Local Mesh Refinement Multilevel Techniques

Abstract

The purpose of this study is to provide criteria for optimizing meshsizes near singularities and develop fast and flexible multigrid methods for creating the nonuniform grids, their difference equations and their solutions. For simplicity, the Poisson problem is studied, with singularities introduced either in the forcing terms (algebraic singularities or sources) or in the shape of the boundaries (re-entrant corners). Local refinements are created by multigrid structures in which some extra finer levels cover increasingly narrower neighborhoods of the singularity, as proposed in [6]. The main innovations here are: (1) Extra local relaxation sweeps near structural singularities (such as re-entrant corners) are employed to restore the asymptotic convergence rates to their values in regular (e.g. infinite) domains. (2) An exchange-rate algorithm ($\lambda $-FMG) is introduced to maintain linear dependence of solution time on number of gridpoints. With these two algorithmic modifications, and with the optimal-mesh criteria from [6, § 8], it is found that singular problems are solved in the same efficiency as regular problems, i.e., the same accuracy in solving the differential equations is obtained in essentially the same computational work. (3) In case of a source singularity it is shown that no local refinement is required to obtain a regular accuracy far from the source; if differencing is in conservation form, it is only important to represent correctly the strength of the source. (4) A technique is developed for retaining conservation forms where local refinements are introduced.