An exact-interface-fitted mesh generator and linearity-preserving finite volume scheme for anisotropic elliptic interface problems

Abstract

Anisotropic elliptic interface problems with non-homogeneous jump conditions are of great importance but hard to solve numerically. In this work, we develop an easy-to-implement exact-interface-fitted mesh generation (EIFMG) algorithm, and then propose a linearity-preserving finite volume (FV) scheme for solving such problems. Firstly, by curving the cell edges to align with the material interface, the EIFMG algorithm produces a structured exact-interface-fitted curved quadrilateral mesh without changing the total number of cells and associated topology. Secondly, by employing the so-called linearity preserving criterion, our derived FV scheme can be exact on curved quadrilateral meshes for constant coefficient problems with a linear solution even with extremely coarse mesh, which is generally not easy for finite element or finite difference methods to achieve. Moreover, a new method for interpolating vertex unknowns with cell-centered ones is also developed for non-homogeneous jump conditions of solutions and normal fluxes. Finally, numerical results are presented to verify the second-order accuracy and linearity-preserving ability of the proposed FV scheme. As far as we know, the algorithm based on FV method in this paper seems to be the best way to handle anisotropic elliptic interface problems with nonhomogeneous jump conditions, judging from the computational performance and efficiency of many typical numerical examples.