A NEW DOMAIN DECOMPOSITION PARALLEL ALGORITHM FOR CONVECTION–DIFFUSION PROBLEM

Jiansong Zhang,Danping Yang

doi:10.3846/13926292.2014.963719

Abstract

Basing on overlapping domain decomposition, we construct a new parallel algorithm combined the method of subspace correction with least-squares procedure for solving time-dependent convection–diffusion problem. This algorithm is fully parallel. We analyze the convergence of approximate solution, and study the dependence of the convergent rate on the spacial mesh size, time increment, iteration number and sub-domains overlapping degree. Both theoretical analysis and numerical results suggest that only one or two iterations are needed to reach to given accuracy at each time step.

Highlights

With the rapid development of super parallel computers and parallel algorithms, parallel computation based upon domain decomposition has become a powerful tool for solving a large-scale system of partial differential equations
One can use any parallel algorithms based on overlapping domain decomposition, which are effective for elliptic problems, to solve these resulted elliptic problems step by step in time
On the basis of the idea of the parallel subspace correction method, the authors established a new family parallel algorithm combined with characteristic finite element scheme and characteristic finite difference scheme for convection–diffusion problem in [26, 27], where the partition functions of unity was used to distribute the corrections in the overlapping domains reasonably

Summary

Introduction

With the rapid development of super parallel computers and parallel algorithms, parallel computation based upon domain decomposition has become a powerful tool for solving a large-scale system of partial differential equations. On the basis of the idea of the parallel subspace correction method, the authors established a new family parallel algorithm combined with characteristic finite element scheme and characteristic finite difference scheme for convection–diffusion problem in [26, 27], where the partition functions of unity was used to distribute the corrections in the overlapping domains reasonably Both theoretical analysis and numerical results suggest that when overlapping degree has a positive lower bound independent of mesh size, only one or two iterations is needed to reach the optimal convergence precision at each time level. It is well known, parallel subspace correction method for the symmetric positive definite system is similar to the Jacobi method. Both theoretical analysis and numerical experiments indicate the full parallelization and very good approximate property of the algorithm

Formulation of Parallel Algorithm

Some Lemmas

Convergence Analysis

Numerical Examples

Conclusion