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Mathematical analysis of Zienkiewicz—Zhu's derivative patch recovery technique
Author(s) -
Zhang Zhimin
Publication year - 1996
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/(sici)1098-2426(199607)12:4<507::aid-num6>3.0.co;2-q
Subject(s) - superconvergence , mathematics , finite element method , boundary value problem , derivative (finance) , polygon mesh , quadratic equation , norm (philosophy) , mathematical analysis , laplace operator , quadrilateral , geometry , physics , political science , financial economics , law , economics , thermodynamics
Zienkiewicz‐Zhu's derivative patch recovery technique is analyzed for general quadrilateral finite elements. Under certain regular conditions on the meshes, the arithmetic mean of the absolute error of the recovered gradient at the nodal points is superconvergent for the second‐order elliptic operators. For rectangular meshes and the Laplacian, the recovered gradient is superconvergent in the maximum norm at the nodal points. Furthermore, it is proved for a model two‐point boundary‐value problem that the recovery technique results in an “ultra‐convergent” derivative recovery at the nodal points for quadratic finite elements when uniform meshes are used. © 1996 John Wiley & Sons, Inc.