Immersed interface method using a finite element formulation

A finite element method is proposed for one dimensional interface problems involving discontinuities in the coefficients of the differential equations and the derivatives of the solutions. The interfaces do not have to be one of grid points. The idea is to construct basis functions which satisfy the interface jump conditions. By constructing an interpolating function of the solution, we are able to give a rigorous error analysis which shows that the approximate solution obtained from the finite element method is second order accurate in the infinity norm. Numerical examples are also provided to support the method and the theoretical analysis. Several numerical approaches are also proposed for dealing with two dimensional problems involving interfaces.