Domain decomposition and the numerical solution of partial differential equations defined on irregular domains using segmented forms of the tau lines method

Abstract

The Tau Lines Method, a numerical technique based on the combination of the Tau Method and the Method of Lines is used, in connection with the domain decomposition technique, to solve problems in partial differential equations defined on irregular domains. Two nontrivial problems have been considered. The first is a curved crack defined on a square domain and the second is defined on a kite-shaped domain. The domain of interest is subdivided into appropriate Semidiscretized elements so to efficiently deal with any appearance of boundary and/or interior singularities. Numerical application is carried out on the Poisson's equation. The approximate solutions are sought along segmented lines as finite expansions in terms of a given orthogonal polynomial basis. Two types of orthogonal expansions have been used alongside second and fourth order accurate Finite Difference Approximations. In both cases good and rapid convergence has been achieved.