An improved non-traditional finite element formulation for solving three-dimensional elliptic interface problems

Solving elliptic equations with interfaces has wide applications in engineering and science. The real world problems are mostly in three dimensions, while an efficient and accurate solver is a challenge. Some existing methods that work well in two dimensions are too complicated to be generalized to three dimensions. Although traditional finite element method using body-fitted grid is well-established, the expensive cost of mesh generation is an issue. In this paper, an efficient non-traditional finite element method with non-body-fitted grids is proposed to solve elliptic interface problems. The special cases when the interface cuts though grid points are handled carefully, rather than perturbing the cutting point away to apply the method for general case. Both Dirichlet and Neumann boundary conditions are considered. Numerical experiments show that this method is approximately second order accurate in the L∞ norm and L2 norm for piecewise smooth solutions. The large sparse matrix for our linear system also has nice structure and properties.