n this papers the asymptotic error expansions of Adini's element for the secondorder imhomogeneous Neumann problem are given and the superconvergence estimations are obtained. Moreover, a numerical example to supp...n this papers the asymptotic error expansions of Adini's element for the secondorder imhomogeneous Neumann problem are given and the superconvergence estimations are obtained. Moreover, a numerical example to support our theoreticalanalysis is reported.展开更多
A highly efficient H1-Galerkin mixed finite element method （MFEM） is presented with linear triangular element for the parabolic integro-differential equation. Firstly, some new results about the integral estimation ...A highly efficient H1-Galerkin mixed finite element method （MFEM） is presented with linear triangular element for the parabolic integro-differential equation. Firstly, some new results about the integral estimation and asymptotic expansions are studied. Then, the superconvergence of order O（h^2） for both the original variable u in H1 （Ω） norm and the flux p = u in H（div, Ω） norm is derived through the interpolation post processing technique. Furthermore, with the help of the asymptotic expansions and a suitable auxiliary problem, the extrapolation solutions with accuracy O（h^3） are obtained for the above two variables. Finally, some numerical results are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method.展开更多
Aiming at the isoparametric bilinear finite volume element scheme,we initially derive an asymptotic expansion and a high accuracy combination formula of the derivatives in the sense of pointwise by employing the energ...Aiming at the isoparametric bilinear finite volume element scheme,we initially derive an asymptotic expansion and a high accuracy combination formula of the derivatives in the sense of pointwise by employing the energy-embedded method on uniform grids.Furthermore,we prove that the approximate derivatives are convergent of order two.Finally,numerical examples verify the theoretical results.展开更多
Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolat...Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.展开更多
In this paper, we discuss a generalized finite element interpolation problem and obtain the asymptotic expansion of the interpolation function. Based on these results, the error asymptotic expansion and superconvergen...In this paper, we discuss a generalized finite element interpolation problem and obtain the asymptotic expansion of the interpolation function. Based on these results, the error asymptotic expansion and superconvergence result of the generalized finite element approximation are derived. Finallym using the Superconvergent Patch Recovery Technique (SPR) proposed by Zienkiewicz & Zhu, we get the superconvergent recovery approximation and the posteriori error estimates to the flux. The numerical test convinced our analysis.展开更多
In this paper we will show that the Richardson extrapolation can be used to enhance the numerical solution generated by a Petrov-Galerkin finite element method for the initial value problem for a nonlinear Volterra in...In this paper we will show that the Richardson extrapolation can be used to enhance the numerical solution generated by a Petrov-Galerkin finite element method for the initial value problem for a nonlinear Volterra integro-differential equation. As by-products, we will also show that these enhanced approximations can be used to form a class of posteriori estimators for this Petrov-Galerkin finite element method. Numerical examples are supplied to illustrate the theoretical results.展开更多
For a class of non-smooth Fredholm integral equations we prove thatRichardson extrapolation method can be used to increase accuracy for finite ele-ment methods.On the basis of local refinement mesh,a superconvergence ...For a class of non-smooth Fredholm integral equations we prove thatRichardson extrapolation method can be used to increase accuracy for finite ele-ment methods.On the basis of local refinement mesh,a superconvergence result isshown.Our theory also covers a class of non-smooth Wiener-Hopf equations andan application includes the calculation in certain linear elastic fracture problems.展开更多
文摘n this papers the asymptotic error expansions of Adini's element for the secondorder imhomogeneous Neumann problem are given and the superconvergence estimations are obtained. Moreover, a numerical example to support our theoreticalanalysis is reported.
基金Project supported by the National Natural Science Foundation of China(Nos.10971203,11271340,and 11101381)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20094101110006)
文摘A highly efficient H1-Galerkin mixed finite element method （MFEM） is presented with linear triangular element for the parabolic integro-differential equation. Firstly, some new results about the integral estimation and asymptotic expansions are studied. Then, the superconvergence of order O（h^2） for both the original variable u in H1 （Ω） norm and the flux p = u in H（div, Ω） norm is derived through the interpolation post processing technique. Furthermore, with the help of the asymptotic expansions and a suitable auxiliary problem, the extrapolation solutions with accuracy O（h^3） are obtained for the above two variables. Finally, some numerical results are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method.
基金supported by NSFC Project(Grant No.11031006,91130002,11171281)the Key Project of Scientific Research Fund of Hunan Provincial Science and Technology Department(Grant No.2011FJ2011)+2 种基金Specialized research Fund for the Doctoral Program of Higher Education(Grant No.20124301110003)Program for Changjiang Scholars and Innovative Research Team in University of China(No.IRT1179)Hunan Provincial Natural Science Foundation of China(Grant No.12JJ3010)。
文摘Aiming at the isoparametric bilinear finite volume element scheme,we initially derive an asymptotic expansion and a high accuracy combination formula of the derivatives in the sense of pointwise by employing the energy-embedded method on uniform grids.Furthermore,we prove that the approximate derivatives are convergent of order two.Finally,numerical examples verify the theoretical results.
基金supported in part by the National Basic Research Program (2007CB814906)the National Natural Science Foundation of China (10471103 and 10771158)+4 种基金Social Science Foundation of the Ministry of Education of China (06JA630047)Tianjin Natural Science Foundation (07JCYBJC14300)Tianjin University of Finance and Economicssupported by the National Basic Research Program under the Grant 2005CB321701the National Natural Science Foundation of China under the Grant 10771211
文摘Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.
文摘In this paper, we discuss a generalized finite element interpolation problem and obtain the asymptotic expansion of the interpolation function. Based on these results, the error asymptotic expansion and superconvergence result of the generalized finite element approximation are derived. Finallym using the Superconvergent Patch Recovery Technique (SPR) proposed by Zienkiewicz & Zhu, we get the superconvergent recovery approximation and the posteriori error estimates to the flux. The numerical test convinced our analysis.
基金This work is supported partially by SRF for ROCS, SEM, NSERC (Canada) and NSF grant DMS-9704621.
文摘In this paper we will show that the Richardson extrapolation can be used to enhance the numerical solution generated by a Petrov-Galerkin finite element method for the initial value problem for a nonlinear Volterra integro-differential equation. As by-products, we will also show that these enhanced approximations can be used to form a class of posteriori estimators for this Petrov-Galerkin finite element method. Numerical examples are supplied to illustrate the theoretical results.
文摘For a class of non-smooth Fredholm integral equations we prove thatRichardson extrapolation method can be used to increase accuracy for finite ele-ment methods.On the basis of local refinement mesh,a superconvergence result isshown.Our theory also covers a class of non-smooth Wiener-Hopf equations andan application includes the calculation in certain linear elastic fracture problems.
