Implementation of a Mapping Procedure for the Domain Decomposition for PDEs in Maple

Abstract

Domain decomposition for partial differential equations(PDEs) is implemented using singular perturbation analysis. The programming work is accomplished in a symbolic programming environment, namely Maple.First we establish the definitions of a domain and boundary conditions in the symbolic programming environment, Maple, then apply the characteristic curve method to transform the reduced equation to a first order system. After a series of transformations, the first order system can be transformed to a third order ODE. If the third order ODE can be solved in Maple, the solutions of the reduced equation are obtained. Thus, an approximate solution of the equation is constructed from a solution of the reduced equation and correction terms corresponding to boundary layers and interior layers. By applying the above methods a set of criteria to determine a stable partition of a domain can be built. Based on the set of criteria, the search for all possible stable partitions is implemented by traversing on the graph of a given domain. Using the similar traversing method on the partitioned domain, each individual subdomain is selected automatically so that it can be distributed to a processor for further processing. The structure of the system implemented in Maple is described in a pseudo-code. In this paper only elliptic equations are discussed.KeywordsDomain DecompositionSingular PerturbationOrder SystemDomain Decomposition MethodBoundary SideThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.