On the domain decomposition method with splitting in subdomains for solving parabolic problems

Abstract

In this work we propose and study an algorithm for decomposing a domain which is a combination of nonintersecting rectangles in the two-dimensional case or parallelepipeds in the threedimensional case. The algorithm is based on the penalization method with the simultaneous use of component-wise splitting in separate domains. We analyse the method and give the results of numerical experiments.In the past years considerable interest has been shown in noniterative domain decomposition methods of solving parabolic boundary value problems. For the first time the domain decomposition techniques for nonstationary problems were supposed to be studied in [14]. Lebedev in [15] gives the equations of the method, which form a basis for our investigation. The noniterative methods both with overlapping subdomains [2,3,5,7,9,10,17,21] and without overlapping subdomains [4,7,11,13,16,21] were considered in the literature. Though the methods with overlapping subdomains have better convergence properties, the methods without overlapping subdomains are preferred from the algorithmic viewpoint. Laevsky in [13] proposes an unconditionally convergent method which uses an approach analogous to that used in the penalization method, when the solution of the Dirichlet problem is approximated by the solution of the third boundary value problem with a small parameter in front of the normal derivative in the boundary condition [1]. However, the algorithm for solving the third boundary value problem in a separate domain was not considered. The direct inversion of the corresponding matrices was supposed to be realized. When the number of nodes in a subdomain is large, it is difficult to solve this problem (from the viewpoint of computer memory and speed). However, if the calculated domain is represented as a combination of a number of rectangles (parallelepipeds) whose sides are orthogonal to coordinate axes, it is logical to combine the domain decomposition with a scheme of splitting in subdomains. It is this kind of algorithm that is proposed in the present paper. A scheme of component-wise splitting for 'rectangular' subdomains is used in the algorithm. This implies the specific form of notation and the method of studying it. Note that, as in [13], an incompatible grid may be used in the algorithm proposed, i.e. grids in separate domains are not connected with each other. The numerical experiments, which are described in Section 6, illustrate the asymptotic behaviour of the penalty parameter and the errors of the method. Moreover, the possibilities for algorithm parallelizing are discussed.We omit the proofs of some assertions. All of them are given in the preliminary publication [11]. Our main concern is with the formulation of the problem, the structure of the method, and the description of numerical experiments.

Domain Decomposition Operator Splittings for the Solution of Parabolic Equations

“…The derivatives of {a k (x, y)} can be estimated in terms of the derivatives of a(x, y) and of the partition of unity functions {χ k (x, y)}. Using the bounds (19) for the partition of unity, we can then obtain that…”

“…For such splittings, we provide in section 4.3 an explicit estimate on how the truncation error depends on the overlap β amongst the subdomains. For related algorithms, we refer the reader to Dryja [13], Laevsky [19,18], Vabishchevich [28] and Vabishchevich and Matus [29], Kuznetsov [17,16], Blum, Lisky, and Rannacher [2], Meurant [21], Dawson, Du, and Dupont [9], and Chen and Lazarov [20].…”

“…Before we proceed further, we define the Hölder norms for integer k ≥ 0 as follows: (19) in the Hölder C m (Ω) norm, where c m > 0 is a constant independent of k and β but dependent on the geometry of the region.…”

“…Using Leibnitz's rule for partial differentiation of a k (x, y) = χ k (x, y)a k (x, y) and applying bounds (19) for the partition of unity, we deduce that…”

“…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].…”