Finite element methods are not always optimal
type: Item
instance of: scholarly article
title: Finite element methods are not always optimal
zbMATH Open document ID: 0633.65107
published in: Advances in Applied Mathematics
publication date: 1987-01-01T00:00:00Z
review text: It is a well known fact that the theory of optimal algorithms for elliptic problems is connected with the theory of N-widths of compacts in Sobolev spaces. The author considers a regular elliptic 2n-th order problem \(Lu=f\) with \(f\in H^ r\), \(r>-m\). He shows that the finite element method with piecewise polynomials of degree k is not asymptotically optimal if \(k<2m-1+r\). A special spline algorithm (a generalized Galerkin method) is recommended in this case. In the complexity analysis the author considers the number of unknowns only as a cast of the algorithm. Usually the computational work to solve grid systems with desired accuracy is considered as the key problem.
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 68Q25
Mathematics Subject Classification ID: 35J40
zbMATH DE Number: 4030904
zbMATH Keywords: optimal algorithms
zbMATH Keywords: N-widths of compacts
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: finite element method
zbMATH Keywords: spline algorithm
zbMATH Keywords: Galerkin method
author: A. G. Werschulz
reviewed by: Evgeniĭ Dyakonov
MaRDI profile type: Publication
full work available at URL: 0196-8858(87)90028-5
OpenAlex ID: W2126763361
cites work: Q3214109
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cites work: Interpolation theory over curved elements, with applications to finite element methods
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cites work: Optimal Error Properties of Finite Element Methods for Second Order Elliptic Dirichlet Problems
DOI: 10.1016/0196-8858(87)90028-5
DOI: 10.7916/D8571M4M
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description: scientific article; zbMATH DE number 4030904