An Analysis of a Broken $P_1$-Nonconforming Finite Element Method for Interface Problems

Abstract

We study some numerical methods for solving a second order elliptic problem with interface. We introduce an immersed finite element method based on the “broken” $P_1$-nonconforming piecewise linear polynomials on interface triangular elements having edge averages as degrees of freedom. These linear polynomials are broken to match the homogeneous jump condition along the interface which is allowed to cut through the element. We prove optimal orders of convergence in the $H^1$- and $L^2$-norm. Next we propose a mixed finite volume method in the context introduced in [S. H. Chou, D. Y. Kwak, and K. Y. Kim, Math. Comp., 72 (2003), pp. 525-539] using the Raviart-Thomas mixed finite element and this “broken” $P_1$-nonconforming element. The advantage of this mixed finite volume method is that once we solve the symmetric positive definite pressure equation (without Lagrangian multiplier), the velocity can be computed locally by a simple formula. This procedure avoids solving the saddle point problem. Furthermore, we show optimal error estimates of velocity and pressure in our mixed finite volume method. Numerical results show optimal orders of error in the $L^2$-norm and broken $H^1$-norm for the pressure and in the $H(\mathrm{div})$-norm for the velocity.