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

This is the second paper on the study of gradient recovery for elliptic interface problem. In our previous work Guo and Yang (2016) [17], we developed a novel gradient recovery technique for finite element method based on the body-fitted mesh. In this paper, we propose new gradient recovery methods for two immersed interface finite element methods: symmetric and consistent immersed finite method (Ji et al. (2014) [23]) and Petrov-Galerkin immersed finite element method (Hou et al. (2004) [22], and Hou and Liu (2005) [20]). Compared to the body-fitted mesh based gradient recovery method, the new 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.