Adaptive mesh refinement for elliptic interface problems using the non-conforming immersed finite element method
C. Wu 1, Z. Li 2, M. Lai 31 Department of Applied Mathematics, National Chiao-Tung University, 1001, Ta Hsueh Road, Hsinchu 300, Taiwan.
2 Department of Mathematics, North Carolina State University, Raleigh, NC. 27695-8205, USA, and School of Mathematical Sciences, Nanjing Normal University, Nanjing, China.
3 Center of Mathematical Modeling and Scientific Computing, National Chiao-Tung University, 1001, Ta Hsueh Road, Hsinchu 300, Taiwan.
Received by the editors December 30, 2010.
In this paper, an adaptive mesh refinement technique is developed and analyzed for the non-conforming immersed finite element (IFE) method proposed in . 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(h^2) and O(h) in L^2 norm and H^1 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 diffusion coefficient beta^- and beta^+, which improves the corresponding result in . We also show that aposteriori error estimate similar to the one obtained by Bernardi and Verf¡Lurth  holds for the IFE solutions. Numerical examples support our theoretical results and show that the adaptive mesh refinement strategy is effective for the IFE approximation.AMS subject classifications: 65M06, 76D45
Key words: interface, immersed finite element, adaptive mesh.
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