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 optima...
Published Version
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