@Article{IJNAM-20-6,
author = {Champike, Attanayake and So-Hsiang, Chou and Quanling, Deng},
title = {High-Order Enriched Finite Element Methods for Elliptic Interface Problems with Discontinuous Solutions},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2023},
volume = {20},
number = {6},
pages = {870--895},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2617-8710},
doi = {https://doi.org/10.4208/ijnam2023-1038},
url = {https://global-sci.com/article/82920/high-order-enriched-finite-element-methods-for-elliptic-interface-problems-with-discontinuous-solutions}
}