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Journal of Scientific Computing

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01-07-2021

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

Authors: Jun Hu, Hua Wang

Published in: Journal of Scientific Computing | Issue 1/2021

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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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Title: An Optimal Multigrid Algorithm for the Combining - Finite Element Approximations of Interface Problems Based on Local Anisotropic Fitting Meshes
Authors: Jun Hu
Hua Wang
Publication date: 01-07-2021
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2021
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-021-01536-6

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