The h-p version of the finite element method for elliptic equations of order 2m
type: Item
instance of: scholarly article
title: The h-p version of the finite element method for elliptic equations of order 2m
zbMATH Open document ID: 0628.65102
DOI: 10.1007/BF01395885
author: Benqi Guo
published in: Numerische Mathematik
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133272
review text: The problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data. It has been shown that the solutions of the second order and fourth order equations belong to the countable normed spaced \(B^{\ell}_{\beta}\) where the weighted Sobolev norms of the k-th derivatives are bounded by \(Cd^{k-\ell}(k-\ell)!\), \(k\geq \ell\), \(\ell =2,3\) where C and d are constants independent of k. In this case the h-p version of the finite element method leads to an exponential rate of convergence measured in the energy norm. The h-p version was implemented in the code PROBE (NOETIC TECHNOLOGIES, Inc., St. Louis, MO) and has been very successfully used in the industry. We discuss the generalization of these results for problems of order 2m. We show also that the exponential rate can be achieved if the exact solution belongs to the countable normed spaces \(B^{\ell}_{\beta}\) where the weighted Sobolev norm of the k-th derivatives is bounded by \(Cd^{k-\ell}(k-\ell)!\), \(k\geq \ell =m+1\), C and d are independent of k. In addition, if the data is piecewise analytic, then in fact the exact solution belongs to the spaces \(B_{\beta}^{m+1}\). Problems of this type are related obviously to many engineering problems, such as problems of plates and shells, and are also important in connection with well known locking problems.
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 74S05
Mathematics Subject Classification ID: 35J40
zbMATH DE Number: 4022551
zbMATH Keywords: piecewise analytic data
zbMATH Keywords: h-p version of the finite element method
zbMATH Keywords: exponential rate of convergence
MaRDI profile type: Publication
cites work: On the Angle Condition in the Finite Element Method
cites work: Regularity of the Solution of Elliptic Problems with Piecewise Analytic Data. Part I. Boundary Value Problems for Linear Elliptic Equation of Second Order
cites work: Regularity of the Solution of Elliptic Problems with Piecewise Analytic Data. II: The Trace Spaces and Application to the Boundary Value Problems with Nonhomogeneous Boundary Conditions
cites work: On the rates of convergence of the finite element method
cites work: Thep-Version of the Finite Element Method
cites work: The h, p and h-p versions of the finite element method in 1 dimension. I. The error analysis of the p-version
cites work: The h-p version of the finite element method. I. The basic approximation results
cites work: A Twelvth Order Theory of Transverse Bending of Transversely Isotropic Plates
cites work: Q4061445
description: scientific article; zbMATH DE number 4022551