Unconditional superconvergence analysis of modified finite difference streamlined diffusion method for nonlinear convection-dominated diffusion equation

Abstract

In this article, a Crank-Nicolson fully discrete finite element scheme of modified finite difference streamlined diffusion (MFDSD) method is developed and investigated for nonlinear convection-dominated diffusion equation, which can get rid of the numerical oscillation appeared in Galerkin finite element method (FEM). The supercloseness and superconvergence estimates of order O(h2+τ2) in H1 norm are derived without the restriction between the time step τ and the mesh size h. Firstly, a time discrete system is established to split the error into two parts - the temporal error and spatial error, and the regularity of the solution of the time discrete system is deduced with the help of mathematical induction. Then the numerical solution is bounded in L∞ norm by the spatial error which leads to the above unconditional supercloseness property, and the global superconvergence result is deduced through interpolation post-processing technique. Lastly, two numerical examples are provided to verify the correctness of the theoretical analysis and to show the big advantage of the proposed MFDSD method over the Galerkin FEM.