Approximated Unfitted Interface Penalty Finite Element Method For Elliptic Interface Problems

We consider the numerical implementation for linear unfitted interface penalty fi-nite element method for elliptic interface problems. A great deal of practical problems in scientific computing and industrial applications can be modeled by elliptic equation-s with discontinuous coefficient. Such problems with discontinuous coefficients are usually called interface problems. In [42], an hp version of interface penalty finite ele-ment method (hp-IPFEM) is proposed for elliptic interface problems in two and three dimensions on unfitted meshes. Error estimations in broken H1-norm, which are op-timal with respect to h, are derived. However, calculating the stiffness matrix on the interface points and surroundings is required mountains of work. In two dimensional case, we use piecewise line interface to approximate the curve interface and the grav-ity center or trapezoidal numerical integration formula to calculate integration. The practical algorithm is obtained and it reduces the amount of computation in numerical experiment. And further, this paper theoretically proves that these approximations do not affect the accuracy of interface penalty finite element solution. In other words, H1-error estimation between finite element solution after approximating numerical in-tegration and the exact solution for the original problem is optimal with respect to h. Finally, we verify the theoretical conclusions and the effectiveness of the approximated interface penalty finite element method by conducting numerical experiment.