Communications in Mathematical Sciences

Volume 21 (2023)

Number 4

Residual-based a posteriori error estimation for elliptic interface problems approximated by immersed finite element methods

Pages: 997 – 1018

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n4.a5

Authors

Yanping Chen (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Jiao Lu (School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, China)

Yang Wang (School of Mathematics and Statistics, Hubei Normal University, Huangshi, China)

Yunqing Huang (School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, China)

Abstract

This paper studies a residual-based a posteriori error estimator for partially penalized immersed finite element (PPIFE) approximation to elliptic interface problems. Utilizing the error equation for the PPIFE approximation, we construct an a posteriori error estimator. Properly weighted coefficients are proposed for the terms in indicators to overcome the dependence of the efficiency constants on the jump of the diffusion coefficients across the interface. The PPIFE method is based on non-body-fitted mesh, and hence we perform detailed analysis on the local efficiency bounds of the estimator on regular and irregular interface elements with different techniques. We introduce a new approach, which does not involve the Helmholtz decomposition, to give the reliability bounds of the estimator with an $L^2$ representation of the true error as the main tool. More importantly, the efficiency and reliability constants are independent of the interface location and the mesh size. Numerical experiments are provided to illustrate the efficiency of the estimator and the adaptive mesh refinement for different jump rates or interface geometries.

Received 24 December 2021

Received revised 13 August 2022

Accepted 5 September 2022

Published 24 March 2023