Bounds on spectral condition numbers of matrices arising in the \(p\)- version of the finite element method
type: Item
instance of: scholarly article
title: Bounds on spectral condition numbers of matrices arising in the \(p\)- version of the finite element method
zbMATH Open document ID: 0834.65034
author: Elwood T. Olsen
author: Jim jun. Douglas
published in: Numerische Mathematik
publication date: 1996-03-31T00:00:00Z
review text: Condition numbers for \(p\)-version finite elements can be quite large and substantial attention has therefore been directed towards finding basis sets for which the growth of the condition number is relatively modest as a function of \(p\). This issue is related to domain decomposition methods; for one many iterative substructuring methods finding well-conditioned basis for the polynomial space is the main practical and theoretical issue. The authors work exclusively with methods based on tensor product spaces, which greatly simplifies the analysis. The reference element is then a square or a cube. Two bases are considered in detail. The first, based on the Lagrangian nodal basis on uniform meshes is shown to lead to a spectacular, exponential growth of the condition number. The second consists of linear functions and the products of \((1 - x^2)\) and Chebyshev polynomials. Upper and lower bounds for the condition number, which are algebraic, and which invites pessimism, are provided. The degree of the growth function is shown to be at least \(5d/2\), where \(d\) is the dimension of the underlying Euclidean space. It is also conjectured that any basis which is hierarchical and also satisfies a local support condition will have condition numbers that grow as fast as the \(4d\)-th power of \(p\). Maybe somewhat surprising the experimental work does not include a study of the basis derived from the integrated Legendre polynomials. For this basis, \textit{Olivier Pourquier} of Lyons, France, in his 1994 thesis, has shown that the condition number grows as the \(4(d - 1)\)th power of \(p\). This appears to disprove the main conjecture of this paper. Pourquier also has good results on the Lagrangian basis using the Gauss-Lobatto- Legendre nodes which are favored by many practitioners.
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65N55
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 739415
zbMATH Keywords: condition numbers
zbMATH Keywords: \(p\)-version finite elements
zbMATH Keywords: domain decomposition
zbMATH Keywords: iterative substructuring methods
zbMATH Keywords: tensor product spaces
zbMATH Keywords: Lagrangian nodal basis
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: Legendre polynomials
reviewed by: Olof B. Widlund
MaRDI profile type: Publication
full work available at URL: s002110050096
OpenAlex ID: W2074250763
DOI: 10.1007/S002110050096
description: scientific article; zbMATH DE number 739415