Extension of NXFEM to nonconforming finite elements

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

“…Therefore, it is necessary to analyze the nonconforming finite element method to solve the interface problems. We can refer to the works [9,[33][34][35] for the nonconforming finite element method implementation. For example, the immersed finite element methods with the nonconforming elements on triangular meshes and rectangular meshes are proposed in [33] and [9] respectively.…”

“…It is shown that optimal a priori error estimates are robust with respect to the geometry and to the diffusion parameters. And then [34] presented the nonconforming Nitsche's extended finite element method on triangular meshes in two cases for an elliptic interface problem. One is the same as that method proposed in [35], and the other is obtained by modifying the nonconforming basis function on the cut elements.…”

“…In this paper, we follow the partition in [34,35], allowing elements cut by the interface and defining the enriched nonconforming finite element space. The difference from [34,35] is that we use the weighted average flux only related to coefficients instead of that related to coefficients and subareas of the interface elements across the interface to guarantee the errors to be independent of the ratio of the discontinuous coefficients. This weighted average flux was used in [18] to solve elliptic interface problems with continuous piecewise finite elements.…”

“…Comparing with the conforming elements, the nonconforming elements exhibit discontinuities across the local cut segments of the edges for the cut elements. Thus, special dispose for the cut segments is needed (see [34,35]). In this paper, we use the arithmetic average flux instead of the weighted average one on the cut segments used in [34,35].…”

“…Thus, special dispose for the cut segments is needed (see [34,35]). In this paper, we use the arithmetic average flux instead of the weighted average one on the cut segments used in [34,35]. Extra stabilization terms are added in the bilinear form, which adopt the techniques in [16,26] (here we use different basis functions) to guarantee the stability and a well conditioning.…”