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Improved L 2 and H 1 error estimates for the Hessian discretization method
Author(s) -
Shylaja Devika
Publication year - 2020
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/num.22460
Subject(s) - superconvergence , mathematics , hessian matrix , discretization , finite volume method , finite element method , quadrature (astronomy) , rate of convergence , mathematical analysis , computer science , physics , key (lock) , thermodynamics , computer security , mechanics , optics
The Hessian discretization method (HDM) for fourth‐order linear elliptic equations provides a unified convergence analysis framework based on three properties namely coercivity, consistency, and limit‐conformity. Some examples that fit in this approach include conforming and nonconforming finite element methods (ncFEMs), finite volume methods (FVMs) and methods based on gradient recovery operators. A generic error estimate has been established in L 2 , H 1 , and H 2 ‐like norms in literature. In this paper, we establish improved L 2 and H 1 error estimates in the framework of HDM and illustrate it on various schemes. Since an improved L 2 estimate is not expected in general for FVM, a modified FVM is designed by changing the quadrature of the source term and a superconvergence result is proved for this modified FVM. In addition to the Adini ncFEM, in this paper, we show that the Morley ncFEM is an example of HDM. Numerical results that justify the theoretical results are also presented.