TY - JOUR
T1 - High-order extended finite element methods for solving interface problems
AU - Xiao, Yuanming
AU - Xu, Jinchao
AU - Wang, Fei
N1 - Funding Information:
The first author was partially supported by the National Key Research and Development Program of China2017YFC0209804 and by the NSF of China grant 11101208. The second author was partially supported by Verne M. Willamen Chair Fund, United States of America and National Science Foundation, United States of America Grant DMS-1819157. The third author was partially supported by the NSF of China grant 11771350.
Publisher Copyright:
© 2020 Elsevier B.V.
PY - 2020/6/1
Y1 - 2020/6/1
N2 - 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.
AB - 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.
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U2 - 10.1016/j.cma.2020.112964
DO - 10.1016/j.cma.2020.112964
M3 - Article
AN - SCOPUS:85081648977
SN - 0045-7825
VL - 364
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
M1 - 112964
ER -