Dongyang Shi

High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equations

Abstract A nonconforming mixed finite element scheme is proposed for Sobolev equations based on a new mixed variational form under semi-discrete and Euler fully-discrete schemes. The corresponding optimal convergence error estimates and superclose property are obtained without using Ritz projection, which are the same as the traditional mixed finite elements. Furthemore, the global superconvergence is obtained through interpolation postprocessing technique. The numerical results show the validity of the theoretical analysis.

Convergence and asymptotic stability of Galerkin methods for a partial differential equation with piecewise constant argument

The paper deals with the convergence and asymptotic stability of Galerkin methods for a partial differential equation with piecewise constant argument. The optimal convergence orders are obtained for the semidiscrete and full discrete (backward Euler) methods respectively. Both the discrete solutions are proved to be asymptotically stable under the condition that the analytical solution is asymptotically stable.

A new second order nonconforming mixed finite element scheme for the stationary Stokes and Navier-Stokes equations

Abstract In this paper, a new stable nonconforming mixed finite element scheme with second order accuracy is proposed for the stationary Stokes and Navier–Stokes equations, in which, a new nonconforming rectangular element is taken for approximating space the velocity and the bilinear element for the pressure. The optimal error estimates for the approximation of both the velocity and the pressure in L 2 -norm are established, as well as one in a broken H 1 -norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis.

Nonconforming quadrilateral finite element method for a class of nonlinear sine-Gordon equations

Nonconforming quadrilateral finite element method (FEM) of the two-dimensional nonlinear sine-Gordon equation is studied for semi-discrete and Crank-Nicolson fully-discrete schemes, respectively. Firstly, we prove a special feature of a new arbitrary quadrilateral element (named modified Quasi-Wilson element), i.e., the consistency error is of order O(h^2) (h denotes the mesh size) in H^1-norm, which leads to optimal order error estimate and superclose result with order O(h^2) for the semi-discrete scheme through a different approach from the existing literature. Secondly, because the consistency error estimate of the new modified Quasi-Wilson element can reach a staggering O(h^3) order, two orders higher than that of interpolation error, the optimal order error estimates of Crank-Nicolson fully-discrete scheme are obtained on arbitrary quadrilateral meshes with Ritz projection. Moreover, a superclose result in H^1-norm is presented on generalized rectangular meshes by a new technique. Thirdly, the global superconvergence results of H^1-norm for both semi-discrete and fully-discrete schemes are derived on rectangular meshes with interpolated postprocessing technique. Finally, a numerical test is carried out to verify the theoretical analysis.

Superconvergence analysis of conforming finite element method for nonlinear Schrödinger equation

The main aim of this paper is to apply the conforming bilinear finite element to solve the nonlinear Schrodinger equation (NLSE). Firstly, the stability and convergence for time discrete scheme are proved. Secondly, through a new estimate approach, the optimal order error estimates and superclose properties in H1-norm are obtained with Backward Euler (B-E) and Crank-Nicolson (C-N) fully-discrete schemes, the global superconvergence results are deduced with the help of interpolation postprocessing technique. Finally, some numerical examples are provided to verify the theoretical analysis.

Quasi-Wilson nonconforming element approximation for nonlinear dual phase lagging heat conduction equations

Accuracy analysis of quasi-Wilson nonconforming element for nonlinear dual phase lagging heat conduction equations is discussed and higher order error estimates are derived under semi-discrete and fully-discrete schemes. Since the nonconforming part of quasi-Wilson element is orthogonal to biquadratic polynomials in a certain sense, it can be proved that the compatibility error of this element is of order O(h^2)/O(h^3) when the exact solution belongs to H^3(@W)/H^4(@W), which is one/two order higher than its interpolation error. Based on the above results, the superclose properties in L^2-norm and broken H^1-norm are deduced by using high accuracy analysis of bilinear finite element. Moreover, the global superconvergence in broken H^1-norm is derived by interpolation postprocessing technique. And then, the extrapolation result with order O(h^3) in broken H^1-norm is obtained by constructing a new extrapolation scheme properly, which is two order higher than the usual error estimate. Finally, optimal order error estimates are deduced for a proposed fully-discrete approximate scheme.

A new robust C0-type nonconforming triangular element for singular perturbation problems ☆

Abstract In this paper, a new robust C 0 triangular element is proposed for the fourth order elliptic singular perturbation problem with double set parameter method and bubble function technique, and a general convergence theorem for C 0 nonconforming elements is presented. The convergence of the new element is proved in the energy norm uniformly with respect to the perturbation parameter. Numerical experiments are also carried out to demonstrate the efficiency of the new element.

ACCURACY ANALYSIS FOR QUASI-CAREY ELEMENT*

In this paper, a new triangular element (Quasi-Carey element) is constructed by the idea of Specht element. It is shown that this Quasi-Carey element possesses a very special property, i.e., the consistency error is of order O(h2), one order higher than its interpolation error when the exact solution belongs to H3(Ω). However, the interpolation error and consistency error of Carey element are of order O(h). It seems that the above special property has never been seen for other triangular elements for the second order problems.

Superclose and superconvergence of finite element discretizations for the Stokes equations with damping

A class of conforming mixed finite element methods (FEMs) are applied to the Stokes equations with damping. With the help of the prior estimates for the exact and approximated solutions and the appropriate choice of the parameters @a,@n and r appeared in the problem, the superclose and superconvergence results for the velocity in H^1-norm and the pressure in L^2-norm are derived. To our best knowledge, this is the first superconvergence analysis of conforming mixed FEMs for this kind nonlinear problem.

Superconvergence Analysis of a Nonconforming Triangular Element on Anisotropic Meshes

The class of anisotropic meshes we conceived abandons the regular assumption. Some distinct properties of Carey’s element are used to deal with the superconvergence for a class of two-dimensional second-order elliptic boundary value problems on anisotropic meshes. The optimal results are obtained and numerical examples are given to confirm our theoretical analysis.