Fast Domain Decomposition Algorithms for Elliptic Problems with Piecewise Variable Orthotropism

Abstract

Second order elliptic equations are considered in the unit square, which is decomposed into subdomains by an arbitrary nonuniform orthogonal grid. For the elliptic operator we assume that the energy integral contains only squares of first order derivatives with coefficients, which are arbitrary positive finite numbers but different for each subdomain. The orthogonal finite element mesh has to satisfy only one condition: it is uniform on each subdomain. No other conditions on the coefficients of the elliptic equation and on the step sizes of the discretization and decomposition are imposed. For the resulting discrete finite element problem, we suggest domain decomposition algorithms of linear total arithmetical complexity, not depending on any of the three factors contributing to the orthotropism of the discretization on subdomains. The main problem of designing such an algorithm is the preconditioning of the inter-subdomain Schur complement, which is related in part to obtaining boundary norms for discrete harmonic functions on the shape irregular domains.KeywordsElliptic ProblemDomain DecompositionDomain Decomposition MethodDomain Decomposition AlgorithmMesh LineThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.