A Locally Modified Parametric Finite Element Method for Interface Problems

We present a modified finite element method that is able to approximate interface problems with high accuracy. We consider interface problems where the solution is continuous; its derivatives, however, may be discontinuous across interface curves within the domain. The proposed discretization is based on a local modification of the finite element basis functions using a fixed quadrilateral mesh. Instead of moving mesh nodes, we resolve the interface locally by an adapted parametric approach. All modifications are applied locally and in an implicit fashion. The scheme is easy to implement and is well suited for time-dependent moving interface problems. We show optimal order of convergence for elliptic problems, and further, we give a bound on the condition number of the system matrix. Both estimates do not depend on the interface location relative to the mesh.