A simple method for matrix-valued coefficient elliptic equations with sharp-edged interfaces

The traditional finite element method has a number of nice properties, and thus it is highly desired for matrix-valued coefficient elliptic equations with sharp-edged interfaces. However, its efficient implementation with body-fitting grids for such problems is highly nontrivial. In this paper, we propose a simple finite element method with body-fitting grids based on semi-Cartesian grid, which makes its implementation fairly straightforward, even for complicated geometry. All possible situations that the interface cuts the grid are considered. Further, the symmetry and positive definiteness of the resulting matrix are ensured for positive definite coefficients. Numerical experiments show that it is second order accurate in the L∞ norm and numerically very stable.