Gradient recovery for elliptic interface problem: II. immersed finite element methods

TL;DR

This paper introduces new gradient recovery methods for elliptic interface problems using immersed finite element methods, demonstrating superconvergence and accurate a posteriori error estimation on regular meshes.

Contribution

It develops gradient recovery techniques based on immersed finite element methods, extending previous work on body-fitted meshes and providing superconvergence results.

Findings

Numerical examples confirm superconvergence of the proposed methods.

The methods provide asymptotically exact a posteriori error estimators.

Immersed finite element methods enable gradient recovery on regular meshes.

Abstract

This is the second paper on the study of gradient recovery for elliptic interface problem. In our previous work [H. Guo and X. Yang, 2016, arXiv:1607.05898], we developed gradient recovery finite element method based on body-fitted mesh. In this paper, we propose new gradient recovery methods based on two immersed interface finite element methods: symmetric and consistent immersed finite method [H. Ji, J. Chen and Z. Li, J. Sci. Comput., 61 (2014), 533--557] and Petrov-Galerkin immersed finite element method [T.Y. Hou, X.-H. Wu and Y. Zhang, Commun. Math. Sci., 2 (2004), 185--205, and S. Hou and X.-D. Liu, J. Comput. Phys., 202 (2005), 411--445]. Compared to body-fitted mesh based gradient recover methods, immersed finite element methods provide a uniform way of recovering gradient on regular meshes. Numerical examples are presented to confirm the superconvergence of both gradient recovery methods. Moreover, they provide asymptotically exact a posteriori error estimators for both immersed finite element methods.