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Two-Grid Methods for a New Mixed Finite Element Approximation of Semilinear Parabolic Integro-Differential Equations

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Abstract

In this paper, we present a two-grid scheme for a semilinear parabolic integro-differential equation using a new mixed finite element method. The gradient for the method belongs to the square integrable space instead of the classical H(div; Ω) space. The velocity and the pressure are approximated by the P02–P1 pair which satisfies the inf-sup condition. Firstly, we solve an original nonlinear problem on the coarse grid in our two-grid scheme. Then, to linearize the discretized equations, we use Newton iteration on the fine grid twice. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = O(H6|lnH|2). As a result, solving such a large class of nonlinear equations will not be much more difficult than the solution of one linearized equation. Finally, a numerical experiment is provided to verify theoretical results of the two-grid method.

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Authors and Affiliations

  1. Institute of Computational Mathematics, Department of Mathematics and Computational Science, Hunan University of Science and Engineering, 425100, Yongzhou, Hunan, China

    C. Liu

  2. School of Mathematics and Statistics, Beihua University, Jilin, 132013, Jilin, China

    T. Hou

Authors
  1. C. Liu
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  2. T. Hou
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Correspondence to C. Liu or T. Hou.

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Russian Text © The Author(s), 2019, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2019, Vol. 22, No. 2, pp. 167–185.

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Liu, C., Hou, T. Two-Grid Methods for a New Mixed Finite Element Approximation of Semilinear Parabolic Integro-Differential Equations. Numer. Analys. Appl. 12, 137–154 (2019). https://doi.org/10.1134/S1995423919020046

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  • DOI: https://doi.org/10.1134/S1995423919020046

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