A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces

Abstract

In this paper, a new finite element and finite difference (FE-FD) method has been developed for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes. In the spatial discretization, the standard $P_1$ FE discretization is applied so that the part of the coefficient matrix is symmetric positive definite, while near the interface, the maximum principle preserving immersed interface discretization is applied. In the time discretization, a modified Crank-Nicolson discretization is employed so that the hybrid FE-FD is stable and second order accurate. Correction terms are needed when the interface crosses grid lines. The moving interface is represented by the zero level set of a Lipschitz continuous function. Numerical experiments presented in this paper confirm second order convergence.

Funding

Simons Foundation(633724); CNSF-Ningxia(2021AAC03234)