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 - <p style="text-align: justify;">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.</p>