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, 2024)
In this paper, a two-grid mixed finite element method (MFEM) of implicit BackwardEuler (BE) formula is presented for the fourth order time-dependent singularly perturbedBi-wave problem for $d$-wave superconductors by the nonconforming $EQ^{rot}_1$ element. In thisapproach, the original nonlinear system is solved on the coarse mesh through the Newtoniteration 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 supercloseand superconvergent estimates in the broken $H^1$-norm are derived, which are independentof 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 ismuch 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.