Domain decomposition techniques for reaction–diffusion equations in two-dimensional regions with re-entrant corners

Abstract

A system of two non-linear reaction–diffusion equations is solved numerically by means of linearized θ-methods and both overlapping and non-overlapping domain decomposition techniques in two-dimensional regions with re-entrant corners. Two numerical methods based on either approximate factorization (AF) or the bi-conjugate-gradient-stabilized (BiCGstab) technique are employed. A study of the effects of the number of overlapping grid lines on both the accuracy and numerical efficiency is presented. For non-overlapping domain decomposition techniques, the unknown values at the common interface between adjacent subdomains have been updated by means of Dirichlet, Neumann and Robin couplings, and combinations thereof. It is shown that non-overlapping domain techniques are less accurate than overlapping ones for domains with re-entrant corners because the interfaces between adjacent subdomains are evaluated by imposing continuity of the unknowns and their normal derivatives there, and, therefore, the partial differential equations are not solved at the interfaces between adjacent subdomains. Nevertheless, the accuracy of these techniques increases as the grid spacing is decreased, although they still exhibit large errors near the re-entrant corners.