Superconvergence analysis of partially penalized immersed finite element method
This work provides theoretical justification for gradient recovery in immersed finite element methods, which is important for practitioners solving interface problems, but the result is incremental.
The paper proves a supercloseness result for the partially penalized immersed finite element method and shows that a gradient recovery technique yields superconvergent gradient approximation with rate O(h^{3/2}), providing an asymptotically exact a posteriori error estimator.
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.