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A Conforming Virtual Element Method Based on Unfitted Meshes for the Elliptic Interface Problem

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Abstract

In this paper, we propose a conforming virtual element method based on an unfitted mesh to solve the elliptic interface problem in two dimensions. The intersecting points of the interface and the edges of triangles are considered as additional nodes of the mesh. Thus each interface triangle is regarded as a polygon with more than three vertices. On each interface polygon, we introduce a virtual element satisfying the interface conditions. On each non-interface triangle, we use the usual linear element. Based on a computable projection-like operator, we introduce our discrete scheme. Both the approximation and consistency errors are analyzed rigorously and all the hidden constants do not depend on how the interface intersects with the meshes. The error between the exact and discrete solution is shown to decrease linear with regard to the mesh size. Some numerical experiments are provided to verify the theoretical results.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

We would like to thank the referees for many valuables comments and suggestions, which lead to a significantly improved presentation of this paper. F. Wang is partially supported by the National Natural Science Foundation of China (Grant No. 12071227), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 20KJA110001) and the National Key Research and Development Program of China (No. 2020YFA0713803); J. Chen is partially supported by the National Natural Science Foundation of China (Grant No. 11871281, 11731007); H. Ji is partially supported by the National Natural Science Foundation of China (Grant No. 11701291, 12101327 and 11801281) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20200848).

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Authors and Affiliations

  1. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Haimei Wang, Feng Wang & Jinru Chen

  2. School of Mathematical Sciences, Jiangsu Second Normal University, Nanjing, 211200, China

    Jinru Chen

  3. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Haifeng Ji

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  1. Haimei Wang
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  2. Feng Wang
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Correspondence to Feng Wang.

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Wang, H., Wang, F., Chen, J. et al. A Conforming Virtual Element Method Based on Unfitted Meshes for the Elliptic Interface Problem. J Sci Comput 96, 21 (2023). https://doi.org/10.1007/s10915-023-02229-y

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