An Immersed Finite Element Method for Anisotropic Elliptic Interface Problems with Nonhomogeneous Jump Conditions

TL;DR

This paper introduces an immersed finite element method for anisotropic elliptic interface problems with nonhomogeneous jump conditions, achieving optimal convergence and robustness without mesh alignment.

Contribution

The paper develops a novel FEM with a modified function space that handles nonhomogeneous jumps on non-aligned meshes, with proven optimal error estimates and an effective preconditioner.

Findings

Optimal error estimates independent of interface location

Robust preconditioner with Gauss-Seidel smoother

Numerical experiments confirm convergence and efficiency

Abstract

A new finite element method (FEM) using meshes that do not necessarily align with the interface is developed for two- and three-dimensional anisotropic elliptic interface problems with nonhomogeneous jump conditions. The degrees of freedom of the proposed method are the same as those of traditional nonconforming FEMs, while the function space is modified to account for the jump conditions of the solution. The modified function space on an interface element is shown to exist uniquely, independent of the element's shape and the manner in which the interface intersects it. Optimal error estimates for the method, along with the usual bound on the condition number of the stiffness matrix, are proven, with the error constant independent of the interface's location relative to the mesh. To solve the resulting linear system, a preconditioner is proposed in which a Gauss-Seidel smoother with the interface correction is employed to ensure robustness against large jumps in the diffusion matrix. Numerical experiments are provided to demonstrate the optimal convergence of the proposed method and the efficiency of the preconditioner.