Quasi-uniform unconditional superconvergent error estimates of FEMs for the time-dependent singularly perturbed Bi-wave problem modeling d-wave superconductors

For the fourth order time-dependent singularly perturbed Bi-wave equation modeling d-wave superconductors, the implicit Backward Euler (BE) and Crank-Nicolson (CN) schemes of Galerkin finite element method (FEM) are studied by Bonner-Fox-Shmite element. Then the quasi-uniform and unconditional superconvergent error estimates of orders $O(h^3+\tau )$ and $O(h^3+{\tau }^2)$ (h, the spatial parameter, and τ, the time step) in the energy norm are derived respectively for the above schemes, which are independent of the negative powers of the perturbation parameter appearing in the model. Finally, some numerical results are provided to verify the theoretical analysis.