The Nonconforming Finite Element Approximation And Convergence Analysis Of Two Evolution Equations With Fully-Discrete Schemes

This paper mainly deals with the error estimates for two kinds of evolution equations with the fully-discrete scheme on anisotropic meshes. The paper is organized as follows. In Section 1, we introduce the preliminaries, basic theory and inequalities. In Section 2, we discuss the lumped mass nonconforming finite element method for nonstationary Navier-Stokes equations. The lowest order Crouzeix-Raviart type stablized mixed finite element scheme and second-order difference scheme arc used for spatial and time space respectively. Without using Navier-Stokes projection, the same error estimates are derived as the traditional finite elements, which extend the application of finite element methods. In Section 3, under the fully-discrete scheme, we study the error analysis of hyperbolic integro-differential equation for nonconforming finite element. For the well-known five nodals rectangular element, by using some special properties of the element, the super-close and superconvergence properties are first obtained under the semi-discrete scheme. Then, by employing numerical integration formula to the integration term appeared in the problem, the convergence result as the traditional method is also presented under the fully-discrete scheme.