Special meshes and domain decomposition methods for evolutionary convection–diffusion–reaction problems

Abstract

AbstractIn this work we propose new parallel numerical methods to solve certain evolutionary singularly perturbed problems in a robust and efficient way. To get this, we firstly consider a semidiscretization in time using a simple fractional step Runge–Kutta method, where the splittings for the convection–diffusion–reaction operator and the source term are subordinated to a decomposition of the spatial domain in many smaller subdomains. Such semidiscretization procedure reduces the original problem to a set of elliptic problems (in smaller subdomains), which can be solved in parallel, and it avoids the use of Schwarz iterations. To discretize in space such problems we have considered classical linear finite elements on certain piecewise uniform meshes which have been constructed with a very simple a priori criterion, similar to the one introduced by Shishkin for one‐dimensional stationary problems of this kind. We show that the use of these meshes permits to obtain uniformly convergent approximations even in the boundary layer regions. Copyright © 2005 John Wiley & Sons, Ltd.