy and numerical eciency is presented. For non-overlapping domain decomposition techniques, the unknown values at the common interface between adjacent subdomains have been updated by means of Dirichlet, Neumann and Robin couplings, and combinations thereof. It is shown that non-overlapping domain techniques are less accurate than overlapping ones for do-mains with re-entrant corners because the interfaces between adjacent subdodo-mains are evaluated by imposing continuity of the unknowns and their normal derivatives there, and, therefore, the partial di€erential equations are not solved at the interfaces between adjacent subdomains. Nevertheless, the accuracy of these techniques increases as the grid spacing is decreased, although they still exhibit large errors near the re-entrant corners. Ó 2001 Elsevier Science Inc. All rights reserved.

Keywords:Domain decomposition methods; Overlapping domains; Approximate factorization; Reaction±di€usion equations; Re-entrant corners

www.elsevier.com/locate/amc

*_{Corresponding author.}

E-mail address:[email protected] (J.I. Ramos).

0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S0096-3003(99)00214-3

1. Introduction

Most of the problems encountered in physics and engineering are charac-terized by irregular domains, interactions between solid and ¯uid systems, several phases, etc. In addition, these problems often have di€erent spatial and temporal scales that make them sti€ in both space and time. In order to ac-curately solve these problems, one must account for the geometry of each domain and the physics of the di€erent phenomena that have to be modelled, and then employ the numerical techniques more accurate and ecient for each domain so that regions with steep gradients, e.g., boundary or internal layers, are appropriately resolved. Moreover, one must take advantage of the emer-gence of parallel computers and their potential for the numerical solution of large-scale or realistic physics and engineering problems.

Domain decomposition methods have recently been the subject of increasing interest due to their ¯exibility for the solution of the linear and non-linear al-gebraic equations that arise upon the discretization of partial di€erential equations, although their origin may be traced back to the work of Schwarz [1]. As pointed out by Smith et al. [1], domain decomposition may mean: (1) the process of distributing the data among the di€erent processors in a distributed memory computer, (2) the separation of the physical domain into di€erent re-gions, and (3) the division of the solution of a large system of linear algebraic equations into smaller systems whose solution can be readily obtained by means of a preconditioner. The domain decomposition techniques considered in this paper belong to the second category above; however, although the problems considered here exhibit steep moving gradients and moderately steep boundary layers, the same equations govern the physical phenomena throughout the whole domain and equally spaced grids are employed in each subdomain.

Domain decomposition methods may also be classi®ed as those which use overlapping subdomains (sometimes referred to as Schwarz's methods) and those which use non-overlapping subdomains (also referred to as substructuring or Schur's complement methods). In this paper, both overlapping and non-overlapping domain decomposition techniques are considered. Overlapping domain decomposition methods have been employed by He [2] to solve the non-linear Schrodinger equation in one dimension; he used a method of lines tech-

nique in space or Rothe's method, i.e., he discretized the time variable, and provided a convergence proof when the interfaces between the two overlapping subdomains are updated by means of Robin's or absorbing boundary condi-tions. Tai [3] provides a convergence proof for an abstract parabolic equation that is solved by means of a two-level ®nite element method and domain de-composition with only one-element overlap. Mathew et al. [4] developed do-main decomposition counterparts of the classical alternating direction and fractional steps methods for solving the large linear systems of equations arising from the implicit time discretization of parabolic equations, and showed that

the errors incurred by their methods deteriorate with smaller overlaps amongst the subdomains unless the time step is decreased. Hebeker and Kuznetsov [5] employed domain decomposition for either an iterative preconditioning of each global implicit time-step or a non-iterative blockwise implicit time-stepping extrapolation for both convection and convection±di€usion equations in two dimensions and showed that the overlap required decreases in time.

Nataf and Nier [6] have used absorbing boundary conditions at the inter-faces between overlapping and non-overlapping subdomains, an additive Schwarz's method that updates all the subdomains simultaneously and ¯ow directed sweeps, and showed that the convergence rate of domain decompo-sition techniques for a two-dimensional linear problem depends on the algo-rithm. Cai [7] considered the solution of linear systems of algebraic equations that arise from the ®nite element discretization of parabolic problems by means of an additive Schwarz's method and a generalized-minimum-residual (GMRES) technique, and evaluated these methods in terms of the time step and amount of overlap for strongly elliptic operators. Benamou [8] has em-ployed a non-overlapping domain decomposition technique with Robin transmission coecients and ®nite element methods for the solution of an el-liptic partial di€erential equation, and shown that the convergence of non-overlapping decomposition depends on the number of subdomains.

Mandel [9] has employed a Neumann±Neumann algorithm as a precondi-tioner for the iterative solution of ®nite element discretizations of elliptic problems; this algorithm su€ers from an inconsistent singular problem, and its convergence deteriorates as the number of subdomains is increased. Rice et al. [10] have also considered non-overlapping domain decomposition for linear, elliptic problems which involves the solution of partial di€erential equations subject to Dirichlet and Neumann boundary conditions in each subdomain coupled with smoothing operations on the interfaces. Yang [11] has developed a parallel iterative non-overlapping domain decomposition method for elliptic problems; his iterative procedure requires that Dirichlet data be passed to one subdomain from the previous iteration, while the other subdomain problem requires that Neumann data be passed to it, uses a Dirichlet±Neumann and a Neumann±Dirichlet iterative procedure at the interfaces between subdomains, and is not valid for domains with corners.

Preconditioning techniques based on domain decomposition or substruc-turing have also been developed for the solution of the ®nite element discret-izations of elliptic partial di€erential equations in non-overlapping subdomains, and analytical estimates may be provided for the convergence of these preconditioned iterative procedures [12,13].

Most of the applications of domain decomposition methods have dealt with linear, elliptic partial di€erential equations because of the large amount of rigorous mathematical results for these equations. When domain decomposi-tion methods are employed for, say, parabolic equadecomposi-tions, a method of lines in

space or discretization of the time variable has been performed in order to obtain an elliptic problem at each time step, e.g., [2,4,7]. Moreover, most of these studies have dealt with rectangular domains which are amenable to the method of separation of variables and, therefore, proofs of convergence for overlapping or non-overlapping subdomains can be readily obtained.

In this paper, we consider a system of two non-linear, reaction±di€usion equations in two-dimensional domains and apply both overlapping and non-overlapping domain decomposition methods. The two geometries considered are a rectangle and a domain with two re-entrant corners (Fig. 1), while the discretization of the governing equations is performed in both space and time by means of a linearizedh-method which provides linear algebraic equations at each time step; however, iterative procedures are employed to determine the solution in each subdomain because of the interface between subdomains. Moreover, two numerical techniques are employed to obtain the solution of the linear algebraic equations: approximate factorization and the BiCGstab method. The non-overlapping domain decomposition methods employed in this study include Dirichlet, Neumann and Robin iterations at the interfaces, and combinations thereof. Special attention is paid to the interface between subdomains when non-overlapping methods are employed.

2. The physical problem and its discretization

The non-linear, partial di€erential equations considered in this paper are

oU

ot ˆ o2_U

ox2 ‡

o2_U

oy2 ‡F…U†; …1†

Fig. 1. Schematic of the domains considered in the paper. The rectangular domain is the union of the subdomains 1, 2, 3 and 4. The domain with re-entrant corners is the union of the subdomains 1 and 2. Subdomains 3 and 4 are sometimes used to de®ne a regular computational domain when analyzing domains with re-entrant corners.

where

Uˆ …u;v†T; Fˆ …ÿuv;uvÿkv†T; …2†

uˆ1 andvˆ0 on the boundaries,t is time,xand ydenote Cartesian coor-dinates,k is a constant (in this paper,kˆ0:5), and the superscript T denotes transpose. Eq. (1) corresponds to a two-dimensional system of reaction±dif-fusion equations which has been previously studied by the senior author [14] without domain decomposition by means of a variety of standard and non-standard or exponential, ®nite di€erence methods.

2.1. Fully linearized, implicith-method

Eq. (1) was discretized by means of an implicit, linearized, h-method in an equally spaced grid where the non-linear term Fn‡1

i;j was approximated by

means of its Taylor polynomial of ®rst degree around…tn_;xi;y

j†to obtain the

following system of linear algebraic equations

DUi;j

k ˆ

1

Dx2 hd2xDUi;j h

‡d2

xUni;j

i ‡Fn

i;j

‡_D1_y2 hd2

yDUi;j

h

‡d2

yUni;j

i ‡hJn

i;jDUi;j; …3†

where

DUi;jˆUin;‡1j ÿUni;j; Jni;jˆ_oo_UF…tn;xi;yj†;

d2

xUi;jˆUi‡1;jÿ2Ui;j‡Uiÿ1;j; d2yUi;jˆUi;j‡1ÿ2Ui;j‡Ui;jÿ1;

…4†

iandjdenotexiandyj, respectively,tndenotes thenth time level,kis the time

step,DxandDy represent the grid spacing in thex- and y-directions, respec-tively, and 0<h61 is the implicitness parameter. In this paper, hˆ1

2, i.e.,

second-order accurate ®nite di€erence methods are employed.

Eq. (3) yields a large system of linear algebraic equations, couples all the dependent variables at tn‡1_{, and was solved by means of the BiCGstab}

algo-rithm [15]. This two-dimensional system may be reduced to a sequence of one-dimensional equations by means of the factorization techniques presented in Section 2.2.

The fully linearized, implicith-method described above is here referred to as

FLand was used in all the domain decomposition methods presented in this paper.

2.2. Approximate factorization method

The two-dimensional ®nite di€erence operators which appear in Eq. (3) may be written in the following approximate factorization form:

I

ÿ_Dk_xh₂d2

xIÿkdhJ

I

ÿ_Dk_yh₂d2

yIÿkhJ

DUˆkRHS; …5†

where the subscript …i;j†denoting the grid point has been eliminated for the sake of conciseness,Idenotes the unit or identity matrix,

d‡ˆ1; …6†

L…DU† ˆkRHSˆ_Dk_x2d2

xUn‡_Dk_y2dy2Un‡kFn; …7†

and the approximate factorization errors, i.e.,EAF, which should be added to the right-hand side of Eq. (5) but have been neglected can be written as [14]

EAFˆk2_h2 1

Dx2_Dy2d2xd2yI

‡_Dx2d2

xJ‡_Dd_y2Jd2yI‡dJ2

DU: …8†

Eq. (5) can also be written as

Lx…DU† ˆ I

ÿ_Dk_xh₂d2

xIÿkdhJ

DU_{ˆkRHS; …9†}

Ly…DU† ˆ I

ÿ kh

Dy2d2yIÿkhJ

DUˆDU_{; …10†}

which represent linear systems of one-dimensional equations in thex- andy -directions, respectively, and can be easily solved by means of a block tridiag-onal matrix algorithm, i.e., the method of Thomas. Note thatEAF are O…k2_†

which is the accuracy of the implicit, linearized method forhˆ1

2; this value ofh

is the one employed in the calculations reported in this paper.

The approximate factorization, implicit h-method described above is here referred to asAFand was used to assess the errors incurred by the approximate factorization.

3. Domain decomposition techniques

As stated in the Introduction, both overlapping and non-overlapping do-main decomposition methods have been used to solve the linear algebraic equations resulting from the fully linearized, implicit method.

3.1. Non-overlapping domain decomposition techniques

The non-overlapping domain decomposition techniques presented here are based on the subdomains X1ˆ …ÿ206x620;ÿ206y620† and X2ˆ …206x660;ÿ206y620†for the rectangular domain (cf. Fig. 1) so that the interface between both subdomains is X1\X2ˆRˆ …xˆ20;ÿ206y620†. For the domain with corners, X1ˆ …ÿ206x620;ÿ206y620† and

X2ˆ …206x640;ÿ106y610†, so that the interface between both subdo-mains is X1\X2ˆRˆ …xˆ20;ÿ106y610†; note that this interface joins the two re-entrant corners shown in Fig. 1. The solution in each subdomain was obtained by solving the corresponding linear, algebraic equations subject to appropriate boundary conditions at the common interface R; these boundary conditions are such that both the solution and their normal deriv-ative at the interface must be continuous, but their implementation results in di€erent methods as discussed below.

Dirichlet method. In this method, the algebraic equations are solved

itera-tively at the interior points of the two non-overlapping subdomainsX1andX2

using values ofUatR; initially, these values correspond to those of the pre-vious time step, but they are updated in successive iterations by imposing that the normal derivative of the solution at the interface be continuous there. Thus, ifuandvdenote the values ofUinX1andX2, respectively, the values ofUat the interface must be determined by imposing that

ou

oxˆ ov

ox; …11†

which may be discretized by means of ®rst-order accurate ®nite di€erences as

p_LuRÿuRÿ1

Dx1 ˆpR

vR‡1ÿvR

Dx2 ; …12†

whereURˆuR ˆvR andRÿ1 and R‡1 denote the grid lines closest to the interface on its left and right, respectively.

Instead of using ®rst-order accurate ®nite di€erences for the derivatives at the interface which degrade the accuracy of the second-order ®nite di€erence methods employed at the interior points, second-order ®nite di€erences for these derivatives result in the following expression:

pL3uRÿ4₂uRÿ1_Dx ‡uRÿ2

1 ˆpR

4vR‡1ÿvR‡2ÿ3vR

2Dx2 ; …13†

which yields

DUR ˆ1₆…4DvR‡1ÿDvR‡2‡4DuRÿ1ÿDuRÿ2† …14†

In both cases, one may determineUR in terms of the values ofUat interior points of the two non-overlapping subdomains once the solution is obtained in both, and use this new value in the next iteration. This iterative procedure was repeated untilkuRÿvRk610ÿ10_{. For the sake of convenience the method}

de-scribed in this paragraph is here referred to as D1 and D2 depending on whether ®rst- or second-order accurate formulae, respectively, are employed to determine the normal derivative at the interface between boundaries.

Neumann method. This method also enforces the continuity of both the

solution and its normal derivative at the boundary as the Dirichlet method, but instead of using uRˆvR when solving for both subdomains, it employs

ouR=oxˆovR=oxand determinesUR using either the ®rst- or the second-order expression for the ®rst-order derivatives at the boundary given by Eq. (12) or Eq. (13), respectively. Since the normal derivatives to the interface are con-tinuous, one may write the values of the unknowns at this interface as functions of these derivatives and replace their values in the ®nite di€erence equations derived in the previous section. In this manner, one can obtain the solution at all interior points ofX1andX2, and determineuR andvRfrom Eq. (12) or Eq. (13). Since during the iterative procedureuR6ˆvR, an updated value ofUR was determined asURˆluR‡ …1ÿl†vR with 06l61 and used to determine the normal derivatives to the interface for the next iteration. This iterative pro-cedure was repeated until the convergence criterion previously speci®ed was satis®ed. In the calculations presented here,lˆ0:5 and, for the sake of con-venience the method described in this paragraph is here referred to asNo1and

No2 depending on whether ®rst- or second-order accurate formulae, respec-tively, are employed to determine the normal derivative at the interface be-tween boundaries.

It must be noted that Eqs. (12) and (13) can also be written, respectively, as

pˆvR‡1ÿuRÿ1

2Dx ; …15†

pˆ4vR‡1ÿ4uRÿ1_4D‡_xuRÿ2ÿvR‡2; …16†

if DxˆDx1ˆDx2 which are based on the continuity of the solution and its

normal derivative at the interface between the subdomains.

Instead of determining the interface values once those at the interior points have been calculated, one may include them in both subdomains as a function of the normal derivative to the boundary (cf. Eq. (12) or Eq. (13)) and thus obtain a larger system of equations than the one forNo1 andNo2. Once the solution of this system is obtained in each subdomain,p is smoothed out for the next iteration as described in the previous paragraph, and the iterative procedure is repeated until the convergence criterion described previously is satis®ed. This method is here referred to asN1andN2depending on whether

®rst- or second-order accurate formulae are employed to determine the normal derivative at the interface between subdomains and requires a simple modi®-cation related to the product of a matrix times a vector in the BiCGstab method when the normal derivative to the interface is evaluated by means of second-order di€erence formulae because, in this case,uR depends onuRÿ1and

uRÿ2, whilevR depends onvR‡1 andvR‡2, i.e., a new diagonal has to be intro-duced in the coecients matrix.

Robin method. In this method, the continuity of the function and of its

normal derivative at the interface are imposed through the use of Robin conditions or absorption or transmission coecients as

ou

ox‡kuˆ ov

ox‡kv on R; …17†

wherek is a constant, the derivatives may be evaluated by means of ®rst- or second-order accurate formulae, and the solution at the interface may be de-termined either once that at interior points is known or together with that at interior points.

If the order derivatives in Eq. (17) are discretized by means of ®rst-order ®nite di€erences, then

Duk‡1

R ˆ_1‡1_kDx DukRÿ1‡1

ÿ

‡Dxbk

R

; …18†

Dvk‡1

R ˆ_1ÿ1_kDx DvkR‡1‡1

ÿ

ÿDxbk

L

; …19†

wherekdenotes thekth iteration within the time step and

bRˆkDvR‡_D1_x…DvR‡1ÿDvR†; …20†

bLˆkDuR‡_D1_x…DuRÿDuRÿ1†: …21†

Eqs. (18) and (19) can be used to eliminateDuR and DvR from the X1andX2

subdomains, respectively, and the resulting equations can be solved at the in-terior points. Then, Eqs. (18) and (19) can be used to determine the solution at the interface; however, since, during the iterative procedure,uR6ˆvR, updated values of UR and b were determined as URˆluR‡ …1ÿl†vR and bˆlbL

‡…1ÿl†bR with 06l61 and used to determine the normal derivatives to the interface for the next iteration. This iterative procedure was repeated until the convergence criterion speci®ed previously was satis®ed. In the calculations presented here,lˆ0:5 and, for the sake of convenience, the method described in this paragraph is here referred to as Ro1 and Ro2depending on whether

®rst- or second-order accurate formulae are employed to determine the normal derivative at the interface between boundaries.

Instead of determining the interface values once those at the interior points have been calculated, one may include in both subdomains the interface values as a function ofbL andbR (cf. Eq. (18) or Eq. (19)) and thus obtain a larger system of equations than the one for Ro1andRo2. Once the solution is ob-tained in each subdomain,bis smoothed out for the next iteration as described in the previous paragraph. This method is here referred to asR1andR2 de-pending on whether ®rst- or second-order accurate formulae are employed to determine the normal derivative at the interface between boundaries and re-quires that the BiCGstab method be modi®ed slightly when the normal de-rivative to the interface is evaluated by means of second-order di€erence formulae as stated before for the Neumann method.

Dirichlet±Neumann method.This method is a combination of the ones

pre-viously described and consists of the following steps in each iteration. Using guessed values at the interface, the solution at interior points of the subdo-mains is obtained and, by using the continuity of the ®rst-order derivative at the interface and a second-order accurate formula, one may calculate, as in the Dirichlet method,UR (cf. Eq. (14)). This value anduRÿ1, uRÿ2,vR‡1 and vR‡2

may then be used to derive a second-order accurate approximation to the derivative normal to the boundary in both subdomains (cf. Eq. (13)). The arithmetic average of these derivatives may then be used to solve the resulting Neumann problem which yields the interface values through Eq. (14). This procedure thus consists of a Dirichlet and a Neumann cycle per iteration and is repeated as many times as necessary until the convergence criterion speci®ed previously is satis®ed. For the sake of convenience this method is here referred to asDN1andDN2 if the ®rst-order derivatives at the interface are evaluated by means of ®rst- and second-order di€erences, respectively.

Neumann±Dirichlet method. This method is similar to the

Dirichlet±Neu-mann one but it ®rst employs a NeuDirichlet±Neu-mann cycle whereby the normal derivatives to the interface are speci®ed, to determine the solution at the interface in both subdomains. These values are then used to determine an average interfacial value as that given by Eq. (14), and the solution is then obtained by means of the Dirichlet method. The solution of the Dirichlet method is used to determine once again the interfacial value with Eq. (14). This iterative procedure which involves a Dirichlet and a Neumann cycle at each iteration is repeated as many times as necessary until the convergence criterion speci®ed previously is satis-®ed. For the sake of convenience this method is here referred to asND1 and

ND2 if the ®rst-order derivatives at the interface are evaluated by means of ®rst- and second-order di€erences, respectively.

Yang's method.This method was developed by Yang [11] and consists of two

steps per iteration. In the ®rst step, one subdomain is solved with Dirichlet boundary conditions at the interface, while the other one is solved with

Neu-mann conditions. In the second step, the subdomain which was solved with Dirichlet conditions is solved with Neumann ones, and vice versa. This itera-tive procedure which involves a Dirichlet and a Neumann cycle at each iter-ation is repeated as many times as necessary until the convergence criterion speci®ed previously is satis®ed. For the sake of convenience this method is here referred to as Y1 and Y2 if the ®rst-order derivatives at the interface are evaluated by means of ®rst- and second-order di€erences, respectively, and its convergence criterion is the same as that discussed previously.

3.2. Overlapping domain decomposition techniques

In this technique, we de®ne the subdomains 1 (X1) and 2 (X2) shown in

Fig. 2. Subdomain 1 is a square region which shares with subdomain 2 what this subdomain penetrates in subdomain 1, while subdomain 2 is a rectangular one; both subdomains share at least two vertical grid lines and one of them corresponds to where the geometry of the domain with re-entrant corners changes its section. Let us denote the interface ofX2 withX1 and the vertical line connecting the two re-entrant corners of the domain shown in Fig. 1 byR1

andR2, respectively. Three iterative solution procedures were used in this paper to solve the system of linear algebraic equations with the BiCGstab method as described in the next paragraphs.

Dirichlet method.In this method, the equations inX1att…n‡1†_{could be ®rst}

solved with the values of the unknowns attn _{on R2, and the values thus}

ob-tained onR1were then employed to obtain the solution inX1; this procedure would be analogous to an iterative Gauss±Seidel technique which is not readily parallelizable. In order to develop overlapping domain decomposition methods which may be easily implemented in parallel computer architectures, we have used an iterative Gauss±Jacobi method that was repeated as many times as required to satisfy the user's speci®ed convergence criterion described

Fig. 2. Schematic of the overlapping domain decomposition technique employed in this paper. Domain 1 contains the vertical line that is the boundary between the domains 1 and 2 shown in Fig. 1.

previously. For the sake of convenience, this overlapping technique is here referred to asODand employs a Dirichlet cycle.

Neumann method. This method is analogous to that for non-overlapping

subdomains, except that the interface is solved in both subdomains so that, when the ®rst-order derivative at the boundary between subdomains is ap-proximated by ®rst-order, forward and backward, ®nite di€erences in the left and right subdomains, respectively, the values ofvR‡1anduRÿ1required by the left and right subdomains are replaced by uR‡Dx2p and vRÿDx1p, respec-tively. The solution obtained from the resulting system of equations is then employed to determine the derivative normal to the interface through Eq. (15) which is the arithmetic average of this derivative on the left and right sides of the interface. This method is also iterative and is here referred to asON1 be-cause it employs ®rst-order di€erences for the evaluation of the normal de-rivative to the boundary. The convergence criterion employed here is the same as in previous sections and the two subdomains share two vertical lines.

If instead of using Eqs. (12) and (15), one uses Eqs. (13) and (16) to obtain second-order accurate ®nite di€erences at the interface, the resulting method is referred to asON2and employs forward and backward di€erences to evaluate the ®rst-order derivative at the interface in the subdomains X1 and X2, re-spectively. If the ®rst-order derivatives at the interface were evaluated with backward and forward di€erences in the subdomainsX1andX2, respectively, one would obtain Eq. (15), i.e., theON1 method described previously.

Robin method. This method is analogous to that for non-overlapping

sub-domains, except that the interface is solved in both subdomains so that the discretization of Eq. (17) using forward and backward di€erences in the left and right subdomains yields

DuR‡1ˆDxbR‡ …1ÿkDx†DuR; …22†

DvRÿ1ˆ ÿDxbL‡ …1‡kDx†DvR; …23†

where

bRˆkDvR‡_D1_x…DvRÿDuRÿ1†; …24†

bLˆkDuR‡_D1_x…DvR‡1ÿDuR†: …25†

Substitution of Eqs. (22) and (23) in the algebraic equations corresponding to the interface for the left and right subdomains, allows one to obtain the so-lution at all points including those on the interface. These values can then be used to obtain new values ofbLandbR. These values can then be used to obtain a smooth new value as

bˆk₂…DuR‡DvR† ‡_2D1_x…DvR‡1ÿDvR‡DuRÿDuRÿ1†; …26†

bˆ k₂…DuR‡DvR† ‡_4D1_x…4DvR‡1ÿ3DvR‡2ÿ3DvR†

‡₄1_Dx…3DuRÿ4DuRÿ1‡DuRÿ2†; …27†

for the next iteration, where Eqs. (26) and (27) correspond to ®rst- and second-order accurate di€erences for the ®rst-second-order derivatives, respectively, and the methods resulting from the application of these formulae are here referred to as

OR1andOR2, respectively.

4. Presentation of results

Eq. (1) was solved in both the rectangular domain and that with re-entrant corners (cf. Fig. 1) subject to the following initial condition:

u…x;y;0† ˆ1; v…x;y;0† ˆeÿ…x2_‡y2_†

; …28†

and a time stepkˆ0:01. For the rectangular domain, the meshes in the left and right subdomains were equally spaced and consisted of 130130 grid points each, the error tolerance in the BiCGstab method was set to 10ÿ12_{, and}

calculations were performed untiltˆ80 when the solution was almost steady. The same tolerance was employed for the domain with re-entrant corners, but only a 6666 grid point mesh was used for the smallest (protruding) subdo-main.

In order to assess the accuracy of the overlapping and non-overlapping domain decomposition methods studied in this paper, an implicit, iterativeh -method without linearization was also employed. This -method is here re-ferred to as NR, does not employ domain decomposition, and its results were compared with those obtained with the AF and FL techniques; the results of NR were taken as the basis for comparisons with those of other techniques. In both NR and FL, two additional subdomains were intro-duced, i.e., subdomains 3 and 4, as shown in Fig. 1 in order to obtain a regular geometry; of course, the boundary conditions speci®ed in Section 2 were imposed at the boundaries between subdomains 1 and 3, 2 and 3, 1 and 4, and 2 and 4.

For the sake of convenience, the results for domains with and without re-entrant corners are discussed separately in the next two sections.

4.1. Rectangular domains

Some sample results illustrating the spatial distributions ofu and vat dif-ferent times are shown in Figs. 3±7 which were obtained with meshes of 6666 points for both subdomains in order to ease the visualization. Fig. 3 shows the initial peak ofvin the left subdomain required to start the reaction process. At

tˆ10 (cf. Fig. 4), u exhibits a valley, whereasvincreases from its boundary value and then decreases owing to the reaction process, and no reaction process can be observed in the right subdomain. Attˆ20, the results shown in Fig. 5 indicate that the reaction front has penetrated in the second subdomain; this is the reason why the tables that will be shown in this section about the accuracy of domain decomposition methods start from this time. Fig. 6 illustrates that at

tˆ40 the largest values ofuoccur at the boundaries of both subdomains and in the second subdomain near its right boundary, whereas attˆ80 (cf. Fig. 7), the spatial distributions ofuandvhave reached steady values characterized by four peaks in vnear the four corners of the domain where the reaction front cannot penetrate.

In order to compare the accuracy and eciency of domain decomposition methods for the numerical solution of two-dimensional, reaction±di€usion equations in rectangular domains and in domains with re-entrant corners, we have monitored the values ofu andvat the corners of the boundary between the subdomains 1 and 2 shown in Fig. 1 for the domain with re-entrant corners and at the corners between subdomains 1 and 3 and subdomains 1 and 4 for the rectangular one, and at the midpoint between the corners referred to above for both rectangular regions and domains with re-entrant corners.

No domain decomposition. When the rectangular domain is solved without

domain decomposition, it has been observed that the numerical errors increase until abouttˆ20 and then decrease for both u and vas shown in Table 1. Although not shown here [16], the largest relative errors of theFLmethod are about 810ÿ6_{and 510}ÿ6_{attˆ20 inuandv, respectively, whereas those of}

theAFmethod are about 510ÿ4 _{and 510}ÿ5 _{at tˆ10 inu andv,}

respec-tively, and these errors are due to the time linearization, and both the time linearization and the approximate factorization for the FLand AFmethods, respectively. On the other hand,NR required 37,041 and 23,628 iterations of BICGstab and Newton±Raphson, respectively, and used 7886 s of CPU time in

a DEC Alpha Server 22,164 at 300 MHz, whileFLrequired 19,880 iterations of BICGstab and 3428 s of CPU time, andAF employed 2048 s of CPU time. Therefore, the accuracy of time linearization with or without approximate factorization can be improved by decreasing the time step, and these techniques may be still more ecient thanNR.

Overlapping domain decomposition. The Dirichlet method was solved with 2,

4, 8, 10, 16 and 32 overlapping columns of grid points betweenX1andX2, and

it was observed that the numerical errors became independent of the overlap when the two subdomains shared more than eight vertical grid lines. For this reason, only the errors corresponding to 2 and 8 vertical lines of overlap are presented in Table 2 which clearly illustrates that the errors decrease as the overlap is increased. For an overlap of two vertical lines, the numerical errors ofODare larger than those of methods based on time linearization (cf. Table 1). Table 3 illustrates that the number of iterations of BiCGstab and the CPU time ®rst decrease and then increase as the overlap is increased whereas those of the Newton±Raphson technique decrease but tend to a constant asymptotic value.

Table 2 also shows that the accuracy of overlapping Neumann and Robin methods is nearly the same and is almost independent of the order of dis-cretization of the interface conditions; however, bothOR1 andOR2failed to converge for kP100;OR1and OR2 each yielded results that di€ered in the relative error in, at most, the sixth decimal digit for kˆ0, 1 and 10 when arithmetic averages of the results for the left and right subdomains were used to update the interface at the next iteration; the eciency of bothOR1andOR2

worsened as k was increased, i.e., OR1 required 7409, 9420 and 15,282 s of CPU forkˆ0, 1 and 10, respectively, whereas OR2required 6365, 9740 and 12,087 s of CPU forkˆ0, 1 and 10, respectively, and these results are con-sistent with those shown in Table 3 forON1 andON2.

Non-overlapping domain decomposition. Table 4 shows that the accuracy of

the Dirichlet and Neumann, non-overlapping domain decomposition methods increases as the accuracy of the discretization of the ®rst-order derivative normal to the interface is increased; the accuracy of Dirichlet methods is similar to that of Neumann ones but these methods do not produce identical errors;Ro1andRo2each yielded results that di€ered in the relative error in, at most, the sixth decimal digit forkˆ0, 1, 10, 102_{, 10}3_{and 10}30_{when arithmetic}

averages of the results for the left and right subdomains were used to update the interface at the next iteration;R1andR2each yielded identical results for

kˆ0 and 1, but these methods failed to converge forkP10; and,DN2yielded exactly the same results asY2, and these results, in turn, di€ered in relative error in, at most, the sixth decimal digit from those ofND2.

The non-overlapping domain decomposition methods for rectangular do-mains were found to be less ecient (cf. Table 5) than overlapping ones when these shared 4, 8 or 10 vertical lines, except forDN2,ND2andY2;ND2 was found to be the most ecient technique for rectangular domains; the e-ciency of D, No, Ro and R was found to increase, whereas that of N was found to decrease as the accuracy of the interface conditions was increased; and, the eciency of bothRo1 andRo2 was found to decrease and increase as k was increased from 0 to about 10 and from 10 to 1030_{, respectively.}

However, the CPU times required byRo1withkˆ1 and 10 were 18,193 and 18,465 s, respectively, whereas those required byRo1withkˆ1 and 10 were 11,990 and 14,174 s, respectively; therefore, the eciency of Robin methods depends on both the value ofkand the order of discretization of the interface conditions.

Ta ble 2 Rela tive errors at di€ erent time s in rectang ular do mains with overlap ping doma in decomp osition M ethod Err or in u Err or in v t ˆ 20 t ˆ 40 t ˆ 80 t ˆ 20 t ˆ 40 t ˆ 80 OD a(2) 4.989 2 ( ) 5) 8.573 5 ( ) 5) 1.870 6 ( ) 5) 1.490 3 ( ) 2) 5.006 7 ( ) 3) 1.979 3 ( ) 3) OD b(2) 3.447 0 ( ) 3) 1.609 8 ( ) 4) 6.448 3 ( ) 4) 5.154 1 ( ) 3) 1.191 0 ( ) 2) 4.508 7 ( ) 3) OD a(8) 1.124 7 ( ) 8) 4.936 4 ( ) 8) 3.494 4 ( ) 8) 8.963 9 ( ) 6) 2.807 0 ( ) 8) 4.596 5 ( ) 7) OD b(8) 2.684 6 ( ) 6) 1.431 3 ( ) 7) 9.546 5 ( ) 8) 5.133 7 ( ) 6) 7.394 2 ( ) 6) 9.544 3 ( ) 7) ON 1 a 2.702 4 ( ) 7) 4.134 5 ( ) 8) 4.102 3 ( ) 8) 7.152 4 ( ) 5) 9.110 8 ( ) 6) 7.227 3 ( ) 7) ON 1 b 1.898 1 ( ) 6) 4.645 0 ( ) 6) 6.790 8 ( ) 7) 3.888 6 ( ) 6) 3.481 0 ( ) 5) 4.246 4 ( ) 6) ON 2 a 3.642 0 ( ) 7) 7.172 2 ( ) 8) 4.306 6 ( ) 8) 9.838 9 ( ) 5) 1.309 9 ( ) 5) 1.118 6 ( ) 6) ON 2 b 3.429 7 ( ) 6) 6.247 6 ( ) 6) 8.750 4 ( ) 7) 3.463 4 ( ) 6) 4.893 2 ( ) 5) 5.352 1 ( ) 6) OR1 a ; c 2.702 4 ( ) 7) 4.134 5 ( ) 8) 4.102 3 ( ) 8) 7.152 4 ( ) 5) 9.110 8 ( ) 6) 7.227 3 ( ) 7) OR1 b ; c 1.898 1 ( ) 6) 4.645 0 ( ) 6) 6.791 1 ( ) 7) 3.888 5 ( ) 6) 3.481 0 ( ) 5) 4.246 5 ( ) 6) OR2 a ; c 3.643 4 ( ) 7) 7.183 8 ( ) 8) 4.307 6 ( ) 8) 9.842 5 ( ) 5) 1.310 9 ( ) 5) 1.120 0 ( ) 6) OR2 a ; c 3.434 1 ( ) 6) 6.251 2 ( ) 6) 8.755 5 ( ) 7) 3.450 0 ( ) 6) 4.896 6 ( ) 5) 5.354 8 ( ) 6) * OD ( i ) indica tes overlapp ing Dirichlet metho d for do mains that sha re i vertical line s of grid po ints. a Indica tes closest grid point to the corne rs. b Indica tes midpoin t betw een the corners. c k ˆ 1 : Ta ble 1 Rela tive errors at di€ erent time s in rectang ular do mains witho ut domain deco mpos ition M ethod Error in u Err or in v t ˆ 20 t ˆ 40 t ˆ 80 t ˆ 20 t ˆ 40 t ˆ 80 AF a 1.0033 ( ) 7) 0 0 3.843 3 ( ) 5) 3.4263 ( ) 6) 4.660 4 ( ) 7) AF b 1.1631 ( ) 6) 8.509 7 ( ) 7) 0 8.746 1 ( ) 6) 6.4574 ( ) 6) 1.030 8 ( ) 6) FL a 0 0 0 8.678 4 ( ) 6) 2.8304 ( ) 6) 4.660 4 ( ) 7) FL b 2.6170 ( ) 6) 0 0 5.247 7 ( ) 6) 7.3300 ( ) 6) 1.030 8 ( ) 6) aIndica tes closest grid point to the corne rs. bIndica tes midpoin t betw een the corners.

4.2. Domains with re-entrant corners

Some sample results illustrating the spatial distributions ofu and vat dif-ferent times are shown in Figs. 8±12 which were obtained with meshes of 6666 and 3434 points for both subdomains in order to ease the visualiza-tion. Fig. 8 shows the initial peak ofvin the left subdomain required to start the reaction process. At tˆ10 (cf. Fig. 9), u exhibits a valley, whereas v in-creases from its boundary value and then dein-creases owing to the reaction process, and no reaction process can be observed in the right subdomain. At

tˆ20, the results shown in Fig. 10 indicate that the reaction front has pene-trated in the second subdomain; this is the reason why the tables that will be shown in this section about the accuracy of domain decomposition methods start from this time. Fig. 11 illustrates that at tˆ40 the largest values of u

occur at the boundaries between subdomains and in the second subdomain near its right boundary, whereas, attˆ80 (cf. Fig. 11), the spatial distributions ofuandvhave reached steady values characterized by four peaks invnear the four corners of the subdomain where the reaction front cannot penetrate.

No domain decomposition. When the domain with re-entrant is solved

without domain decomposition, it has been observed that the numerical errors increase until about tˆ20 and then decrease for both u and v as shown in Table 6. The results shown here and those of [16] indicate that the largest relative errors of theFLmethod are about 310ÿ6_{and 710}ÿ6_{attˆ20 inu}

andv, respectively, whereas those of theAF method are about 110ÿ6 _and

210ÿ2_{attˆ20 inuandv, respectively, and these errors are due to the time}

linearization, and both the time linearization and the approximate factoriza-tion for theFLandAFmethods, respectively. On the other hand,NRrequired 33,653 and 21,343 iterations of BiCGstab and Newton±Raphson, respectively,

Table 3

Iterations and CPU time for rectangular domains with overlapping domain decomposition Iterations OD(2) _{OD(4) OD(8) OD(16) OD(32)} BiCGstab 173,501 106,002 95,803 96,299 97,003 Newton 34,849 18,447 15,493 15,469 15,469

CPU time 13,028 7649 6845 8588 9141

ON1 ON2 OR1a _OR2a _OR2b BiCGstab 101,291 86,229 141,455 106,596 154,539 Newton 21,361 20,110 30,934 24,414 31,389

CPU time 6104 5659 9420 9740 12,087

*_{OD (i) indicates overlapping Dirichlet method for domains that share ivertical lines of grid} points.

a_kˆ1. b_kˆ10.

Ta ble 4 Rela tive errors at di€ erent time s in rectang ular do mains with non-o verlapp ing doma in decomp osition M ethod Err or in u Error in v t ˆ 20 t ˆ 40 t ˆ 80 t ˆ 20 t ˆ 40 t ˆ 80 D1 a 2.081 9 ( ) 4) 7.301 1 ( ) 5) 1.2447 ( ) 5) 6.1179 ( ) 2) 2.366 8 ( ) 3) 1.057 2 ( ) 3) D1 b 1.011 8 ( ) 2) 4.250 6 ( ) 4) 3.8505 ( ) 4) 7.2208 ( ) 3) 3.274 1 ( ) 3) 3.190 8 ( ) 3) D2 a 6.019 9 ( ) 7) 6.339 6 ( ) 7) 1.0372 ( ) 7) 1.4381 ( ) 4) 6.107 7 ( ) 5) 7.922 6 ( ) 6) D2 b 2.893 2 ( ) 5) 2.425 3 ( ) 5) 3.6189 ( ) 6) 9.0960 ( ) 5) 2.204 2 ( ) 4) 1.958 5 ( ) 5) No1 a 3.906 9 ( ) 4) 4.032 0 ( ) 4) 8.5468 ( ) 5) 1.1567 ( ) 1) 2.166 0 ( ) 2) 8.769 8 ( ) 3) No1 b 2.283 1 ( ) 2) 2.876 3 ( ) 4) 2.8864 ( ) 3) 2.5994 ( ) 2) 4.910 6 ( ) 2) 2.087 5 ( ) 2) No2 a 7.023 2 ( ) 7) 5.283 0 ( ) 7) 0 1.4195 ( ) 4) 6.122 6 ( ) 5) 5.359 4 ( ) 6) No2 b 2.893 2 ( ) 5) 2.425 3 ( ) 5) 4.3426 ( ) 6) 8.8336 ( ) 5) 2.192 0 ( ) 4) 2.319 3 ( ) 5) N1 a 2.082 0 ( ) 4) 7.298 4 ( ) 5) 1.2460 ( ) 5) 6.1179 ( ) 2) 2.366 6 ( ) 3) 1.057 2 ( ) 3) N1 b 1.011 8 ( ) 2) 4.253 4 ( ) 4) 3.8518 ( ) 4) 7.2208 ( ) 3) 3.273 1 ( ) 3) 3.191 1 ( ) 3) N2 a 5.775 3 ( ) 7) 6.564 4 ( ) 7) 5.6011 ( ) 8) 1.4410 ( ) 4) 6.104 9 ( ) 5) 7.912 5 ( ) 6) N2 b 2.885 8 ( ) 5) 2.419 3 ( ) 5) 3.6115 ( ) 6) 9.0885 ( ) 5) 2.203 3 ( ) 4) 1.947 9 ( ) 5) Ro1 a ; c 2.082 0 ( ) 4) 7.298 5 ( ) 5) 1.2459 ( ) 5) 6.1179 ( ) 2) 2.366 8 ( ) 3) 1.057 2 ( ) 3) Ro1 b ; c 1.011 8 ( ) 2) 4.252 2 ( ) 4) 3.8515 ( ) 4) 7.2209 ( ) 3) 3.274 1 ( ) 3) 3.190 9 ( ) 3) Ro2 a ; c 5.775 2 ( ) 7) 6.564 3 ( ) 7) 5.6000 ( ) 8) 1.4410 ( ) 4) 6.104 8 ( ) 5) 7.912 5 ( ) 6) Ro2 b ; c 2.885 7 ( ) 5) 2.419 2 ( ) 5) 3.6112 ( ) 6) 9.0883 ( ) 5) 2.203 2 ( ) 4) 1.952 0 ( ) 5) Ro2 a ; d 5.783 1 ( ) 7) 6.573 6 ( ) 7) 5.4984 ( ) 8) 1.4434 ( ) 4) 6.111 5 ( ) 5) 7.912 0 ( ) 6) Ro2 b ; d 2.896 5 ( ) 5) 2.423 9 ( ) 5) 3.6172 ( ) 6) 9.1071 ( ) 5) 2.206 8 ( ) 4) 1.930 1 ( ) 5) R1 a ; c 2.082 0 ( ) 4) 7.298 5 ( ) 5) 1.2459 ( ) 5) 6.1179 ( ) 2) 2.366 8 ( ) 3) 1.057 2 ( ) 3) R1 b ; c 1.011 8 ( ) 2) 4.252 3 ( ) 4) 3.8516 ( ) 4) 7.2209 ( ) 3) 3.274 0 ( ) 3) 3.190 7 ( ) 3) R2 a ; c 5.775 2 ( ) 7) 6.563 5 ( ) 7) 5.5897 ( ) 8) 1.4410 ( ) 4) 6.103 9 ( ) 5) 7.962 2 ( ) 6) R2 b ; c 2.885 9 ( ) 5) 2.420 0 ( ) 5) 3.6290 ( ) 6) 9.0886 ( ) 5) 2.202 8 ( ) 4) 9.316 9 ( ) 6) DN2 a ; c 6.019 9 ( ) 7) 6.339 6 ( ) 7) 1.0372 ( ) 7) 1.4381 ( ) 4) 6.107 7 ( ) 5) 7.922 6 ( ) 6) DN2 b ; c 2.878 7 ( ) 5) 2.425 3 ( ) 5) 3.6189 ( ) 6) 9.0960 ( ) 5) 2.202 5 ( ) 4) 1.958 5 ( ) 5) aIndica tes closest grid point to the corne rs. bIndica tes midpoin t betw een the corners. ck ˆ 1. dk ˆ 1000.

and used 4206 s of CPU time, whileFLrequired 18,136 iterations of BiCGstab and 2221 s of CPU time; AF employed 1142 s of CPU time. Therefore, the accuracy of time linearization with or without approximate factorization can be improved by decreasing the time step, and these techniques may be still more ecient thanNR.

Overlapping domain decomposition. The Dirichlet method was solved with 2,

4, 8, 10, 12, 16 and 32 overlapping columns between subdomains 1 and 2, and it was observed that the numerical errors became independent of the overlap when the two subdomains shared more than eight vertical grid lines. For this reason, only the errors corresponding to 2 and 8 vertical lines of overlap are presented in Table 7 which clearly illustrates that the errors decrease as the overlap is increased. For an overlap of two vertical lines, the numerical errors ofODare larger than those of methods based on time linearization (cf. Table 6). Table 8 illustrates that the number of iterations of BiCGstab and the CPU time ®rst decrease and then increase as the overlap is increased whereas those of the Newton±Raphson technique decrease but tend to a constant asymptotic value.

Table 7 also shows that the accuracy of overlapping Neumann and Robin methods is nearly the same and is almost independent of the order of dis-cretization of the interface conditions; however, bothOR1 andOR2failed to converge for kP100;OR1and OR2 each yielded results that di€ered in the relative error in, at most, the sixth decimal digit for kˆ0, 1 and 10 when arithmetic averages of the results for the left and right subdomains were used to

Table 5

Iterations and CPU time for rectangular domains with non-overlapping domain decomposition

Iter./Meth. D1 D2 No1 No2 N1 N2

BiCGstab 160,513 169,954 215,085 113,663 187,576 199,867 Newton 32,525 34,182 28,116 28,805 35,464 31,742 CPU time 12,102 10,346 11,839 8554 11,431 14,095

Ro1a _Ro1b _Ro1c _Ro2a _Ro2b _Ro2c BiCGstab 228,110 231,310 149,444 147,386 183,671 122,632 Newton 32,534 32,581 19,794 33,442 36,150 21,078 CPU time 18,193 18,465 13,374 11,990 14,174 8905

R1a _R2d _R2a _{DN2 ND2 Y2} BiCGstab 309,585 210,904 298,939 97,328 74,273 135,719 Newton 51,641 33,778 44,807 30,984 30,984 30,580 CPU time 19,277 13,418 18,203 6853 5125 6956 a_kˆ1.

b_kˆ10. c_kˆ1000. d_kˆ0.

update the interface at the next iteration; the eciency of bothOR1andOR2

worsened askwas increased, i.e.,OR1required 5134, 5387 and 8250 s of CPU forkˆ0, 1 and 10, respectively, whereasOR2required 4517, 5068 and 5902 s of CPU forkˆ0, 1 and 10, respectively, and these results are consistent with those shown in Table 3 forON1andON2.

Non-overlapping domain decomposition. Table 9 shows that the accuracy of

the Dirichlet and Neumann, non-overlapping domain decomposition methods increases (but not as much as for rectangular domains, cf. Table 4) as the accuracy of the discretization of the ®rst-order derivative normal to the in-terface is increased; the accuracy of Dirichlet methods is similar to that of Neumann ones but these methods do not produce identical errors;No1 and

N1 yielded results that di€ered in the ®fth decimal ®gure from those of N1;

Ro1 and R1 yielded results that di€ered in the seventh decimal ®gure from those of N1; Ro1 and Ro2 each yielded results that di€ered in the relative error in, at most, the sixth decimal digit for kˆ0, 1, 10, 102_{, 10}3 _{and 10}30

when arithmetic averages of the results for the left and right subdomains were used to update the interface at the next iteration; R1 and R2 each yielded

identical results for kˆ0 and 1, but these methods failed to converge for

kP10; and,DN2 yielded exactly the same results asY2, and these results, in turn, di€ered in relative error in, at most, the sixth decimal digit from those of ND2.

The non-overlapping domain decomposition methods for domains with re-entrant corners were found to be less ecient than overlapping ones when these share 4, 8 or 10 vertical lines, except forNo1andNo2(cf. Table 10);ON1was found to be the most ecient technique for domains with re-entrant corners; the eciency ofD,N,RoandRwas found to increase, whereas that ofNowas found to decrease as the accuracy of the interface conditions was increased; and, the eciency of bothRo1andRo2was found to decrease and increase ask

was increased from to 0 to about 10 and from 10 to 1030_{, respectively.}

How-ever, the CPU times required byRo1withkˆ1 and 10 were 7171 and 7794 s, respectively, whereas those required byRo1withkˆ1 and 10 were 7540 and 8931 s, respectively; therefore, the eciency of Robin methods depends on both the value ofk and the order of discretization of the interface conditions.

Fig. 9. Spatial distribution ofu(top) andv (bottom) in the domain with re-entrant corners at

A comparison between Tables 4 and 9 clearly shows that the accuracy of domain decomposition methods depends strongly on the regularity of the domain's geometry, and that the errors of these methods increase when the domain has re-entrant corners if the boundaries between subdomains are de-®ned where abrupt changes in geometry occur as in the study presented here (cf. Fig. 2); this is due to several reasons. First, the re-entrant corners are singular points. Second, the instantaneous isocontours of u and v are not perpendicular to the boundary between adjacent subdomains. Third, and more importantly, the continuity and smoothness conditions imposed at the inter-face between subdomains only require continuity of the function and ¯uxes, whereas the partial di€erential equations should be satis®ed there. For one-dimensional problems, it is a simple exercise to show that [16], when ®rst-order discretizations are employed for the derivative at the interface, the continuity of ¯uxes is equivalent to solve only steady di€usion processes which only in the limit Dxˆ0 correspond to the solution of the partial di€erential equation because both the accumulation and the reaction rate tend to zero as the computational cell volume tends to zero. Moreover, since one cannot achieve

Fig. 10. Spatial distribution ofu(top) andv(bottom) in the domain with re-entrant corners at

the limit valueDxˆ0, the errors incurred by the discretization and the ®nite value ofDxdecrease asDxis decreased. This can be better appreciated in Fig. 13 that shows the spatial distributions of the errors inuandvin both subdomains attˆ40 [16], and clearly indicate that the largest errors occur at the interface between subdomains and, especially, near the re-entrant corners. The results presented in this ®gure were obtained with No2 and meshes of 6666 and 3434 grid points for X1 and X2, respectively, to ease the visualizations,

whereas the errors shown in the tables were obtained with meshes of 130130 and 6666 grid points, respectively.

The results presented in Tables 7 and 9 show that, for domains with re-entrant corners, overlapping domain decomposition techniques are more ac-curate than non-overlapping ones.

5. Conclusions

A variety of fully linearized, implicit domain decomposition methods has been used to study a system of two non-linear reaction±di€usion equations in

Fig. 11. Spatial distribution ofu(top) andv(bottom) in the domain with re-entrant corners at

rectangular domains and domains with re-entrant corners, and the results have been compared in terms of both accuracy and eciency with those obtained from an approximate factorization technique that does not use domain de-composition and solves one-dimensional problems at each time step, and with those of a Newton±Raphson method that solves the non-linear equations in an arti®cially enlarged domain without domain decomposition.

It has been shown that the approximate factorization technique is much more ecient than the Newton±Raphson method, fully implicit, linearized methods, and overlapping and non-overlapping domain decompositions de-spite the fact that this technique su€ers from both linearization and approxi-mate factorization errors which can be decreased by decreasing the time step employed in the calculations.

For rectangular regions and regions with re-entrant corners and overlapping domain decomposition, it has been found that the accuracy of the results im-proves as the number of overlapping lines is increased up to about eight; greater overlaps do not improve the accuracy. For an overlap of two grid lines, it has been found that the accuracy of domain decomposition methods that

Fig. 12. Spatial distribution ofu(top) andv(bottom) in the domain with re-entrant corners at

Ta ble 6 Rela tive errors at di€ erent time s in domains with re-e ntrant corners w ithout doma in decomp osition M ethod Error in u Err or in v t ˆ 20 t ˆ 40 t ˆ 80 t ˆ 20 t ˆ 40 t ˆ 80 AF a 7.4021 ( ) 7) 1.119 7 ( ) 7) 0 8.407 9 ( ) 6) 3.5168 ( ) 6) 5.989 4 ( ) 7) AF b 1.1740 ( ) 6) 3.711 8 ( ) 7) 0 0 3.9182 ( ) 6) 9.751 6 ( ) 7) FL a 4.2298 ( ) 7) 1.119 7 ( ) 7) 0 6.637 8 ( ) 6) 2.5994 ( ) 6) 4.492 1 ( ) 7) FL b 2.7616 ( ) 6) 3.711 8 ( ) 7) 0 5.250 4 ( ) 6) 4.8138 ( ) 6) 1.114 5 ( ) 6) a Indica tes closest grid point to the corne rs. b Indica tes midpoin t betw een the corners. Ta ble 7 Rela tive errors at di€ erent time s in domains with re-e ntrant corners w ith overlapp ing doma in de compo sition M ethod Err or in u Error in v t ˆ 20 t ˆ 40 t ˆ 80 t ˆ 20 t ˆ 40 t ˆ 80 OD a(2) 3.742 3 ( ) 4) 9.863 8 ( ) 5) 2.0656 ( ) 5) 5.5612 ( ) 3) 3.895 1 ( ) 3) 1.426 8 ( ) 3) OD b(2) 3.436 9 ( ) 3) 7.697 0 ( ) 4) 1.0894 ( ) 4) 5.1441 ( ) 3) 7.655 6 ( ) 3) 3.644 4 ( ) 3) OD a(8) 3.751 0 ( ) 7) 7.407 1 ( ) 8) 3.2640 ( ) 8) 6.7818 ( ) 6) 2.612 3 ( ) 6) 4.936 0 ( ) 7) OD b(8) 2.720 6 ( ) 6) 3.090 5 ( ) 7) 3.2336 ( ) 8) 5.3422 ( ) 6) 4.794 6 ( ) 6) 1.134 9 ( ) 6) ON 1 a 7.276 0 ( ) 6) 9.563 2 ( ) 7) 1.2575 ( ) 7) 8.8975 ( ) 5) 3.848 1 ( ) 5) 8.183 7 ( ) 6) ON 1 b 1.880 6 ( ) 6) 2.591 5 ( ) 6) 5.7387 ( ) 8) 4.0698 ( ) 6) 2.649 2 ( ) 5) 5.822 8 ( ) 6) ON 2 a 9.831 0 ( ) 6) 1.300 7 ( ) 6) 1.7875 ( ) 7) 1.2095 ( ) 4) 5.222 1 ( ) 5) 1.108 8 ( ) 5) ON 2 b 3.418 2 ( ) 6) 3.354 5 ( ) 6) 8.7835 ( ) 8) 3.6357 ( ) 6) 3.695 9 ( ) 5) 8.156 4 ( ) 6) OR1 a, c 7.276 0 ( ) 6) 9.563 2 ( ) 7) 1.2575 ( ) 7) 8.8975 ( ) 5) 3.848 1 ( ) 5) 8.183 7 ( ) 6) OR1 b ; c 1.880 6 ( ) 6) 2.591 5 ( ) 6) 5.7387 ( ) 8) 4.0698 ( ) 6) 2.649 2 ( ) 5) 5.822 9 ( ) 6) OR2 a ; c 9.821 3 ( ) 6) 1.299 4 ( ) 6) 1.7853 ( ) 7) 1.2079 ( ) 4) 5.217 3 ( ) 5) 1.107 7 ( ) 5) OR2 b ; c 3.422 4 ( ) 6) 3.356 0 ( ) 6) 8.8172 ( ) 8) 3.6226 ( ) 6) 3.698 0 ( ) 5) 8.161 6 ( ) 6) *OD ( i ) indica tes overlapp ing Dirichlet metho d for do mains that sha re i vertical line s of grid po ints. aIndica tes closest grid point to the corne rs. bIndica tes midpoin t betw een the corners. ck ˆ 1.

employ Neumann or Robin boundary conditions is better than that of those which use Dirichlet ones, but is not very much sensitive to the accuracy of discretization of the ®rst-order derivative at the subdomain interface. The ef-®ciency of methods that employ Neumann boundary conditions at the sub-domain interface was found to be larger than those based on Dirichlet and Robin ones with the same overlap, and the eciency of overlapping techniques was found to be largest for eight lines of overlap.

For rectangular regions, it was found that the accuracy of non-overlapping domain decomposition techniques improves as the accuracy of the discretiza-tion of the ®rst-order derivative at the subdomain interface is increased, and is nearly independent of the transmission coecient for Robin boundary con-ditions or the order in which Dirichlet and Neumann boundary concon-ditions are implemented in methods that use two iterations per cycle. The eciency of non-overlapping domain decomposition increases as the accuracy of discreti-zation of the ®rst-order derivative at the subdomain interface was increased for Dirichlet, Neumann (for interior points) and Robin, but it decreased for Neumann (for both interior and interfacial points) methods. The eciency of Robin methods for interior points was found to be a function of the trans-mission coecients, and the most ecient of the non-overlapping techniques was the Neumann±Dirichlet method. It was also found that Robin methods that evaluate the interface together with the solution at interior points may not converge for large values of the transmission coecient for both overlapping and non-overlapping decomposition techniques.

For regions with re-entrant corners, it was found that the accuracy of non-overlapping domain decomposition was much lower than that for rectangular ones, especially near the re-entrant corners where there is a mathematical singularity, the isocontours of the dependent variables are not orthogonal to

Table 8

Iterations and CPU time in domain with re-entrant corners with overlapping domain decompo-sition

Iterations OD(2) _{OD(4) OD(8) OD(16) OD(32)} BiCGstab 177,154 108,960 106,570 107,185 108,127 Newton 36,851 18,696 17,687 17,687 17,687

CPU time 8935 4388 4305 5934 6258

ON1 ON2 OR1a _OR2a _OR2b BiCGstab 92,748 96,312 138,542 107,722 155,171 Newton 23,581 23,448 31,648 24,495 31,564

CPU time 3417 4116 5387 5068 5902

*_{OD (i) indicates overlapping Dirichlet method for domains that share ivertical lines of grid} points.

a_kˆ1. b_kˆ10.

Ta ble 9 Rela tive errors at di€ erent time s in domains with re-e ntrant corners w ith non-ove rlap ping doma in de compo sition M ethod Err or in u Error in v t ˆ 20 t ˆ 40 t ˆ 80 t ˆ 20 t ˆ 40 t ˆ 80 D1 a 1.943 4 ( ) 2) 3.442 4 ( ) 2) 2.7410 ( ) 2) 3.2300 ( ) 1) 2.454 9 ( ) 1) 2.905 5 ( ) 1) D1 b 1.013 2 ( ) 2) 2.864 8 ( ) 3) 5.8278 ( ) 3) 7.2613 ( ) 3) 1.061 5 ( ) 2) 2.139 2 ( ) 2) D2 a 1.042 1 ( ) 2) 2.024 2 ( ) 2) 1.5823 ( ) 2) 1.8103 ( ) 1) 1.668 1 ( ) 1) 1.855 0 ( ) 1) D2 b 1.511 6 ( ) 5) 9.108 9 ( ) 4) 9.1972 ( ) 4) 6.5630 ( ) 5) 2.889 2 ( ) 3) 6.673 0 ( ) 3) No2 a 1.040 5 ( ) 2) 2.024 5 ( ) 2) 1.5822 ( ) 2) 1.8081 ( ) 1) 1.669 0 ( ) 1) 1.854 8 ( ) 1) No2 b 1.526 2 ( ) 5) 9.116 3 ( ) 4) 9.1882 ( ) 4) 6.3880 ( ) 5) 2.884 4 ( ) 3) 6.677 3 ( ) 3) N2 a 1.042 1 ( ) 2) 2.024 2 ( ) 2) 1.5823 ( ) 2) 1.8103 ( ) 1) 1.668 1 ( ) 1) 1.855 0 ( ) 1) N2 b 1.506 6 ( ) 5) 9.109 8 ( ) 4) 9.1974 ( ) 4) 6.5815 ( ) 5) 2.889 2 ( ) 3) 6.673 1 ( ) 3) Ro2 a ; c 1.040 8 ( ) 2) 2.024 5 ( ) 2) 1.5822 ( ) 2) 1.8085 ( ) 1) 1.668 9 ( ) 1) 1.854 8 ( ) 1) Ro2 b ; c 1.516 7 ( ) 5) 9.117 0 ( ) 4) 9.1946 ( ) 4) 6.5993 ( ) 5) 2.884 9 ( ) 3) 6.679 2 ( ) 3) Ro2 a ; d 1.042 0 ( ) 2) 2.024 2 ( ) 2) 1.5823 ( ) 2) 1.8103 ( ) 1) 1.668 1 ( ) 1) 1.855 0 ( ) 1) Ro2 b ; d 1.513 2 ( ) 5) 9.109 7 ( ) 4) 9.1974 ( ) 4) 6.5928 ( ) 5) 2.889 0 ( ) 3) 6.673 1 ( ) 3) R2 a ; c 1.042 1 ( ) 2) 2.024 2 ( ) 2) 1.5823 ( ) 2) 1.8103 ( ) 1) 1.668 1 ( ) 1) 1.855 0 ( ) 1) R2 b ; c 1.506 6 ( ) 5) 9.109 8 ( ) 4) 9.1975 ( ) 4) 6.5816 ( ) 5) 2.889 3 ( ) 3) 6.673 1 ( ) 3) DN2 a 1.040 6 ( ) 2) 2.024 5 ( ) 2) 1.5822 ( ) 2) 1.8081 ( ) 1) 1.669 0 ( ) 1) 1.854 8 ( ) 1) DN2 b 1.526 2 ( ) 5) 9.120 0 ( ) 4) 9.1950 ( ) 4) 6.5630 ( ) 5) 2.884 0 ( ) 3) 6.680 5 ( ) 3) ND2 a 1.042 1 ( ) 2) 2.024 2 ( ) 2) 1.5823 ( ) 2) 1.8103 ( ) 1) 1.668 1 ( ) 1) 1.855 0 ( ) 1) ND2 b 1.511 6 ( ) 5) 9.108 9 ( ) 4) 9.1972 ( ) 4) 6.5630 ( ) 5) 2.889 2 ( ) 3) 6.673 0 ( ) 3) Y2 a 1.040 6 ( ) 2) 2.024 5 ( ) 2) 1.5822 ( ) 2) 1.8081 ( ) 1) 1.669 0 ( ) 1) 1.854 8 ( ) 1) Y2 b 1.497 1 ( ) 5) 9.131 1 ( ) 4) 9.2017 ( ) 4) 6.5630 ( ) 5) 2.885 2 ( ) 3) 6.683 0 ( ) 3) a Indica tes closest grid point to the corne rs. b Indica tes midpo int between the corne rs. ck ˆ 1. dk ˆ 1000.

the subdomain interface, and the continuity and smoothness of the solution at the interface imply that the discretized form of the partial di€erential equation is not satis®ed there except when the computational cell volume shrinks to zero, because these conditions impose continuity of the ¯uxes normal to the subdomain interface. Moreover, the accuracy of non-overlapping domain de-composition methods for regions with re-entrant corners is much lower than that of overlapping ones.

For regions with re-entrant corners, it was also found that the accuracy of non-overlapping domain decomposition techniques is almost insensitive to the accuracy of discretization of the ®rst-order derivative at the subdomain in-terface, and is nearly independent of the transmission coecient for Robin boundary conditions or the order in which Dirichlet and Neumann boundary conditions are implemented in methods that use two iterations per cycle. The eciency of non-overlapping domain decomposition increases as the order of discretization of the ®rst-order derivative at the subdomain interface was in-creased for Dirichlet, Neumann (for interior points) and Robin, but it de-creased for Neumann (for both interior and interfacial points) methods. The eciency of Robin methods for interior points was found to be a function of the transmission coecients, and the most ecient non-overlapping technique was the Neumann method that evaluates the interface separately from the interior points.

Table 10

Iterations and CPU time in domains with re-entrant corners with non-overlapping domain decomposition

Iter./Meth. D1 D2 No1 No2 N1 N2

BiCGstab 159,344 169,196 121,196 121,087 191,479 228,414 Newton 32,678 34,427 27,668 28,122 34,140 34,735

CPU time 5703 6121 4442 4384 8242 9346

Ro1a _Ro1b _Ro1c _Ro2a _Ro2b _Ro2c BiCGstab 233,733 240,515 154,564 145,057 192,718 135,069 Newton 33,171 32,730 20,011 33,686 39,511 24,144

CPU time 7171 7794 6234 7540 8931 7224

R1a _R2d _R2a _{DN2 ND2 Y2} BiCGstab 302,603 241,066 309,933 198,426 192,114 195,894 Newton 49,875 34,735 44,041 30,984 30,984 30,580 CPU time 11,725 7993 11,629 8801 8442 7855 a_kˆ1.

b_kˆ10. c_kˆ1000. d_kˆ0.

Acknowledgements

The research reported in this paper was supported by Project PB94-1494 from the D.G.I.C.Y.T. and Project PB97-1086 from the D.G.E.S.I.C. of Spain.

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