A hybridizable discontinuous Galerkin method for elliptic interface problems in the formulation of boundary integral equations

Abstract

A novel numerical method is presented to solve the general elliptic interface problem with piecewise constant diffusion coefficients. To overcome the difficulties caused by piecewise constant diffusion coefficients, a combination of the kernel-free boundary integral (KFBI) method and the unfitted mesh-based hybridizable discontinuous Galerkin (HDG) method is considered. To this end, we first reformulate the elliptic interface problem into a Fredholm boundary integral equation of the second kind, and further solve the well-conditioned discrete integral equation iteratively with the GMRES method. In each iteration, evaluation of boundary and volume integrals is made by interpolating numerical solutions to equivalent but simple interface problems, which is efficiently calculated with a HDG method on unfitted meshes. The advantage of using the unfitted HDG method to treat the equivalent but simple interface problem is that the interpolation along boundary integrals on the interface can be easily evaluated via a linear Lagrange polynomial element by element, which is an important improvement from Ying and Henriquez (2007). Numerical experiments are given to verify the effectiveness of the novel method, which has second-order accuracy even in the case of high contrast diffusion coefficients.