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EXPLICIT ERROR ESTIMATES FOR COURANT, CROUZEIX-RAVIART AND RAVIART-THOMAS FINITE ELEMENT METHODS

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Journal of Computational Mathematics 2012年第4期30卷 337-353页

作者： carsten carstensen Joscha Gedicke Donsub Rim Institut fur Mathematik Humboldt-Universitiit zu Berlin Unter den Linden 6 10099 Berlin Germany Department of Computational Science and Engineering Yonsei University 120-749 Seoul Korea Institut fur Mathematik Humboldt-Universitat zu Berlin Unter den Linden 6 10099 Berlin Germany Yonsei School of Business and Department of Computational Science and Engineering Yonsei University 120-749 Seoul Korea

The elementary analysis of this paper presents explicit expressions of the constants in the a priori error estimates for the lowest-order Courant, Crouzeix-Raviart nonconforming and P^viart-Thomas mixed finite element methods in the Poisson model problem. The three constants and their dependences on some maximal angle in the triangulation are indeed all comparable and allow accurate a priori error control.

关键词： Error estimates Conforming Nonconforming Mixed Finite element method.

SATURATION AND RELIABLE HIERARCHICAL A POSTERIORI MORLEY FINITE ELEMENT ERROR CONTROL

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Journal of Computational Mathematics 2018年第6期36卷 833-844页

作者： carsten carstensen Dietmar Gallistl Yunqing Huang Humboldt-Universitat zu Berlin Institut fiir Mathematik Unter den Linden 6 10099 Berlin Germany Department of Applied Mathematics University of Twente P.O. Box 217 7500 AE Enschede Institute for Computational and Applied Mathematics and Hunan Key Laboratory for Computation & Simulation in Science & Engineering Xiangtan University Xiangtan 411105 China

This paper proves the saturation assumption for the nonconforming Morley finite ele- ment discretization of the biharmonic equation. This asserts that the error of the Morley approximation under uniform refinement is strictly reduced by a contraction factor smaller than one up to explicit higher-order data approximation terms. The refinement has at least to bisect any edge such as red refinement or 3-bisections on any triangle. This justifies a hierarchical error estimator for the Morley finite element method, which simply compares the discrete solutions of one mesh and its red-refinement. The related adaptive mesh-refining strategy performs optimally in numerical experiments. A remark for Crouzeix-Raviart nonconforming finite element error control is included.

关键词： Saturation Hierarchical error estimation Finite element Nonconforming Biharmonie Morley Kirchhoff plate Crouzeix-Raviart

ERROR REDUCTION IN ADAPTIVE FINITE ELEMENT APPROXIMATIONS OF ELLIPTIC OBSTACLE PROBLEMS

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Journal of Computational Mathematics 2009年第2期27卷 148-169页

作者： Dietrich Braess carsten carstensen Ronald H.W.Hoppe Faculty of Mathematics Ruhr University BochumD-44780 BochumGermany Institute of Mathematics Humboldt University of BerlinD-10099 BerlinGermany Department of Mathematics University of HoustonHoustonTX 77204-3008USA Institute of Mathematics University at AugsburgD-86159 AugsburgGermany

We consider an adaptive finite element method （AFEM） for obstacle problems associated with linear second order elliptic boundary value problems and prove a reduction in the energy norm of the discretization error which leads to R-linear convergence. This result is shown to hold up to a consistency error due to the extension of the discrete multipliers （point functionals） to H^-1 and a possible mismatch between the continuous and discrete coincidence and noncoincidence sets. The AFEM is based on a residual-type error estimator consisting of element and edge residuals. The a posteriori error analysis reveals that the significant difference to the unconstrained case lies in the fact that these residuals only have to be taken into account within the discrete noncoincidence set. The proof of the error reduction property uses the reliability and the discrete local efficiency of the estimator as well as a perturbed Galerkin orthogonality. Numerical results are given illustrating the performance of the AFEM.

关键词： Adaptive finite element methods Elliptic obstacle problems Convergence analysis

Sino-German Computational and Applied Mathematics

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Journal of Computational Mathematics 2020年第1期38卷 I0001-I0002页

作者： carsten carstensen Dietmar Gallistl Jun Hu Xuejun Xu Humboldt-Universitat zu Berlin 10099 BerlinGermany Friedrich-Schiller-Universitat Jena 07737 JenaGermany LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China LSEC ICMSECAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

This short article is the epilog of the 11 preceding papers in this issue of *** contributions are authored by participants of the 6th Sino-German Workshop on Computational and Applied Mathematics at Shanghai.

关键词： Sino Applied German

Preface Special Issue for“the 2nd Sino-German Workshop on Computational and Applied Mathematics”

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Journal of Computational Mathematics 2009年第2期27卷 115-115页

作者： carsten carstensen,Humboldt University Rolf Rannacher,University of Heidelberg Zhongci Shi,Chinese Academy of Sciences Tao Tang,Hong Kong Baptist University Special Issue for "the 2nd Sino-German Workshop on Computational and Applied Mathematics"

The 2nd Sino-German Workshop on Computational and Applied Mathematics took place in Hangzhou, China, from October 9-13, 2007. The long list of senior Chinese numerical analysts who had spent a year or more somewhere in Germany as Humboldt fellows had led to the first Sino-German Workshop in Berlin held at the Humboldt-Universitat zu Berlin in 2005. The particular purpose of the second German-Chinese Workshop on Computational and Applied Mathematics was to attract more junior Chinese scientists to the actual research activities in Germany. A summer school in Beijing on adaptive finite element methods with carsten carstensen and Roll Rannacher piror to the Hangzhou workshop underlined this activity to foster the collaboration of the new generations in the fields of computational and applied mathematics. This special issue reflects the present topics therein in both countries and can be summarised under five headings （i）-（v）.

关键词： Preface Special Issue for the 2nd Sino-German Workshop on Computational and Applied Mathematics