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An Unfitted Finite Element Method by Direct Extension for Elliptic Problems on Domains with Curved Boundaries and Interfaces

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Abstract

We propose and analyze an unfitted finite element method of arbitrary order for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. We apply a non-symmetric bilinear form and the boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. The optimal convergence rate under the energy norm is derived, and \(O(h^{-2})\)-upper bounds of the condition numbers are shown for the final linear systems. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.

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Acknowledgements

The authors would like to thank the anonymous referee sincerely for the constructive comments that improve the quality of this paper. This work was supported by National Natural Science Foundation of China (12201442, 12171340, 11971041) and National Key R &D Program of China (2020YFA0714000).

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  1. School of Mathematics, Sichuan University, Chengdu, 610064, China

    Fanyi Yang & Xiaoping Xie

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  1. Fanyi Yang
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Correspondence to Xiaoping Xie.

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Yang, F., Xie, X. An Unfitted Finite Element Method by Direct Extension for Elliptic Problems on Domains with Curved Boundaries and Interfaces. J Sci Comput 93, 75 (2022). https://doi.org/10.1007/s10915-022-02035-y

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