Heterogeneous Domain Decomposition for Singularly Perturbed Elliptic Boundary Value Problems

Abstract

A heterogeneous domain-decomposition method is presented for the numerical solution of singularly perturbed elliptic boundary value problems. The method, which is parallelizable at various levels, uses several ideas of asymptotic analysis. The subdomains match the domains of validity of the local (inner and outer) asymptotic expansions, and cut-off functions are used to match solutions in neighboring subdomains. The positions of the interfaces, as well as the mesh widths, depend on the small parameter $\varepsilon$. On the subdomains, iterative solution techniques are used, which may vary from one subdomain to another. The global convergence rate depends on $\varepsilon$; it generally increases like some power of $(\log (\varepsilon ^{-1}))^{-1}$ as $\varepsilon \downarrow 0$. The method is illustrated on several two-dimensional singular perturbation problems.