A Space Decomposition Method for Parabolic Equations

A convergence proof is given for an abstract parabolic equation using general space decomposition techniques. The space decomposition technique may be a domain decomposition method, a multilevel method, or a multigrid method. It is shown that if the Euler or Crank-Nicolson scheme is used for the parabolic equation, then by suitably choosing the space decomposition, only O(jlog j) steps of iteration at each time level are needed, where is the time step size. Applications to overlapping domain decomposition and to a two-level method are given for a second order parabolic equation. The analysis shows that only a one-element overlap is needed. Discussions about iterative and non-iterative methods for parabolic equations are presented. A method that combines the two approaches and utilizes some of the good properties of the two approaches is tested numerically. x 1. Introduction In this article, we use space decomposition techniques for the abstract parabolic equation 8 > < > : @u @t ; v S + a(u; v) = (f; v) S ; 8v 2 V ; t 2 0; T] ; u(0) = u 0 2 S: (1.1) Here, S is a Hilbert space and a(u; v) is a bounded bilinear, symmetric, and positive deenite form on the Hilbert space V. A space decomposition method refers to a method that decomposes the Hilbert space V into a sum of subspaces, i. It was observed by Xu 31], see also Tai 22], that domain decomposition methods, multilevel methods, multigrid methods and substructuring methods can all be viewed in some way as space decomposition techniques. The above mentioned space decomposition methods have been widely used for elliptic problems and in this context give two beneets. Firstly, by suitably using the space decomposition methods, the original elliptic problem is reduced to a number of smaller problems, and these smaller problems can be computed in parallel. Secondly, the space decomposition methods are iterative methods. In the iteration procedure, they produce good preconditioners. So, in order to reach a certain accuracy, the iteration number can be greatly reduced, see 1], 3], 4], 5], 6], 8], 31] etc. It is possible to use space decomposition methods for parabolic problems in two ways. Firstly, one could use a time stepping scheme to discretize the time variable. Then, at each time level an elliptic equation must be solved, and one could use the space decomposition method for this elliptic equation. We shall call …