(School of Mathematics and Computing Science,Anqing Teacher’s College,Anqing 246133,China)

Overlapping Domain Decomposition Finite Diference Algorithm for Compact Diference Scheme of the Heat Conduction Equation

ZHANG Hong-mei

(School of Mathematics and Computing Science,Anqing Teacher’s College,Anqing 246133,China)

In this paper,a modifed additive Schwarz fnite diference algorithm is applied in the heat conduction equation of the compact diference scheme.The algorithm is on the basis of domain decomposition and the subspace correction.The basic train of thought is the introduction of the units function decomposition and reasonable distribution of the overlap of correction.The residual correction is conducted on each subspace while the computation is completely parallel.The theoretical analysis shows that this method is completely characterized by parallel.

heat equation;compact diference scheme;domain decomposition;partition of unity;subspace correction

§1. Introduction

[1]proposes a new parallel fnite diference algorithm based on overlapping domain decomposition method for solving heat conduction equation numerically.The algorithm is based upon the domain decomposition method and the modifcation of classical parallel subspace correction method.The residual is modifed on each subspace and the computation is completely parallel. The proposed method is applied to heat conduction equation of compact diference scheme so as to draw some new and constructive conclusions.

§2.Compact Diference Scheme of Heat Equation

We consider initial boundary value problem for the heat equation in one dimension

Let a,b be two endpoints,Ω=(a,b),a＜ b.It’s to be solved with fnite diference method.First,the domain(a,b)is decomposed into N subdomains,assuming that the domain is discretized uniformly with spacing h=(b−a)/N.N is a positive integer while the full set of grid points on Ω is denoted Ωh={xi;xi=a+ih,0≤i≤N}and the full set of interior point as Ω0h={xi;xi=a+ih,1≤i≤N−1}.It’s also to denote the boundary of the set of points as Γh={x0,xN},time step is set as τ and tn=nτ.For the grid function v(x,t),it’s defned that vin=v(xi,tn).Given the following diferential operator

These schemes are stable absolutely.And the optimal error estimate is O(τ2+h4),which has been described in[3].For the grid functions un,vnof Ωh,We defne the discrete inner product and a discrete norm

The following briefy illustrates thatis symmetric

The introduction of grid function space

The solution of diference scheme(2)is equivalent to seek Un∈vh,satisfy the equation

For any arbitrary V ∈vh.On both ends of the equation(3)Vih is multiplied and seek summation of i.

By the discrete Green formula it can be concluded as the following

for any u,v∈vhwe defne the bilinear form

for that A(u,v)is bounded and symmetric,so we can defne the norm

The diference scheme(2)can be written as a discrete variational form seek Un∈vhsatisfy

The following will construct the parallel domain decomposition algorithm for the solution of this equation.

§3.Parallel Domain Decomposition Finite Diference Algorithm

We construct a discrete domain decomposition domain decomposition,defne

We consider parallel iterative algorithm from the subspace correction.Letbe the approximate solution of the diference scheme(2).The residuals

If roldis small enough,we will takeotherwise,on each subdomain Ωjrespectively approximate solving residual equation

Set the solution for the︿ej,then the correction of the approximate solutions be

Thus we get improved additive Schwarz algorithms.

Parallel Domain Decomposition Finite Diference AlgorithmThe number of iterations in each time level is m,W0is the initial function approximation.For n≥1,assume that Wk(k=0,1,···,n−1)is known.Give˜wn0=Wn−1,seek Wnby three steps as follows.

Step 1For i=1,2,···,m,parallel iteration in J sub region as the synchronous. Seeksatisfying

Step 2Sum of j(j=1,2,···,J)

Step 3Letgo to the next time layer iteration.

NoteIn the subspace correction algorithm[4],correction of the item is defned as

Due to the fact that the amount of correction on the overlapping area is twice the original damage quantity.Consequently,classic subdomains correction algorithm is not convergence, while the algorithm presented in this paper difers from it:In the second step of correction,the unit decomposition functions{φj}Jj=1,is introduced and then it’s to distribute the correction on the overlapping domain to each subspace reasonably.Additionally the algorithm is convergent quickly.The following theorem is given to prove the convergence of the algorithm.

Theorem 1Let u be the exact solution of the problem(1),which is smooth enough and Wnbe the solution of the parallel domain decomposition fnite diference algorithm.Then when the τ is sufciently small with the following error estimate

where m ≥2 is the iterative number at each time step and H is the overlapping degree of the domain decomposition.c0and c1is positive constant dependent upon norms of u,but independent of m,h,τ and H.

§4.Convergence Analysis of Parallel Domain Decomposition Finite Diference Algorithm

To prove convergence mentioned above,we give an error estimate of compact diference scheme.

Lemma 1Let u be the exact solution of the problem(1),which is smooth enough and Unbe the solution of the compact diference scheme(2),such that

The Proof of Theorem 1We write

By the Lemma 1,to prove theorem(8),we only need to consider the error between the parallel subspace correction algorithm solution Wnand the compact diferential solution Un.The error equation is derived at frst.For any arbitrary V∈vh,multiply Eq(6)with Vkh and obtain the sum of k,we have

Parallel subspace correction algorithm is used in iterative method to solve the equation of initial value Wn−1

It follows from(10)and(11)that

It follows from(5)and(11)that

We see that

and then

For m≥2,applying a discrete Bellman’s inequality,we get the a prior error estimate

We have proved Theorem 1

It is obvious that theoretical analysis indicated the absolute parallelism of the algorithm. It needs only one or two iteration steps to get the optimal convergence precision at each time level.

[1]TIAN Min,YANG Dan-ping.Two order parallel domain decomposition fnite diference algorithm for heat equation[J].Journal of Shandong University(Natural Science),2006,41(5):12-19.

[2]TIAN Min,YANG Dan-ping.New overlapping domain decomposition parallel algorithm for the heat equation[J].Numerical Mathematics A Journal of Chinese Universities,2008,30(1):14-25.

[3]HU Jian-wei,TANG Huai-min.Numerical Methods for Diferential Equations[M].Beijing:Science Press, 1999.

[4]XU Jin-chao.Iterative methods by space decomposition and subspace correction[J].Siam Review,1992, 34(4):581-613.

O241.82

A

1002–0462(2015)04–0495–08

Received date:2013-05-22

Foundation item:Supported by the School Youth Foundation Project Funding of Anqing Teacher’s College (KJ201108)

Biography:ZHANG Hong-mei(1979-),female,native of Baoji,Shaanxi,a lecturer of Anqing Teacher’s College,M.S.D.,engages in numerical solution of partial diferential equations.

2000 MR Subject Classifcation:65M06,65M55