On the accuracy of finite element approximations to a class of interface problems

We define piecewise linear and continuous finite element methods for a class of interface problems in two dimensions. Correction terms are added to the right-hand side of the natural method to render it second-order accurate. We prove that the method is second-order accurate on general quasiuniform meshes at the nodal points. Finally, we show that the natural method, although non-optimal near the interface, is optimal for points O(√h log(1/h)) away from the interface.