Finite element method for elliptic problems with edge singularities
type: Item
instance of: scholarly article
title: Finite element method for elliptic problems with edge singularities
zbMATH Open document ID: 0936.65130
DOI: 10.1016/S0377-0427(99)00062-X
author: Jean Lubuma
author: Serge Nicaise
published in: Journal of Computational and Applied Mathematics
publication date: 2000-05-03T00:00:00Z
review text: The authors consider numerical approximations of tangentially regular solutions of the Dirichlet problem for homogeneous elliptic partial differential equations in three dimensions. The solution domain is a bounded polygonal domain in two dimensions extended infinitely in the third. Theoretical results employ Fourier transforms in the infinite direction with numerical results attained by truncated Fourier series. The bounded polygonal cross-section is treated using finite elements. For this configuration, the solution can exhibit singularities arising from corners in the cross-section and yet maintain regularity in the third direction. This situation can lead to slow convergence rates. Convergence rates can be improved by augmenting the finite element spaces with certain singular functions. These singular functions are easily constructed in two dimensions but their extension to three dimensions is more delicate. The authors construct the singular functions first in two space and one Fourier dimension and then use inverse transforms to return to three space dimensions. The transform variable adds an infinite dimension to the normally finite-dimensional finite element spaces, but such important features as completeness remain true. Error estimates using the finite element method augmented with singular functions are proved and demonstrate the effectiveness of the method.
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 65N12
zbMATH DE Number: 1334653
zbMATH Keywords: Fourier-finite element method
zbMATH Keywords: singular function method
zbMATH Keywords: regularizing kernel
zbMATH Keywords: error estimate
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Fourier transform
zbMATH Keywords: numerical results
zbMATH Keywords: finite elements
zbMATH Keywords: convergence
reviewed by: Myron Sussman
MaRDI profile type: Publication
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full work available at URL: s0377-0427(99)00062-x
OpenAlex ID: W2080044180
description: scientific article; zbMATH DE number 1334653