A Cell-Centered Multigrid Solver for the Finite Volume Discretization of Anisotropic Elliptic Interface Problems on Irregular Domains (Journal of Computational Mathematics, 2024)
The aim of this paper is to develop a fast multigrid solver for interpolation-free finitevolume (FV) discretization of anisotropic elliptic interface problems on general boundeddomains that can be described as a union of blocks. We assume that the curved interfacefalls exactly on the boundaries of blocks. The transfinite interpolation technique is appliedto generate block-wise distorted quadrilateral meshes, which can resolve the interface withfine geometric details. By an extensive study of the harmonic average point method,an interpolation-free nine-point FV scheme is then derived on such multi-block grids foranisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover,for the resulting linear algebraic systems from cell-centered FV discretization, a high-orderprolongation operator based fast cascadic multigrid solver is developed and shown to berobust with respect to both the problem size and the jump of the diffusion coefficients.Various non-trivial examples including four interface problems and an elliptic problem incomplex domain without interface, all with tens of millions of unknowns, are provided toshow that the proposed multigrid solver is dozens of times faster than the classical algebraicmultigrid method as implemented in the code AMG1R5 by Stüben.
Keywords: Elliptic interface problem; Discontinuous coefficients; Anisotropic coefficients; Cascadic multigrid method; Richardson extrapolation.