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Volume 39, Issue 3
Uniform Convergence of Multigrid V-Cycle on Adaptively Refined Finite Element Meshes for Elliptic Problems with Discontinuous Coefficients

Haijun Wu & Weiying Zheng

Commun. Math. Res., 39 (2023), pp. 437-475.

Published online: 2023-04

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  • Abstract

The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered. Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm, some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours. The multigrid V-cycle algorithm uses $\mathcal{O}(N)$ operations per iteration and is optimal.

  • AMS Subject Headings

65N55, 65N12

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COPYRIGHT: © Global Science Press

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@Article{CMR-39-437, author = {Wu , Haijun and Zheng , Weiying}, title = {Uniform Convergence of Multigrid V-Cycle on Adaptively Refined Finite Element Meshes for Elliptic Problems with Discontinuous Coefficients}, journal = {Communications in Mathematical Research }, year = {2023}, volume = {39}, number = {3}, pages = {437--475}, abstract = {

The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered. Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm, some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours. The multigrid V-cycle algorithm uses $\mathcal{O}(N)$ operations per iteration and is optimal.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2022-0047}, url = {http://global-sci.org/intro/article_detail/cmr/21610.html} }
TY - JOUR T1 - Uniform Convergence of Multigrid V-Cycle on Adaptively Refined Finite Element Meshes for Elliptic Problems with Discontinuous Coefficients AU - Wu , Haijun AU - Zheng , Weiying JO - Communications in Mathematical Research VL - 3 SP - 437 EP - 475 PY - 2023 DA - 2023/04 SN - 39 DO - http://doi.org/10.4208/cmr.2022-0047 UR - https://global-sci.org/intro/article_detail/cmr/21610.html KW - Multigrid, adaptive finite elements, elliptic problems, discontinuous coefficients, uniform convergence. AB -

The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered. Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm, some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours. The multigrid V-cycle algorithm uses $\mathcal{O}(N)$ operations per iteration and is optimal.

Haijun Wu & Weiying Zheng. (2023). Uniform Convergence of Multigrid V-Cycle on Adaptively Refined Finite Element Meshes for Elliptic Problems with Discontinuous Coefficients. Communications in Mathematical Research . 39 (3). 437-475. doi:10.4208/cmr.2022-0047
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