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Superconvergence of partially penalized immersed finite element methods. (English) Zbl 1462.65187

Summary: The contribution of this article contains two parts: first, we prove a supercloseness result for the partially penalized immersed finite element (PPIFE) methods in [T. Lin et al., SIAM J. Numer. Anal. 53, No. 2, 1121–1144 (2015; Zbl 1316.65104)] and then based on the supercloseness result, we show that the gradient recovery method proposed in our previous work [H. Guo and X. Yang, J. Comput. Phys. 338, 606–619 (2017; Zbl 1415.65256)] can be applied to the PPIFE methods and the recovered gradient converges to the exact gradient with a superconvergent rate \(\mathcal O(h^{1.5})\). Hence, the gradient recovery method provides an asymptotically exact a posteriori error estimator for the PPIFE methods. Several numerical examples are presented to verify our theoretical results.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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