Superconvergence analysis of partially penalized immersed finite element method

TL;DR

This paper proves a supercloseness result for the PPIFE method and demonstrates that a gradient recovery technique yields superconvergent and accurate gradient estimates, enhancing error estimation for interface problems.

Contribution

The paper establishes a supercloseness property for the PPIFE method and applies a gradient recovery technique to achieve superconvergent gradient approximation.

Findings

Supercloseness of PPIFE method proved.

Gradient recovery achieves $ ext{O}(h^{3/2})$ superconvergence.

Numerical examples confirm theoretical results.

Abstract

The contribution of this paper contains two parts: first, we prove a supercloseness result for the partially penalized immersed finite element (PPIFE) method in [T. Lin, Y. Lin, and X. Zhang, SIAM J. Numer. Anal., 53 (2015), 1121--1144]; then based on the supercloseness result, we show that the gradient recovery method proposed in our previous work [H. Guo and X. Yang, arXiv: 1608.00063] can be applied to the PPIFE method and the recovered gradient converges to the exact gradient with a superconvergent rate $\mathcal{O}(h^{3/2})$. Hence, the gradient recovery method provides an asymptotically exact a posteriori error estimator for the PPIFE method. Several numerical examples are presented to verify our theoretical result.