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HANGING NODES IN THE UNIFYING THEORY OF a posteriori FINITE ELEMENT ERROR CONTROL

Journal of Computational Mathematics 2009年第2期27卷 215-236页

作者： C. Carstensen Jun Hu Institut für Mathematik Humboldt Universitaet zu Berlin Rudower Chaussee 25 D -12489 Berlin Germany LMAM and School of Mathematical Sciences Peking University Beijing 100871 China

A unified a posteriori error analysis has been developed in [18, 21-23] to analyze the finite element error a posteriori under a universal roof. This paper contributes to the finite element meshes with hanging nodes which are required for local mesh-refining. The twodimensional 1-irregular triangulations into triangles and parallelograms and their combinations are considered with conforming and nonconforming finite element methods named after or by Courant, Q1, Crouzeix-Raviart, Poisson, Stokes and Navier-Lamé equations Han, Rannacher-Turek, and others for the The paper provides a unified a priori and a posteriori error analysis for triangulations with hanging nodes of degree ≤ 1 which are fundamental for local mesh refinement in self-adaptive finite element discretisations.

关键词： a posteriori A priori Finite element Hanging node Adaptive algorithm.

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A Priori and a posteriori Error Analysis of the Discontinuous Galerkin Methods for Reissner-Mindlin Plates

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Advances in Applied Mathematics and Mechanics 2011年第6期3卷 649-662页

作者： Jun Hu Yunqing Huang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105HunanChina

In this paper,we apply an a posteriori error control theory that we develop in a very recent paper to three families of the discontinuous Galerkin methods for the Reissner-Mindlin plate *** derive robust a posteriori ... 详细信息

关键词： a posteriori error analysis Reissner-Mindlin plate finite element reduction integration

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A Review of Unified a posteriori Finite Element Error Control

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Numerical Mathematics(Theory,Methods and Applications) 2012年第4期5卷 509-558页

作者： C.Carstensen M.Eigel R.H.W.Hoppe C.Löbhard Department of Mathematics Humboldt Universität zu BerlinD-10099 BerlinGermany Department of Computational Science and Engineering Yonsei UniversitySeoul 120-749Korea Department of Mathematics University of HoustonHouston TX 77204-3008USA. Institute of Mathematics University of AugsburgD-86159 AugsburgGermany

This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential *** the abstract setting of mixed formulations,a generalised formulation of the corresponding residuals is proposed which then allows for the unified estimation of the respective dual ***,this can be done with an approach which is applicable in the same way to conforming,nonconforming and mixed ***,the unified approach is applied to various model *** particular,we consider the Laplace,Stokes,Navier-Lamé,and the semi-discrete eddy current equations.

关键词： a posteriori error analysis finite element method nonconforming finite element method mixed finite element method adaptive algorithm Poisson equation Laméequations Stokes equations Maxwell equations unified a posteriori error analysis discontinuous Galerkin residual estimator

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Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs

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Applied Mathematics 2016年第12期7卷 1359-1407页

作者： Karan S. Surana A. D. Joy J. N. Reddy Department of Mechanical Engineering Univeristy of Kansas Lawrence KS USA Department of Mechanical Engineering Texas A&M University College Station TX USA

This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. a posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. a posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation prop

关键词： Finite Element Error Estimation Convergence Rate A Priori a posteriori BVP Variationally Consistent Integral Form Variationally Inconsistent Integral Form Differential Operator Classification Self-Adjoint Non-Self-Adjoint Nonlinear

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