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Accuracy Enhancement Using Spectral postprocessing for Differential Equations and Integral Equations

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Communications in Computational Physics 2009年第2期5卷 779-792页

作者： Tao Tang Xiang Xu Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong School of Mathematics Fudan UniversityShanghai 200433China.

It is demonstrated that spectral methods can be used to improve the accuracy of numerical solutions obtained by some lower order *** precisely,we can use spectral methods to postprocess numerical solutions of initial value differential *** a few number of iterations(say 3 to 4),the errors can decrease to a few orders of magnitude *** iteration uses the Gauss-Seidel type strategy,which gives an explicit way of *** examples for ODEs,Hamiltonian system and integral equations are *** all indicate that the spectral processing technique can be a very useful way in improving the accuracy of the numerical *** particular,for a Hamiltonian system accuracy is only one of the issues;some other conservative properties are even more important for large time *** spectral postprocessing with the coarse-mesh symplectic initial guess can not only produce high accurate approximations but can also save a significant amount of computational time over the standard symplectic schemes.

关键词： postprocessing spectral methods rate of convergence Hamiltonian system integral equations

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Sobolev型方程有限元的后处理方法

高等学校计算数学学报 1998年第3期20卷 193-200页

作者：李潜陈焕祯山东师范大学数学系山东师范大学数学系济南 250014

1 引言考虑下述Sobolev型方程的混合问题（a）（b）（c）其中Ω为R;中具有边界的矩形域,a,b,f,u;。为适当光滑且有界的已知函数,a（x,t）有正下界a;. Sobolev型方程是重要的数学物理方程之一,文[1]导出了问题（1.1）标准有限元方... 详细信息

1 引言考虑下述Sobolev型方程的混合问题（a）（b）（c）其中Ω为R;中具有边界的矩形域,a,b,f,u;。为适当光滑且有界的已知函数,a（x,t）有正下界a;. Sobolev型方程是重要的数学物理方程之一,文[1]导出了问题（1.1）标准有限元方法的最优L;（2≤P<∞）估计.本文研究矩形剖分上的双k次有限元方法,用插值算子对近似解进行后处理,仅增加极少工作量,使整体收敛性提高一阶.本文证明了误差及任意阶时间导进行后处理,仅增加极少工作量,使整体收敛性提高一阶.本文证明了误差及任意阶时间导数的H;,W;,L;和L;的超收敛估计.若采用文[2]的预处理方法构造最优剖分,可将本文结果推广到一般区域（仍超收敛1/2阶）.这样,采用低次有限元可获得高阶精度,从而大大节省了计算量.

关键词： postprocessing equations of Sobolev type superconvergence finite element.

Contact-Stress-Based Stress Recovery Methods for Discontinuous Deformation Analysis

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Acta Mechanica Solida Sinica 2020年第4期33卷 508-520页

作者： Ruoyu Guan Shean Bie State Key Laboratory of Hydraulic Engineering Simulation and Safety School of Civil EngineeringTianjin UniversityTianjin 300072China

Discontinuous deformation analysis(DDA)has been widely applied for the simulation of block systems that have many discontinuous *** penalty method is utilized to ensure that there are no penetrations between blocks.A linear polynomial function for displacement leads to a constant stress for a block,which cannot precisely describe the stress field within the ***,a high-order polynomial displacement function and a flue mesh are always used to improve the precision of the stress ***,these means are not practical for simulating block systems that have many *** this paper,the contact-stress-based stress recovery methods are proposed for ***-precision solutions for the contact stresses on the boundaries of the blocks are *** first-order Gaussian point of a block is the block's centroid,where the constant stress obtained via DDA is of higher *** high-precision solutions for the stresses are utilized in the least squares method to recover a single block's inner stress *** proposed methods enhance the resolution of the stress field inside a single block without increasing the computational effort in the main iterative process for displacement in *** examples are simulated using both the finite element method(FEM)with a fine mesh and the proposed DDA *** recovered DDA results can accurately describe the distribution of the stresses in a single block and,in some areas,have the same precision as the FEM ***,the precision of the proposed methods improves as the gradient of the contact stress on the boundary decreases.

关键词： Contact stress postprocessing Stress recovery method Discontinuous deformation analysis

Superconvergence of a Galerkin FEM for Higher-Order Elements in Convection-Diffusion Problems

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Numerical Mathematics(Theory,Methods and Applications) 2014年第3期7卷 356-373页

作者： Sebastian Franz H.-G.Roos Institut für Numerische Mathematik Technische Universit¨at Dresden01062 DresdenGermany

In this paper we present a first supercloseness analysis for higher-order Galerkin FEM applied to a singularly perturbed convection-diffusion *** a solution decomposition and a special representation of our finite element space,we are able to prove a supercloseness property of p+1/4 in the energy norm where the polynomial order p≥3 is odd.

关键词： Singular perturbation layer-adapted meshes superconvergence postprocessing

A NEW APPROACH TO RECOVERY OF DISCONTINUOUS GALERKIN

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Journal of Computational Mathematics 2009年第6期27卷 697-712页

作者： Sebastian Franz Lutz Tobiska Helena Zarin Institut für Numerische MathematikTechnische Universitt Dresden Institut für Analysis und NumerikOtto-von-Guericke Universitt Magdeburg Department of Mathematics and Informatics University of Novi SadTrg Dositeja Obradovica 421 000 Novi Sad

A new recovery operator P ：Qn^disc（T）→Qn＋1^disc（M） for discontinuous Galerkin is derived. It is based on the idea of projecting a discontinuous, piecewise polynomial solution on a given mesh T into a higher order polynomial space on a macro mesh M. In order to do so, we define local degrees of freedom using polynomial moments and provide global degrees of freedom on the macro mesh. We prove consistency with respect to the local L2-projection, stability results in several norms and optimal anisotropic error estimates. As an example, we apply this new recovery technique to a stabilized solution of a singularly perturbed convection-diffusion problem using bilinear elements.

关键词： Discontinuous Galerkin postprocessing Recovery.

Computing the lower and upper bounds of Laplace eigenvalue problem:by combining conforming and nonconforming finite element methods

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Science China Mathematics 2012年第5期55卷 1069-1082页

作者： LUO FuSheng LIN Qun XIE HeHu LSEC ICMSECAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

We introduce some ways to compute the lower and upper bounds of the Laplace eigenvalue *** using the special nonconforming finite elements,i.e.,enriched Crouzeix-Raviart element and extended Q1ro t,we get the lower bound of the ***,we use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue,which only needs to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is ***,we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only *** numerical results are also presented to demonstrate our theoretical analysis.

关键词： lower bound upper bound ECR EQ1ro t eigenvalue problem postprocessing

Superconvergence and L^(∞)-Error Estimates of RT1Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint

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Numerical Mathematics(Theory,Methods and Applications) 2012年第3期5卷 423-446页

作者： Yanping Chen Tianliang Hou School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina Hunan Key Laboratory for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtan 411105HunanChina

In this paper,we investigate the superconvergence property and the L∞-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral *** state and co-state are approximated by the order one Raviart-Thomas mixed finite element space and the control variable is approximated by piecewise constant functions or piecewise linear *** derive some superconvergence results for the control variable and the state variables when the control is approximated by piecewise constant ***,we derive L∞-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear ***,some numerical examples are given to demonstrate the theoretical results.

关键词： Semilinear elliptic equations optimal control problems superconvergence L∞-error estimates mixed finite element methods postprocessing

Superconvergence of RT1 mixed finite element approximations for elliptic control problems

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Science China Mathematics 2013年第2期56卷 267-281页

作者： HOU TianLiang CHEN YanPing Department of Mathematics Xiangtan University School of Mathematical Sciences South China Normal University

In this paper,we investigate the superconvergence property of the numerical solution to a quadratic elliptic control problem by using mixed finite element *** state and co-state are approximated by the order k=1 Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant *** prove the superconvergence error estimate of h3/2 in L2-norm between the approximated solution and the average L2 projection of the ***,by the postprocessing technique,a quadratic superconvergence result of the control is derived.

关键词： elliptic equations optimal control problems superconvergence mixed finite element methods postprocessing

Accuracy Enhancement of Discontinuous Galerkin Method for Hyperbolic Systems

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Numerical Mathematics(Theory,Methods and Applications) 2014年第2期7卷 214-233页

作者： Tie Zhang Jingna Liu Department of Mathematics and the State Key Laboratory of Synthetical Automation for Process Industries Northeastern UniversityShenyang 110004China

We study the enhancement of accuracy,by means of the convolution postprocessing technique,for discontinuous Galerkin(DG)approximations to hyperbolic *** investigations have focused on the superconvergence obtained by this technique for elliptic,time-dependent hyperbolic and convection-diffusion *** this paper,we demonstrate that it is possible to extend this postprocessing technique to the hyperbolic problems written as the Friedrichs’systems by using an upwind-like DG *** prove that the L2-error of the DG solution is of order k+1/2,and further the post-processed DG solution is of order 2k+1 if Qkpolynomials are *** key element of our analysis is to derive the(2k+1)-order negative norm error *** experiments are provided to illustrate the theoretical analysis.

关键词： Discontinuous Galerkin method hyperbolic problem accuracy enhancement postprocessing negative norm error estimate