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Volume 42, Issue 2
Uniform Superconvergence Analysis of a Two-Grid Mixed Finite Element Method for the Time-Dependent Bi-Wave Problem Modeling $D$-Wave Superconductors

Yanmi Wu & Dongyang Shi

DOI: 10.4208/jcm.2203-m2021-0058

J. Comp. Math., 42 (2024), pp. 415-431.

Published online: 2024-01

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  • Abstract

In this paper, a two-grid mixed finite element method (MFEM) of implicit Backward Euler (BE) formula is presented for the fourth order time-dependent singularly perturbed Bi-wave problem for $d$-wave superconductors by the nonconforming $EQ^{rot}_1$ element. In this approach, the original nonlinear system is solved on the coarse mesh through the Newton iteration method, and then the linear system is computed on the fine mesh with Taylor’s expansion. Based on the high accuracy results of the chosen element, the uniform superclose and superconvergent estimates in the broken $H^1$-norm are derived, which are independent of the negative powers of the perturbation parameter appeared in the considered problem. Numerical results illustrate that the computing cost of the proposed two-grid method is much less than that of the conventional Galerkin MFEM without loss of accuracy.

  • Keywords

Time-dependent Bi-wave problem, Two-grid mixed finite element method, Uniform superclose and superconvergent estimates.

  • AMS Subject Headings

65M60, 65N12, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-415, author = {Wu , Yanmi and Shi , Dongyang}, title = {Uniform Superconvergence Analysis of a Two-Grid Mixed Finite Element Method for the Time-Dependent Bi-Wave Problem Modeling $D$-Wave Superconductors}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {2}, pages = {415--431}, abstract = {

In this paper, a two-grid mixed finite element method (MFEM) of implicit Backward Euler (BE) formula is presented for the fourth order time-dependent singularly perturbed Bi-wave problem for $d$-wave superconductors by the nonconforming $EQ^{rot}_1$ element. In this approach, the original nonlinear system is solved on the coarse mesh through the Newton iteration method, and then the linear system is computed on the fine mesh with Taylor’s expansion. Based on the high accuracy results of the chosen element, the uniform superclose and superconvergent estimates in the broken $H^1$-norm are derived, which are independent of the negative powers of the perturbation parameter appeared in the considered problem. Numerical results illustrate that the computing cost of the proposed two-grid method is much less than that of the conventional Galerkin MFEM without loss of accuracy.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2203-m2021-0058}, url = {http://global-sci.org/intro/article_detail/jcm/22887.html} }
TY - JOUR T1 - Uniform Superconvergence Analysis of a Two-Grid Mixed Finite Element Method for the Time-Dependent Bi-Wave Problem Modeling $D$-Wave Superconductors AU - Wu , Yanmi AU - Shi , Dongyang JO - Journal of Computational Mathematics VL - 2 SP - 415 EP - 431 PY - 2024 DA - 2024/01 SN - 42 DO - http://doi.org/10.4208/jcm.2203-m2021-0058 UR - https://global-sci.org/intro/article_detail/jcm/22887.html KW - Time-dependent Bi-wave problem, Two-grid mixed finite element method, Uniform superclose and superconvergent estimates. AB -

In this paper, a two-grid mixed finite element method (MFEM) of implicit Backward Euler (BE) formula is presented for the fourth order time-dependent singularly perturbed Bi-wave problem for $d$-wave superconductors by the nonconforming $EQ^{rot}_1$ element. In this approach, the original nonlinear system is solved on the coarse mesh through the Newton iteration method, and then the linear system is computed on the fine mesh with Taylor’s expansion. Based on the high accuracy results of the chosen element, the uniform superclose and superconvergent estimates in the broken $H^1$-norm are derived, which are independent of the negative powers of the perturbation parameter appeared in the considered problem. Numerical results illustrate that the computing cost of the proposed two-grid method is much less than that of the conventional Galerkin MFEM without loss of accuracy.

Wu , Yanmi and Shi , Dongyang. (2024). Uniform Superconvergence Analysis of a Two-Grid Mixed Finite Element Method for the Time-Dependent Bi-Wave Problem Modeling $D$-Wave Superconductors. Journal of Computational Mathematics. 42 (2). 415-431. doi:10.4208/jcm.2203-m2021-0058
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