UNCONDITIONAL SUPERCONVERGENT ANALYSIS OF QUASI-WILSON ELEMENT FOR BENJAMIN-BONA-MAHONEY EQUATION

This article aims to study the unconditional superconvergent behavior of nonconforming quadrilateral quasi-Wilson element for nonlinear Benjamin Bona Mahoney(BBM)equation.For the generalized rectangular meshes including rectangular mesh,deformed rectangular mesh and piecewise deformed rectangular mesh,by use of the special character of this element,that is,the conforming part(bilinear element)has high accuracy estimates on the generalized rectangular meshes and the consistency error can reach order O(h^(2)),one order higher than its interpolation error,the superconvergent estimates with respect to mesh size h are obtained in the broken H^(1)-norm for the semi-/fully-discrete schemes.A striking ingredient is that the restrictions between mesh size h and time stepτrequired in the previous works are removed.Finally,some numerical results are provided to confirm the theoretical analysis.

APPROXIMATION OF NONCONFORMING QUASI-WILSON ELEMENT FOR SINE-GORDON EQUATIONS

In this paper, nonconforming quasi-Wilson finite element approximation to a class of nonlinear sine-Gordan equations is discussed. Based on the known higher accuracy results of bilinear element and different techniques from the existing literature, it is proved that the inner product △↓（u - Ih^1u）, △↓vh） and the consistency error can be estimated as order O（h^2） in broken H^1 - norm/L^2 - norm when u ∈ H^3（Ω）/H^4（Ω）, where Ih^1u is the bilinear interpolation of u, Vh belongs to the quasi-Wilson finite element space. At the same time, the superclose result with order O（h^2） for semi-discrete scheme under generalized rectangular meshes is derived. Furthermore, a fully-discrete scheme is proposed and the corresponding error estimate of order O（h^2 ＋ τ^2） is obtained for the rectangular partition when u ∈ H^4（Ω）, which is as same as that of the bilinear element with ADI scheme and one order higher than that of the usual analysis on nonconforming finite elements.

EQ^(rot)_1 Nonconforming Finite Element Method for Nonlinear Dual Phase Lagging Heat Conduction Equations

EQrot 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 Hi-norm are obtained. At the same time, the global superconvergence in broken Hi-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 EQrot element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.

High Accuracy Analysis of the Lowest Order H1-Galerkin Mixed Finite Element Method for Nonlinear Sine-Gordon Equations

The lowest order H1-Galerkin mixed finite element method （for short MFEM） is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart- Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O（h2）/O（h2 ＋ r2） in Hi-norm and H（div; Ω）-norm axe deduced for the semi-discrete and the fully-discrete schemes, where h, r- denote the mesh size and the time step, respectively, which improve the results in the previous literature.