Uniform superconvergent analysis of a new mixed finite element method for nonlinear Bi-wave singular perturbation problem

Abstract

Uniform superconvergent analysis of a new low order nonconforming mixed finite element method (MFEM) is studied for solving the fourth order nonlinear Bi-wave singular perturbation problem (SPP) by EQ1rot element. On one hand, the existence, uniqueness and stability of the numerical solutions are proved. On the other hand, with the help of the special characters of this element, uniform superclose result of order O(h2) for the original variable in the broken H1 norm and uniform optimal order estimate of order O(h2) for the intermediate variable in L2 norm are deduced irrelevant to the real perturbation parameter δ appearing in the considered problem, respectively. In which, the nonlinear term in the Bi-wave SPP, which is the main difficulty in the whole error analysis, is dealt with rigorously through a novel splitting technique. Moreover, the global uniform superconvergent estimate is obtained with the interpolated postprocessing approach. Finally, some numerical results are provided to confirm the theoretical analysis. Here h is the subdivision parameter.