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Quasi-uniform and unconditional superconvergence analysis of Ciarlet–Raviart scheme for the fourth order singularly perturbed Bi-wave problem modeling d-wave superconductors

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  • Wu, Yanmi
  • Shi, Dongyang

Abstract

In this paper, two implicit Backward Euler (BE) and Crank-Nicolson (CN) formulas of Ciarlet–Raviart mixed finite element method (FEM) are presented for the fourth order time-dependent singularly perturbed Bi-wave problem arising as a time-dependent version of Ginzburg-Landau-type model for d-wave superconductors by the bilinear element. The well-posedness of the weak solution and the approximation solutions of the considered problem are proved through Faedo-Galerkin technique and Brouwer fixed point theorem, respectively. The quasi-uniform and unconditional superconvergent estimates of O(h2+τ) and O(h2+τ2)(h, the spatial parameter, and τ, the time step) in the broken H1- norm are obtained for the above formulas independent of the negative powers of the perturbation parameter δ. Some numerical results are provided to illustrate our theoretical analysis.

Suggested Citation

  • Wu, Yanmi & Shi, Dongyang, 2021. "Quasi-uniform and unconditional superconvergence analysis of Ciarlet–Raviart scheme for the fourth order singularly perturbed Bi-wave problem modeling d-wave superconductors," Applied Mathematics and Computation, Elsevier, vol. 397(C).
    • Handle: RePEc:eee:apmaco:v:397:y:2021:i:c:s0096300320308778
    DOI: 10.1016/j.amc.2020.125924
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    References listed on IDEAS

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    1. Shi, Dongyang & Wu, Yanmi, 2020. "Uniformly superconvergent analysis of an efficient two-grid method for nonlinear Bi-wave singular perturbation problem," Applied Mathematics and Computation, Elsevier, vol. 367(C).
    2. Shi, Dongyang & Wang, Junjun, 2017. "Unconditional superconvergence analysis of conforming finite element for nonlinear parabolic equation," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 216-226.
    3. Shi, Dongyang & Yang, Huaijun, 2017. "Unconditional optimal error estimates of a two-grid method for semilinear parabolic equation," Applied Mathematics and Computation, Elsevier, vol. 310(C), pages 40-47.
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