Accurate gradient computations at interfaces using finite element methods

TL;DR

This paper introduces new finite element methods for elliptic interface problems that achieve high accuracy in both solutions and derivatives at interfaces in 1D and 2D, with rigorous analysis and numerical validation.

Contribution

The paper proposes novel finite element techniques that accurately compute solutions and derivatives at interfaces, including a mixed approach in 2D, with proven convergence properties.

Findings

Second order convergence for solutions in 1D and 2D.

Super-convergence of gradients at interfaces in 2D.

Method maintains low computational cost similar to standard FEM.

Abstract

New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea from \cite{wheeler1974galerkin}. For 2D interface problems, the idea is to introduce a small tube near the interface and introduce the gradient as part of unknowns, which is similar to a mixed finite element method, except only at the interface. Thus the computational cost is just slightly higher than the standard finite element method. We present rigorous one dimensional analysis, which show second order convergence order for both of the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface.