Highly accurate finite element method for one-dimensional elliptic interface problems

A high order finite element method for one-dimensional elliptic interface problems is presented. Due to presence of these interfaces the problem will contain discontinuities in the coefficients and singularities in the right hand side that are represented by delta functional with the support on the interfaces. As a result, the solution to the interface problem and its derivatives may have jump discontinuities. The proposed method is specifically designed to handle this features of the solution using non-body fitted grids, i.e. the grids are not aligned with the interfaces.The finite element method will be based on third order Hermitian interpolation. The main idea is to modify the basis functions in the vicinity of the interface such that the jump conditions are well approximated. A rigorous error analysis shows that the presented finite element method is fourth order accurate in L2 norm. The numerical results agree well with the theoretical analysis. The basic idea can easily be generalized to other finite element ansatz functions.