Uniform superconvergence analysis of Ciarlet‐Raviart scheme for Bi‐wave singular perturbation problem

Abstract

Uniform superconvergence analysis of the Ciarlet‐Raviart mixed finite element scheme is discussed for solving the fourth‐order Bi‐wave singular perturbation problem (SPP) by the bilinear element. Firstly, the existence and uniqueness of the approximation solution are proved. Secondly, with the help of the special characters of this element, uniform superclose result of order O(h2) for the original variable in H1 norm and uniform optimal order estimate of order O(h2) for the intermediate variable in L2 norm are deduced with respect to the real perturbation parameter δ appearing in the considered problem. Furthermore, the global uniform superconvergent estimate is obtained through the interpolated postprocessing approach. Finally, some numerical results are provided to verify the theoretical analysis. Here, h denotes the mesh size.

Uniform Superconvergence Analysis of a Two-Grid Mixed Finite Element Method for the Time-Dependent Bi-Wave Problem Modeling $D$-Wave Superconductors

“…Hence, 0 < δ < 1 is expected to be small for d-wave superconductors and problem (1.1) degenerates into the semilinear parabolic equation when δ → 0. In recent years, there are some theoretical analysis and numerical simulations about FEMs, such as optimal order error estimates of conforming Galerkin FEMs and the modified Morleytype discontinuous Galerkin FEMs in [10,11], uniform superconvergence error estimates of Ciarlet-Raviart schemes with the conforming and nonconforming elements in [12][13][14]. But these work mainly focused on the stationary singularly perturbed Bi-wave problems.…”

Quasi‐uniform convergence analysis of a modified penalty finite element method for nonlinear singularly perturbed bi‐wave problem

“…For example, two conforming Galerkin finite element methods (FEMs) and the modified nonconforming Morley-type Galerkin FEMs were proposed for the linear bi-wave equations, and convergence error estimates were derived in [2,3], respectively. The Ciarlet-Raviart mixed FEMs were analyzed in [4][5][6] for the linear bi-wave equations and problem (1), and the uniform superconvergence estimates in the broken H 1 norm were deduced, respectively. We mention that the piecewise polynomials space was required to be C 1 continuous in [2], which would be expensive and less efficient.…”