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An Optimal Multigrid Algorithm for the Combining \(P_1\)-\(Q_1\) Finite Element Approximations of Interface Problems Based on Local Anisotropic Fitting Meshes

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Abstract

A new finite element method is proposed for second order elliptic interface problems based on a local anisotropic fitting mixed mesh. The local anisotropic fitting mixed mesh is generated from an interface-unfitted mesh by simply connecting the intersected points of the interface and the underlying mesh successively. Optimal approximation capabilities on anisotropic elements are proved, the convergence rates are linear and quadratic in \(H^1\) and \(L^2\) norms, respectively. The discrete system is usually ill-conditioned due to anisotropic and small elements near the interface. Thereupon, a new multigrid method is presented to handle this issue. The convergence rate of the multigrid method is shown to be optimal with respect to both the coefficient jump ratio and mesh size. Numerical experiments are presented to demonstrate the theoretical results.

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, China

    Jun Hu & Hua Wang

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  1. Jun Hu
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  2. Hua Wang
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Correspondence to Hua Wang.

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In this research, Jun Hu was supported by NSFC projects 11625101 and 11421101; Hua Wang was supported by China Postdoctoral Science Foundation Grand 2019M660277 and Jiangsu Key Lab for NSLSCS Grant 201906.

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Hu, J., Wang, H. An Optimal Multigrid Algorithm for the Combining \(P_1\)-\(Q_1\) Finite Element Approximations of Interface Problems Based on Local Anisotropic Fitting Meshes. J Sci Comput 88, 16 (2021). https://doi.org/10.1007/s10915-021-01536-6

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  • DOI: https://doi.org/10.1007/s10915-021-01536-6

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