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An Efficient Iterative Method for Solving Parameter-Dependent and Random Convection–Diffusion Problems

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Abstract

This paper develops and analyzes a general iterative framework for solving parameter-dependent and random convection–diffusion problems. It is inspired by the multi-modes method and the ensemble method and extends those methods into a more general and unified framework. The main idea of the framework is to reformulate the underlying problem into another problem with parameter-independent convection and diffusion coefficients and a parameter-dependent (and solution-dependent) right-hand side, a fixed-point iteration is then employed to compute the solution of the reformulated problem. The main benefit of the proposed approach is that an efficient direct solver and a block Krylov subspace iterative solver can be used at each iteration, allowing to reuse the LU matrix factorization or to do an efficient matrix-matrix multiplication for all parameters, which in turn results in significant computation saving. Convergence and rates of convergence are established for the iterative method both at the variational continuous level and at the finite element discrete level under some structure conditions. Several strategies for establishing reformulations of parameter-dependent and random diffusion and convection–diffusion problems are proposed and their computational complexity is analyzed. Several 1-D and 2-D numerical experiments are also provided to demonstrate the efficiency of the proposed iterative method and to validate the theoretical convergence results.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like thank the referee for his/her valuable comments and the suggestion of adding the numerical experiments of Sects. 6.3 and 6.4 as well as for pointing out the reference [15].

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Funding for each author has been given earlier.

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Authors and Affiliations

  1. Department of Mathematics, The University of Tennessee, Knoxville, TN, 37996, USA

    Xiaobing Feng & Liet Vo

  2. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, China

    Yan Luo

  3. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Zhu Wang

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  1. Xiaobing Feng
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  2. Yan Luo
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  3. Liet Vo
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  4. Zhu Wang
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Correspondence to Xiaobing Feng.

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No conflict of interest is involved with this research, both data and code are available from the corresponding author on reasonable request.

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Xiaobing Feng and Liet Vo: The work of these authors was partially supported by the NSF Grants DMS-1620168 and DMS-2012414. Yan Luo: The work of this author was supported by the Young Scientists Fund of the National Natural Science Foundation of China grant 11901078 and by the Fundamental Research Funds for the Central Universities Grant ZGX2020J021. Zhu Wang: The work of this author was partially supported by the NSF Grant DMS-1913073.

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Feng, X., Luo, Y., Vo, L. et al. An Efficient Iterative Method for Solving Parameter-Dependent and Random Convection–Diffusion Problems. J Sci Comput 90, 72 (2022). https://doi.org/10.1007/s10915-021-01737-z

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  • DOI: https://doi.org/10.1007/s10915-021-01737-z

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