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Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems

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Abstract

In this paper we prove the uniform convergence of the standard multigrid V-cycle algorithm with the Gauss-Seidel relaxation performed only on the new nodes and their “immediate” neighbors for discrete elliptic problems on the adaptively refined finite element meshes using the newest vertex bisection algorithm. The proof depends on sharp estimates on the relationship of local mesh sizes and a new stability estimate for the space decomposition based on the Scott-Zhang interpolation operator. Extensive numerical results are reported, which confirm the theoretical analysis.

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Authors and Affiliations

  1. Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing, 100080, China

    Wu Haijun & Chen Zhiming

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  1. Wu Haijun
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  2. Chen Zhiming
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Correspondence to Wu Haijun.

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Wu, H., Chen, Z. Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems. SCI CHINA SER A 49, 1405–1429 (2006). https://doi.org/10.1007/s11425-006-2005-5

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  • DOI: https://doi.org/10.1007/s11425-006-2005-5

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