A Domain Decomposition Finite Difference Method for Singularly Perturbed Elliptic Equations in Composed Domains

Abstract

Numerical modelling of stationary heat and mass transfer processes in composite materials often leads to singularly perturbed problems in composed domains, that is, to elliptic equations with discontinuous coefficients and a small parameter ɛ multiplying the highest derivatives. The concentrated source acts on the interface boundary. For such problems the application of domain decomposition (DD) methods seems quite reasonable: the original domain is naturally partitioned into several non-overlapping subdomains with smooth coefficients. Due to the presence of transition and boundary layers, standard numerical methods yield large errors for small ɛ. By this reason, we need for special methods whose errors are independent of the parameter ɛ. To construct such DD schemes possessing the property of ɛ-uniform convergence, we use standard finite difference approximations on piecewise uniform grids, which are a priori refined in the transition and boundary layers.

This research was supported by the Russian Foundation for Basic Research (grant No. 98-01-00362) and partially by the NWO grant (dossiernr. 047.008.007).