High-Order Extended Finite Element Methods for Solving Interface Problems

Abstract

In this paper, we study arbitrary order extended finite element (XFE) methods based on two discontinuous Galerkin (DG) schemes in order to solve elliptic interface problems in two and three dimensions. Optimal error estimates in the piecewise $H^1$-norm and in the $L^2$-norm are rigorously proved for both schemes. In particular, we have devised a new parameter-friendly DG-XFEM method, which means that no "sufficiently large" parameters are needed to ensure the optimal convergence of the scheme. To prove the stability of bilinear forms, we derive non-standard trace and inverse inequalities for high-order polynomials on curved sub-elements divided by the interface. All the estimates are independent of the location of the interface relative to the meshes. Numerical examples are given to support the theoretical results.

References in corpus (1)

• Estimate of the Convergence Rate of Finite Element Solutions to Elliptic Equations of Second Order with Discontinuous Coefficients

Cited by in corpus (6)

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• An immersed $CR$-$P_0$ element for Stokes interface problems and the optimal convergence analysis
• A Trilinear Immersed Finite Element Method for Solving Elliptic Interface Problems
• Extended HDG methods for second order elliptic interface problems
• A Nitsche-eXtended finite element method for distributed optimal control problems of elliptic interface equations