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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.

In this paper, a modified penalty finite element method (FEM) for solving the nonlinear singularly perturbed bi‐wave problem with a rectangular Morley element is presented. The second order term, the nonlinear term, and the source term are approximated by the bilinear interpolation part (conforming part) instead of the whole function on each element, respectively. The well‐posedness of the approximation solution is proved through Brouwer fixed point theorem. Quasi‐uniform optimal error estimate in the energy norm is derived independent of the negative powers of the real perturbation parameter δ appearing in the considered problem, which improves the corresponding result in the existing literature. Finally, some numerical results are given to verify the theoretical analysis.

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