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Fast Multilevel Solvers for a Class of Discrete Fourth Order Parabolic Problems

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Abstract

In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigate the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems.

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Acknowledgments

B. Zheng would like to acknowledge the support by NSF Grant DMS-0807811 and a Laboratory Directed Research and Development (LDRD) Program from Pacific Northwest National Laboratory. L.P. Chen was supported by the National Natural Science Foundation of China under Grant No. 11501473. L. Chen was supported by NSF Grant DMS-1418934 and in part by NIH Grant P50GM76516. R.H. Nochetto was supported by NSF under Grants DMS-1109325 and DMS-1411808. J. Xu was supported by NSF Grant DMS-1522615 and in part by US Department of Energy Grant DE-SC0014400. Computations were performed using the computational resources of Pacific Northwest National Laboratory (PNNL) Institutional Computing cluster systems. The PNNL is operated by Battelle for the US Department of Energy under Contract DE-AC05-76RL01830.

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Authors and Affiliations

  1. Fundamental and Computational Sciences, Pacific Northwest National Laboratory, Richland, WA, 99352, USA

    Bin Zheng

  2. School of Mathematics, Southwest Jiaotong University, Chengdu, 611756, China

    Luoping Chen

  3. Department of Mathematics, Tufts University, Medford, MA, 02155, USA

    Xiaozhe Hu

  4. Department of Mathematics, University of California, Irvine, CA, 92697, USA

    Long Chen

  5. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

  6. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Jinchao Xu

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  1. Bin Zheng
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  2. Luoping Chen
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  3. Xiaozhe Hu
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Correspondence to Bin Zheng.

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Zheng, B., Chen, L., Hu, X. et al. Fast Multilevel Solvers for a Class of Discrete Fourth Order Parabolic Problems. J Sci Comput 69, 201–226 (2016). https://doi.org/10.1007/s10915-016-0189-6

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