High-order extended finite element methods for solving interface problems

• Published in: Computer methods in applied mechanics and engineering Vol. 364; p. 112964
• Main Authors: Xiao, Yuanming, Xu, Jinchao, Wang, Fei
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.06.2020; Elsevier BV
• Subjects: Convergence; Elliptic interface problems; Extended finite element; Finite element method; Galerkin method; High order; Multigrid solver; Parameters; Polynomials; Unfitted mesh; Extended finite element; Multigrid solver; Unfitted mesh; Elliptic interface problems; High order
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2020.112964

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 H1-norm and L2-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. This paper is adapted from the work originally post on arXiv.com by the same authors (arXiv:1604.06171). New ingredients are an optimal multigrid solver for the generated linear system and its analysis. This multigrid method converges uniformly with respect to the mesh size, and is independent of the location of the interface relative to the meshes, just like all the other estimates in this paper. Numerical examples are given to support the theoretical results. •The methods are of arbitrary high order with optimal convergence rates.•Parameters need not be “sufficiently large” to ensure the stability of the scheme.•A multigrid solver is proved to converge uniformly with respect to the mesh size.•Estimates are independent of the location of the interface relative to the meshes.