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

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 ( h 2 + τ ) and O ( h 2 + τ 2 ) ( h , the spatial parameter, and τ , the time step) in the broken H 1 - 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.