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Optimal Finite Element Methods for Interface Problems

  • Conference paper
Domain Decomposition Methods in Science and Engineering XXII

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 104))

Abstract

There are many physical problems such as multiphase flows and fluid-structure interactions whose solutions are piecewise smooth but may have discontinuity across some curved interfaces. The direct application of standard finite element method may not perform well. In this paper, we study some special finite element methods for this type of problems. For simplicity of exposition, we consider the case that there is only one interface which is smooth. Let \(\varOmega,\varOmega _{1} \subset \mathbb{R}^{2}\) be two bounded domains with \(\varOmega _{1} \subset \varOmega\). We assume that Γ = ∂ Ω 1 is sufficiently smooth, and \(\varGamma \cap \partial \varOmega =\emptyset\). To be focused on the influence of Γ, we assume \(\varOmega = (-1,1)^{2}\).

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Notes

  1. 1.

    Both authors are supported by the Department of Energy (DOE) Grants DE-SC0006903 and DE-SC0009249 (through the Applied Mathematics Program within the DOE Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4)) and the Center for Computational Mathematics and Applications of the Pennsylvania State University. The second author is also supported by the NSFC Grant 11101415 and SRF for ROCS by SEM of P. R. China.

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Acknowledgement

The authors would like to thank Dr. Xiaozhe Hu for his help on the numerical examples.

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Correspondence to Jinchao Xu .

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Xu, J., Zhang, S. (2016). Optimal Finite Element Methods for Interface Problems. In: Dickopf, T., Gander, M., Halpern, L., Krause, R., Pavarino, L. (eds) Domain Decomposition Methods in Science and Engineering XXII. Lecture Notes in Computational Science and Engineering, vol 104. Springer, Cham. https://doi.org/10.1007/978-3-319-18827-0_7

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