Nonconforming modified Quasi-Wilson finite element method for convection–diffusion-reaction equation

Abstract

The nonconforming modified Quasi-Wilson finite element method (FEM) is employed to investigate the unconditional superconvergence behavior of the convection–diffusion-reaction equation for the linearized energy stable Backward-Euler (BE) and the second order Backward differentiation formula (BDF2) fully discrete schemes on arbitrary quadrilateral meshes. Then, the energy stabilities of the discrete solutions for the proposed schemes are demonstrated through the mathematical induction, which is crucial to proving the unique solvabilities with Brouwer fixed point theorem. Based on the above achievements and the two typical characteristics of the element: one is that its consistency error in the broken H1-norm can be estimated as order O(h2) when the exact solution belongs to H3(Ω), which is one order higher than its interpolation error, and the other is that the nonconforming part of the function in the finite element space is orthogonal to the bilinear polynomials on each element of the subdivision in the integration sense, the superclose estimates without any restriction between spatial partition parameter h and the time step τ are obtained. Furthermore, the superconvergence properties for the aforementioned schemes are derived by the interpolation post-processing technique. Finally, the theoretical analysis is justified by the provided numerical experiments.