MONOTONE FINITE DIFFERENCE DOMAIN DECOMPOSITION ALGORITHMS AND APPLICATIONS TO NONLINEAR SINGULARLY pdf

DECOMPOSITION ALGORITHMS ANDAPPLICATIONS TO NONLINEAR SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEMS IGOR BOGLAEV AND MATTHEW HARDY Received 16 September 2004; Revised 21 December 2004

DECOMPOSITION ALGORITHMS AND

APPLICATIONS TO NONLINEAR SINGULARLY

PERTURBED REACTION-DIFFUSION PROBLEMS

IGOR BOGLAEV AND MATTHEW HARDY

Received 16 September 2004; Revised 21 December 2004; Accepted 11 January 2005

This paper deals with monotone finite difference iterative algorithms for solving linear singularly perturbed reaction-diffusion problems of elliptic and parabolic types.Monotone domain decomposition algorithms based on a Schwarz alternating methodand on box-domain decomposition are constructed These monotone algorithms solveonly linear discrete systems at each iterative step and converge monotonically to the ex-act solution of the nonlinear discrete problems The rate of convergence of the mono-tone domain decomposition algorithms are estimated Numerical experiments are pre-sented

non-Copyright © 2006 Hindawi Publishing Corporation All rights reserved

whereμ is a small positive parameter, c ∗ > 0 is a constant, ∂ω is the boundary of ω If

f and g are sufficiently smooth, then under suitable continuity and compatibility

condi-tions on the data, a unique solutionu of (1.1) exists (see [6] for details) Furthermore,forμ 1, problem (1.1) is singularly perturbed and characterized by boundary layers(i.e., regions with rapid change of the solution) of widthO(μ |lnμ |) near∂ω (see [1] fordetails)

Hindawi Publishing Corporation

Advances in Di fference Equations

Volume 2006, Article ID 70325, Pages 1 38

DOI 10.1155/ADE/2006/70325

2 Monotone domain decomposition algorithms

The second problem considered corresponds to the singularly perturbed diffusion problem of parabolic type

reaction-− μ2 

u xx+u yy

+f (x, y,t,u) + u t =0, (x, y) ∈ ω, t ∈(0,T],

f u ≥0, (x, y,t,u) ∈ ω ×[0,T] ×(−∞,∞), (1.2)where ω = {0< x < 1 } × {0< y < 1 } and μ is a small positive parameter The initial-

boundary conditions are defined by

u(x, y,0) = u0(x, y), (x, y) ∈ ω, u(x, y,t) = g(x, y,t), (x, y,t) ∈ ∂ω ×(0,T]. (1.3)

The functions f , g, and u0are sufficiently smooth Under suitable continuity and patibility conditions on the data, a unique solutionu of (1.2) exists (see [5] for details).Forμ 1, problem (1.2) is singularly perturbed and characterized by the boundary lay-ers of widthO(μ |lnμ |) at the boundary∂ω (see [2] for details) We mention that theassumption f u ≥0 in (1.2) can always be obtained via a change of variables

com-In solving such nonlinear singularly perturbed problems by the finite differencemethod, the corresponding discrete problem is usually formulated as a system of non-linear algebraic equations One then requires a reliable and efficient computational algo-rithm for computing the solution A fruitful method for the treatment of these nonlinearsystems is the method of upper and lower solutions and its associated monotone itera-tions (in the case of unperturbed problems with reaction-diffusion equations see [8,9]and the references therein) Since the initial iteration in the monotone iterative method

is either an upper or lower solution constructed directly from the difference equationswithout any knowledge of the exact solution (see [3,4] for details), this method elimi-nates the search for the initial iteration as is often needed in Newton’s method This gives

a practical advantage in the computation of numerical solutions

Iterative domain decomposition algorithms based on Schwarz-type alternating dures have received much attention for their potential as efficient algorithms for parallelcomputing In [3,4], for solving the nonlinear problems (1.1) and (1.2), respectively,

proce-we proposed discrete iterative algorithms which combine the monotone approach and

an iterative domain decomposition method based on the Schwarz alternating procedure.The spatial computational domain is partitioned into many nonoverlapping subdomains(vertical strips) with interface γ Small interfacial subdomains are introduced near the

interfaceγ, and approximate boundary values computed on γ are used for solving

prob-lems on nonoverlapping subdomains Thus, this approach may be considered as a ant of a block Gauss-Seidel iteration (or in the parallel context as a multicoloured al-gorithm) for the subdomains with a Dirichlet-Dirichlet coupling through the interfacevariables In this paper, we generalize the monotone domain decomposition algorithmsfrom [3,4] and employ a box-domain decomposition of the spatial computational do-main This leads to vertical and horizontal interfacesγ and ρ, and corresponding vertical

vari-and horizontal interfacial subdomain problems provide Dirichlet data onγ and ρ for the

problems on the nonoverlapping box-subdomains

InSection 2, we introduce the classical nonlinear finite difference schemes for the merical solution of (1.1) and (1.2) Iterative methods by which each of these schemesmay be solved are presented in [3,4] From an arbitrary initial mesh function, one mayconstruct a sequence of functions which converges monotonically to the exact solution

nu-of the nonlinear difference scheme Each function in the sequence is generated as the lution of a linear difference problem InSection 3, we consider the elliptic problem andextend the monotone method to a box-decomposition of the computational domain

so-We show that monotonic convergence is maintained under the proposed decompositionand associated algorithm Further, we develop estimates of the rate of convergence Thebox-decomposition of the spatial domain is applied to the parabolic nonlinear differencescheme inSection 4 Numerical experiments are presented inSection 5 These confirmthe theoretical estimates of the earlier sections Suggestions are made regarding futureparallel implementation

2 Difference schemes for solving ( 1.1 ) and ( 1.2 )

Onω and [0,T] introduce nonuniform meshes ω h = ω hx × ω hyandω τ:

4 Monotone domain decomposition algorithms

To approximate the parabolic problem (1.2), we use the implicit difference scheme

ᏸhτ U(P,t) + f (P,t,U) = τ −1U(P,t − τ), (P,t) ∈ ω h × ω τ,

ᏸhτ U(P,t) ≡ᏸh U(P,t) + τ −1U(P,t), U(P,0) = u0(P), P ∈ ω h, U(P,t) = g(P,t), (P,t) ∈ ∂ω h × ω τ,

where β = 0 for (2.2 ) and β = 1 for (2.5 ).

The proof of the lemma can be found in [11]

3 Monotone domain decomposition algorithm for the elliptic problem ( 1.1 )

We consider a rectangular decomposition of the spatial domain ¯ω into (M × L)

y l−1

y e l−1

ω ml

y b l

y l

y e

x b m−1 x e

m

ω m,l+l

Figure 3.1 Fragment of the domain decomposition.

and (L −1) interfacial subdomainsϑ l,l =1, ,L −1 (horizontal strips):

ϑ l = ω x × ϑ l y = {0< x < 1 } ×y b

l < y < y e

l

, ϑ l −1∩ ϑ l = ∅,

ρ b

l =0≤ x ≤1, y = y b

l, ρ e

l =0≤ x ≤1, y = y e

l,

3.1 Statement of domain decomposition algorithm We consider the following domain

decomposition approach for solving (2.2) On each iterative step, we first solve problems

on the nonoverlapping subdomainsω h

ml,m =1, ,M, l =1, ,L with Dirichlet

bound-ary conditions passed from the previous iterate Then Dirichlet data are passed from thesesubdomains to the vertical and horizontal interfacial subdomainsθ h m,m =1, ,M −1andϑ h l,l =1, ,L −1, respectively Problems on the vertical interfacial subdomains arecomputed Then Dirichlet data from these subdomains are passed to the horizontal inter-facial subdomains before the corresponding linear problems are solved Finally, we piecetogether the solutions on the subdomains

Step 1 Initialization: On the whole mesh ω h, choose an initial mesh functionV(0)(P),

P ∈ ω hsatisfying the boundary conditionsV(0)(P) = g(P) on ∂ω h

6 Monotone domain decomposition algorithms

m ∩ ω h m+1,l,l =1, ,L,

l \ θ h

∩ ω h m,l+1,m =1, ,M;

Step 5 Compute the mesh function V(n+1)(P), P ∈ ω hby piecing together the solutions

Algorithm (3.5)–(3.10) can be carried out by parallel processing Steps2, 3, and 4

must be performed sequentially, but on each step, the independent subproblems may beassigned to different computational nodes

Remark 3.1 We note that the original Schwarz alternating algorithm with overlapping

subdomains is a purely sequential algorithm To obtain parallelism, one needs a main colouring strategy, so that a set of independent subproblems can be introduced.The modification of the Schwarz algorithm (3.5)–(3.10) can be considered as an additiveSchwarz algorithm

subdo-3.2 Monotone convergence of algorithm ( 3.5 )–( 3.10 ) Additionally, we assume that f

from (1.1) satisfies the two-sided constraints

0< c ∗ ≤ f u ≤ c ∗, c ∗,c ∗ =const. (3.11)

We say thatV(P) is an upper solution of (2.2) if it satisfies the inequalities

ᏸh V + f (P,V) ≥0, P ∈ ω h,V ≥ g on ∂ω h (3.12)

Similarly,V(P) is called a lower solution if it satisfies the reversed inequalities Upper and

lower solutions satisfy the following inequality

where f u(P) ≡ f u[P,V(P) + Θ(P)δV(P)], 0 < Θ(P) < 1 In view of the maximum

princi-ple inLemma 2.1, we conclude (3.13)

The following convergence property of algorithm (3.5)–(3.10) holds true

8 Monotone domain decomposition algorithms

Proof We consider only the case of the upper sequence Let V(n)

be an upper solution.Then by the maximum principle inLemma 2.1, from (3.5) we conclude that

Z(n+1)

ml (P) ≤0, P ∈ ω h

ml,m =1, ,M, l =1, ,L. (3.16)

Using the mean-value theorem and the equation forZ(n+1)

ml (P), we obtain the difference

Taking into account (3.16) andV(n)

is an upper solution, by the maximum principle

inLemma 2.1, from (3.6) and (3.8) it follows that

m ∩ ω h m+1,l,l =1, ,L,

(3.19)

l \ θ h

∩ ω h m,l+1,m =1, ,M;

We check this inequality in the case of the left interfacial boundaryγhb

ml, since the casewithγhe

mlis checked in a similar way From (3.5), (3.6), and (3.18), we conclude that themesh functionW(n+1)

10 Monotone domain decomposition algorithms

Thus, using (3.17), we conclude

l is checked in a similar way From (3.5), (3.8), (3.18), and (3.23), we concludethat the mesh functionW (n+1)

ml = V(n+1)

m −  V(n+1)

l satisfies thedifference problem

Thus, using (3.19), we conclude

G(n+1)(P) ≥ᏸh V(n+1)

m +fP,V(n+1)

≥0, P ∈  ρ hb ml (3.33)From here and (3.29), we conclude the required inequality onρ hb



P b ml



− V(n+1) ml



P bx − ml



h b xm −

,

m,y b

l ± h b ±,

yl =2−1

h b −+h b+

yl,

(3.35)whereh b+

xm,h b −

xmare the mesh step sizes on the left and right fromP b

ml, andh b+

yl,h b −are themesh step sizes on the top and bottom fromP b

ml From here, (3.17), (3.23) and (3.27), weconclude

For arbitrary P ∈ ω h, it follows from (3.16), (3.18), and (3.13) that the sequence

{ V(n)

(P) } is monotonically decreasing and bounded below by V(P), where V is any

lower solution Therefore, the sequence is convergent and it follows from (3.5)–(3.8) thatlimZ(n)

ml =0, limZ(n)

l =0 and limZ (n)

l =0 asn → ∞ Now by linearity of the operatorᏸh

and the continuity of f , we have also from (3.5)–(3.8) that the mesh functionU defined

12 Monotone domain decomposition algorithms

By (2.8),δU =0 which leads to the uniqueness of the solution to (2.2) This proves the

3.3 Convergence analysis of algorithm ( 3.5 )–( 3.10 ) We now establish convergence

Theorem 3.5 For algorithm (3.5 )–( 3.10 ), the following estimate holds true

Z(n+1)

θ h m ≤max

1

Z(n+1) ω h ≤ 1

c ∗ G(n) ω h . (3.48)From (3.17), (3.19) and (3.20) at the iterative stepn, and using the definition of Z(n),

we estimateG(n)as follows

G(n)(P) = −c ∗ − f(n)

u (P)Z(n)(P), P ∈  ω h, ωh = ω h \γh ∪ ρ h

, (3.49)whereγhandρ hare defined in (3.21) By (3.11),

1

c ∗ G(n)



ω h ≤ q Z(n) ω h . (3.50)Now we estimateG(n)onγh Onγhb

ml = { x i = x b

m,y e

l −1< y j < y b

l }, we representG(n)inthe form

14 Monotone domain decomposition algorithms

From here, (3.17) and taking into account thatZ(n)

ml(P) = Z(n)(P), P ∈  γ hb

ml, we get theestimate

1

c ∗ G(n)

hb ≤q + κ b

xm Z(n) ω h . (3.53)Similarly, we can prove the estimate

1

c ∗ G(n)

hb ≤q + κ b

yl Z(n) ω h . (3.58)Similarly, we can prove the estimate

From here and (3.19), and taking into account thatZ(n)

m (P) = Z(n)(P), P ∈  ρ hb ml, we get theestimate

1

c ∗ G(n)

hb ≤q + κ b

yl Z(n) ω h . (3.62)Similarly, we can prove the estimate

, P ml by+ =x b

l , and onρ hwe get the estimate1

c ∗ G(n) ρ h ≤

q + q I+q II Z(n) ω h (3.69)

16 Monotone domain decomposition algorithms

From here, (3.50) and (3.55), we conclude the estimate

1

c ∗ G(n) ω h ≤

q + q I+q II Z(n) ω h . (3.70)

Remark 3.6 For the undecomposed algorithm, with M =1 andL =1, one hasωh = ω h

in (3.50) which together with (3.48) gives estimate (3.44) withq= q.

Without loss of generality, we assume that the boundary condition in (1.1) is zero,that is,g(P) =0 This assumption can always be obtained via a change of variables Letthe initial functionV(0)be chosen in the form of (3.39) withR(P) =0, that is,V(0)is thesolution of the following difference problem



ᏸh+c ∗V(0)= ν f (P,0) , P ∈ ω h,

V(0)(P) =0, P ∈ ∂ω h, ν =1,−1. (3.71)

Then the functionsV(0)

(P), V(0)(P) corresponding to ν =1 andν = −1 are upper andlower solutions, respectively

Theorem 3.7 Let the factor q in ( 3.44 ) satisfy the condition q < 1 Suppose that the initial

upper or lower solution V(0)is chosen in the form of ( 3.71 ) Then for algorithm ( 3.5 )–( 3.10 ), the following estimate holds true

V(n) − U ω h ≤ c0(q) n

(1−  q) f (P,0) ω h, c0=3c ∗+c ∗

c ∗ c ∗ , (3.72)

where U is the solution to ( 2.2 ).

Proof Using (3.44), we have

V(n) − U ω h ≤ (q) n

1−  q Z(1) ω h . (3.74)From (3.48), (3.71) and the mean-value theorem

From here and estimatingV(0)from (3.71) with (2.8),

V(0) ω h ≤ c1∗

we conclude the estimate onZ(1)in the form

Z(1) ω h ≤ c0 f (P,0) ω h, (3.77)wherec0is defined in (3.72) Thus, from here and (3.74), we prove (3.72) 

Remark 3.8 In the next section, we present sufficient conditions to guarantee the equalityq < 1 required in Theorem 3.7

in-3.4 Uniform convergence of the monotone domain decomposition algorithm ( 3.5 )– ( 3.10 ) Here we analyze a convergence rate of algorithm (3.5)–(3.10) applied to the dif-ference scheme (2.2) defined on the piecewise uniform mesh introduced in [7] On thismesh, the difference scheme (2.2) convergesμ-uniformly to the solution of (1.1).The piecewise uniform mesh is formed in the following manner We divide each of theintervalsω x =[0, 1] andω y =[0, 1] into three parts each [0,σ x], [σ x, 1− σ x], [1− σ x, 1],and [0,σ y], [σ y, 1− σ y], [1− σ y, 1], respectively Assuming thatN x,N yare divisible by 4,

in the parts [0,σ x], [1− σ x, 1] and [0,σ y], [1− σ y, 1] we use a uniform mesh withN x /4 + 1

and N y /4 + 1 mesh points, respectively, and in the parts [σ x, 1− σ x], [σ y, 1− σ y] with

N x /2 + 1 and N y /2 + 1 mesh points, respectively This defines the piecewise equidistant

mesh in thex- and y-directions condensed in the boundary layers at x =0, 1 andy =0, 1:

x,yare very small relative toμ This is unlikely in practice, and in this

case the difference scheme (2.2) can be analysed using standard techniques We therefore

18 Monotone domain decomposition algorithms

The difference scheme (2.2) on the piecewise uniform mesh (3.80) converges

μ-uni-formly to the solution of (1.1):

where constantC is independent of μ and N The proof of this result can be found in [7]

Theorem 3.9 Let the interfacial subdomains θ h m , m =1, ,M − 1 and ϑ h l , l =1, ,L −1

be located in the x- and y-directions, respectively, outside the boundary layers Assume μ ≤

μ0 1, and the following condition

N ≤ αc ∗ /2

μ0 , N =max

N x,N y, 0< α < 1, α =const. (3.82)

If the initial upper or lower solution V(0)is chosen in the form of ( 3.71 ), then the tone domain decomposition algorithm ( 3.5 )–( 3.10 ) on the piecewise uniform mesh ( 3.80 ) converges μ-uniformly to the solution of the problem ( 1.1 ):

where constant C and the factor Q are independent of μ and N.

Proof Under the above assumption on N, the factor q in ( 3.44) satisfies the condition



q < 1 Indeed, since the interfacial subdomains are located outside the boundary layers,

where the step sizesh x andh y are in use, then using (3.80),q I andq II from (3.44) areestimated as follows

Thus,q < Q < 1, and we can apply Theorem 3.7 From here, (3.72) and (3.81), we

Remark 3.10 Such domain decompositions, in which the interfacial subdomains are side the boundary layers, are said to be unbalanced, since the distribution of mesh points among the nonoverlapping main subdomains is uneven By contrast, a balanced domain

out-decomposition is one in which the mesh points are equally distributed among the mainsubdomains For balanced decompositions, the first and last interfacial subdomains eachoverlap the boundary layer

4 Monotone domain decomposition algorithm for the parabolic problem ( 1.2 )

For solving the nonlinear difference scheme (2.5), we construct and investigate a lel domain decomposition algorithm based on the domain decomposition of the spatialdomainω introduced inSection 3

paral-4.1 Statement of domain decomposition algorithm for solving ( 2.5 ) On each time

levelt ∈ ω τ, we calculaten ∗iteratesV(n)(P,t), P ∈ ω h,n =1, ,n ∗as follows

Step 1 Initialization: on the mesh ω h, chooseV(0)(P,t), P ∈ ω hsatisfying the boundaryconditionV(0)(P,t) = g(P,t) on ∂ω h

For n = 0 to n ∗ − 1 do Steps 2 – 5

Step 2 On subdomains ω h

ml,m =1, ,M, l =1, ,L, compute mesh functions V(n+1)

ml (P, t), m =1, ,M, l =1, ,L satisfying the following difference problems

m ∩ ω h m+1,l,l =1, ,L,