Superconvergence of a two‐grid method for fourth‐order dispersive‐dissipative wave equations

In this paper, the superconvergence analysis of a two‐grid method (TGM) with low‐order finite elements is presented for the fourth‐order dispersive‐dissipative wave equations for a second order fully discrete scheme. The superclose estimates in the H¹‐norm on the two grids are obtained by the combination technique of the interpolation and Ritz projection. Then, with the help of the interpolated postprocessing technique, the global superconvergence properties are deduced. Finally, numerical results are provided to show the performance of the proposed TGM for conforming bilinear element and nonconforming element, respectively. It shows that the TGM is an effective method to the problem considered of our paper compared with the traditional Galerkin finite element method (FEM).     

Unconditionally Superclose Analysis of a New Mixed Finite Element Method for Nonlinear Parabolic Equation

This paper develops a framework to deal with the unconditional superclose analysis of nonlinear parabolic equation. Taking the finite element pair $Q_{11}/Q_{01} × Q_{10}$ as an example, a new mixed finite element method (FEM) is established and the $τ$ -independent superclose results of the original variable $u$ in $H^1$-norm and the flux variable $\mathop{q} \limits ^{\rightarrow}= −a(u)∇u$ in $L^2$-norm are deduced ($τ$ is the temporal partition parameter). A key to our analysis is an error splitting technique, with which the time-discrete and the spatial-discrete systems are constructed, respectively. For the first system, the boundedness of the temporal errors is obtained. For the second system, the spatial superclose results are presented unconditionally, while the previous literature always only obtain the convergent estimates or require certain time step conditions. Finally, some numerical results are provided to confirm the theoretical analysis, and show the efficiency of the proposed method.     

Superconvergence of a Nonconforming Finite Element Approximation to Viscoelasticity Type Equations on Anisotropic Meshes

The main aim of this paper is to study the approximation to viscoelasticity type equations with a Crouzeix-Raviart type nonconforming finite element on the anisotropic meshes. The superclose property of the exact solution and the optimal error estimate of its derivative with respect to time are derived by using some novel techniques. Moreover, employing a postprocessing technique, the global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself is studied.     

Superconvergence analysis of finite element method for Poisson–Nernst–Planck equations

This article concerns with the superconvergence analysis of bilinear finite element method (FEM) for nonlinear Poisson–Nernst–Planck (PNP) equations. By employing high accuracy integral identities together with mean value technique, the superclose estimates in H¹‐norm are derived for the semi‐discrete and the backward Euler fully‐discrete schemes, which improve the suboptimal error estimate in L²‐norm in the previous literature. Furthermore, the global superconvergence results in H¹‐norm are obtained through interpolation postprocessing approach. Finally, a numerical example is provided to confirm the theoretical analysis.     

EQ 1 rot nonconforming finite element method for nonlinear dual phase lagging heat conduction equations

EQ rot 1 nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O(h2 ) one order higher than its interpolation error O(h), the superclose results of order O(h2 ) in broken H1 -norm are obtained. At the same time, the global superconvergence in broken H1 -norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O(h4 ) is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQ rot 1 element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.     

Superconvergence analysis of splitting positive definite nonconforming mixed finite element method for pseudo-hyperbolic equations

In this paper, a new splitting positive definite nonconforming mixed finite element method is proposed for pseudo-hyperbolic equations, in which a quasi-Wilson quadrilateral element is used for the flux p, and the bilinear element is used for u. Superconvergence results in ｜｜·｜｜div,h norm for p and optimal error estimates in L2 norm for u are derived for both semi-discrete and fully discrete schemes under almost uniform meshes.     

Sobolev方程一个新的H~1-Galerkin混合有限元分析

研究了Sobolev方程的H~1-Galerkin混合有限元方法.利用不完全双二次元Q_2~-和一阶BDFM元,建立了一个新的混合元模式,通过Bramble-Hilbert引理,证明了单元对应的插值算子具有的高精度结果.进一步,对于半离散和向后欧拉全离散格式,分别导出了原始变量u在H~1-模和中间变量p在H(div)-模意义下的超逼近性质.     

三维矩形域上泊松方程四面体线元的超逼近与外推

改进三角元的积分恒等式,使之适用于拟一致四面体元,借此证明了泊松方程四面体线元梯度有超逼近现象,函数值Richardson外推可以提高精度.     

椭圆型方程四面体线元的超逼近与外推

重新讨论了三角线元的积分恒等式,使之适用于三维区域的拟一致四面体元,借此证明了椭圆型方程有限元解梯度有超逼近现象,函数值Richardson外推可以提高精度.