A Well-Conditioned, Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems

Abstract

In this paper, we introduce a nonconforming Nitsche's extended finite element method (NXFEM) for elliptic interface problems on unfitted triangulation elements. The solution on each side of the interface is separately expanded in the standard nonconforming piecewise linear polynomials with the edge averages as degrees of freedom. The jump conditions on the interface and the discontinuities on the cut edges (the segment of edges cut by the interface) are weakly enforced by the Nitsche's approach. In the method, the harmonic weighted fluxes are used and the extra stabilization terms on the interface edges and cut edges are added to guarantee the stability and the well conditioning. We prove that the convergence order of the errors in energy and L 2 norms are optimal. Moreover, the errors are independent of the position of the interface relative to the mesh and the ratio of the discontinuous coefficients. Furthermore, we prove that the condition number of the system matrix is independent of the interface position. Numerical examples are given to confirm the theoretical results.

A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems

“…In order to estimate the error of our method, the following trace inequality is needed for the cut segments totally contained in Ω i . We have proved in [22].…”

“…This Nitsche's XFE method (NXFEM) was originally considered in [19] to solve the elliptic interface problems. Then a large number of related methods have been developed, such as [2,6,9,10,22,24,32,34,37,38] for elliptic interface problems, [3,11,20,25,35,36] for Stokes interface problems and [31] for Oseen problems.…”

“…Therefore, the natural idea is that same finite element spaces would be adequate to solve Stokes interface problem by the NXFEM formulation. In [22], we have studied a nonconforming NXFEM to solve elliptic interface problems. Thus, we want to apply it to solve Stokes interface problems.…”

“…In this paper, we will use the nonconforming NXFEM of [22] and propose an accurate and stable extended finite element method for Stokes interface problems based on nonconforming-P 1 /P 0 shape functions with the unfitted-interface mesh. Although the same spaces considered in this paper (compared to [13,35]), we mention the following contributions of this paper.…”

“…where χ i = χ| Ωi , i = 1, 2 and we use the so-called "harmonic weights" as adopted by [7,8,14,21,22,24],…”