Domain Decomposition Operator Splittings for the Solution of Parabolic Equations

Abstract

We study domain decomposition counterparts of the classical alternating direction implicit (ADI) and fractional step (FS) methods for solving the large linear systems arising from the implicit time stepping of parabolic equations. In the classical ADI and FS methods for parabolic equations, the elliptic operator is split along coordinate axes; they yield tridiagonal linear systems whenever a uniform grid is used and when mixed derivative terms are not present in the differential equation. Unlike coordinate-axes-based splittings, we employ domain decomposition splittings based on a partition of unity. Such splittings are applicable to problems on nonuniform meshes and even when mixed derivative terms are present in the differential equation and they require the solution of one problem on each subdomain per time step, without iteration. However, the truncation error in our proposed method deteriorates with smaller overlap amongst the subdomains unless a smaller time step is chosen. Estimates are presented for the asymptotic truncation error, along with computational results comparing the standard Crank-Nicolson method with the proposed method.Key words. domain decomposition, alternating direction implicit method, fractional step method, operator splitting, parabolic equation, partition of unity AMS subject classifications. 65N20, 65F10PII. S1064827595288206 1. Introduction. The numerical solution of parabolic partial differential equations by implicit time stepping procedures requires the solution of large systems of linear equations. These linear systems need only be solved approximately, provided that the inexact solutions obtained by using approximate solvers preserve the stability and local truncation error of the original scheme. Though iterative methods with preconditioners are a popular way for solving such linear systems, see [5,6,3,8,15,33,4,7], in this paper we consider only approximate solvers that do not involve iteration at each time step. Such approximate noniterative solution methods for parabolic equations include the alternating direction implicit (ADI) methods of Peaceman and Rachford [22] and Douglas and Gunn [12], the fractional step methods (FS) of Bagrinovskii and Godunov [1], Yanenko [34], and Strang [26], and also more recent one-iteration domain decomposition solvers of Kuznetsov [16, 17], Meurant [21], Dryja [13], Blum, Lisky, and Rannacher [2], Dawson, Du, and Dupont [9], Laevsky [19, 18] and Vabishchevich [28] and Vabishchevich and Matus [29], and Chen and Lazarov [20].The method proposed here uses the same framework as the classical ADI and FS methods for solving parabolic equations of the form u t + Lu = f using an operator splitting L = L 1 + · · · + L q ; however, the splittings chosen here are based on domain decomposition, unlike classical splittings along coordinate directions. Furthermore, they are applicable to problems with mixed derivative terms and on nonuniform grids. The basic idea is simple. Given a smooth partition of unity {χ k } k=1,...,q subordinate to a decomposition ...

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