Domain Decomposition Scheme for First-Order Evolution Equations with Nonselfadjoint Operators

Abstract

Domain decomposition iterative methods and implicit schemes are usually used for solving evolution equations. An alternative approach is based on constructing non-iterative method based on special schemes of splitting into subdomains. Such regional-additive schemes are based on the general theory of additive operator-difference schemes. Domain decomposition analogues of the classical schemes of alternating direction method, locally one-dimensional schemes, factored schemes, and regularized vector-additive schemes are used here. The main results in the literature are obtained for time-dependent problems with selfadjoint second-order elliptic operators. This paper discusses the Cauchy problem for first-order evolution equations with nonnegative nonselfadjoint operators in a finite-dimensional Hilbert space. Based on the partition of unity, we have constructed nonnegativity preserving decomposition operators for the respective operator term in the equation. We construct unconditionally stable additive domain decomposition schemes based on the principle of regularization of operator-difference schemes and vector-additive schemes.KeywordsFirst-order evolution equationsParabolic partial differencial equationDomain decomposition methodDifference scheme.