Adaptive mesh refinement for elliptic interface problems using the non-conforming immerse finite element method

In this paper, an adaptive mesh refinement technique is developted and analyzed for the non-conforming immersed finite element (IFE) method proposed in [25]. The IFE method was developed for solving the second order elliptic boundary value problem with interfaces across which the coefficient may be discontinuous. The IFE method was based on a triangulation that does not need to fit the interface. One of the key ideas of IFE method is to modify the basis functions so that the natural jump conditions are satisfied across the interface. The IFE method has shown to be order of O(h2) and O(h) in L2 norm and H1 norm, respectively. In order to develop the adaptive mesh refinement technique, additional priori and posterior error estimations are derived in this paper. Our new a priori error estimation shows that the generic constant is only linearly proportional to ratio of the diffusive coefficients β− and β+, which improves the corresponding result in [25]. ∗Corresponding author. Department of Applied Mathematics, National Chiao-Tung University, 1001, Ta Hsueh Road, Hsinchu 300, Taiwan. [email protected] †Center for Research in Scientific Computation & Department of Mathematics, North Carolina State University, Raleigh, NC. 27695-8205. ‡Center of mathematical modeling and Scientific computing, National Chiao-Tung University, 1001, Ta Hsueh Road, Hsinchu 300, Taiwan.