High-Order Enriched Finite Element Methods for Elliptic Interface Problems with Discontinuous Solutions
Year: 2023
Author: Champike Attanayake, So-Hsiang Chou, Quanling Deng
International Journal of Numerical Analysis and Modeling, Vol. 20 (2023), Iss. 6 : pp. 870–895
Abstract
Elliptic interface problems whose solutions are $C^0$ continuous have been well studied over the past two decades. The well-known numerical methods include the strongly stable generalized finite element method (SGFEM) and immersed FEM (IFEM). In this paper, we study numerically a larger class of elliptic interface problems where their solutions are discontinuous. A direct application of these existing methods fails immediately as the approximate solution is in a larger space that covers discontinuous functions. We propose a class of high-order enriched unfitted FEMs to solve these problems with implicit or Robin-type interface jump conditions. We design new enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. A linear enriched method in 1D was recently developed using one enrichment function and we generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. The natural tensor product extension to the 2D case is demonstrated. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. We also establish superconvergence at all discretization nodes (including exact nodal values in special cases). Numerical examples are provided to confirm the theory. Finally, to prove the efficiency of the method for practical problems, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ijnam2023-1038
International Journal of Numerical Analysis and Modeling, Vol. 20 (2023), Iss. 6 : pp. 870–895
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 26
Keywords: Generalized finite element method elliptic interface implicit interface jump condition Robin interface jump condition linear and quadratic finite elements.