Unconditional superconvergent error estimates of second-order linearized nonconforming quadrilateral FEMs for nonlinear nuclear reactor model

Abstract

The focus of this paper is to establish the linearized Crank-Nicolson (C-N) and two-step Backward Differentiation Formula (BDF2) fully discrete schemes for the nonlinear nuclear reactor model, and investigate their superclose and superconvergent behaviors without the restrictions on the relationship between mesh size h and time step τ on quadrilateral meshes of nonconforming modified quasi-Wilson element. First, we will show that the consistency error of this element can reach $O\left(h^3\right)$ for homogeneous Neumann boundary condition which is the same as the quasi-Wilson element on rectangular meshes for the Dirichlet boundary condition. Then, based on the combination technique of interpolation and projection, mathematical induction and interpolation post-processing approach, the superclose and superconvergent results $O(h^2+{\tau }^2)$ in the broken $H^1$-norm are derived. Finally, a numerical illustration is executed to validate the theoretical findings.