Multi-dimensional quadrature of singular and discontinuous functions
type: Item
instance of: scholarly article
title: Multi-dimensional quadrature of singular and discontinuous functions
zbMATH Open document ID: 1021.65010
DOI: 10.1023/A:1021988001059
author: Anna-Karin Tornberg
published in: BIT
publication date: 2003-10-09T00:00:00Z
review text: This paper is concerned with the problem of evaluating integrals with non-smooth integrands \(F:\Omega\subset\mathbb R^2\to\mathbb R\). The singularities of \(F\) are all located in a given curve \(\gamma\in \Omega\) which separates \(\Omega\) into two regions \(\Omega_1\) and \(\Omega_2\). The class of integrands is described as follows. Let \(d\) be the signed distance function in \(\Omega\) which is defined as \(d(\mathbf{x})=(-1)^{i+1}\)dist\((\mathbf{x},\gamma)\) if \(\mathbf{x}\in \Omega_i\), \(i=1,2\). Let \(G(\mathbf{x})\) be an arbitrary smooth function in \(\Omega\). The functions \(F(\mathbf{x})\) to be integrated have the form \(F(\mathbf{x})=f(d(\mathbf{x}))G(\mathbf{x})\), where \(f\) is one of the two following. 1. \(f(t)=\delta(t)\), where \(\delta\) is the Dirac delta function (singular functions). 2. \(f(t)=H(t)\), where \(H(t)\) is the Heaviside function (discontinuous functions). In general, the author suggests an approach which consists in a regularization of the integrand by introducing a smooth approximating function \(f_w(d(\mathbf{x}))\) to the function \(f(d(\mathbf{x}))\). Thus, the whole error in the integration of \(F(\mathbf{x})\) is the sum \(E_{G}(f)=E_{w,G}(f_w)+E_{quad,G}(f_w)\), where \(E_{w,G}(f_w)\) is the analytical error made by replacing \(f(d(\mathbf{x}))\) with \(f_w(d(\mathbf{x}))\), and \(E_{quad,G}(f_w)\) is the numerical error made by applying a quadrature procedure to the integral of \(f_w(d(\mathbf{x}))\). By introducing the moments of a function, some results on the analytical error are obtained for both types of integrands. Moreover, regularization techniques to problems for which \(\gamma\) is close to the boundary, are discussed. Numerical tests are shown to illustrate the behavior of the method.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 1886395
zbMATH Keywords: multi-dimensional quadrature
zbMATH Keywords: cubature formulas
zbMATH Keywords: singular integrands
zbMATH Keywords: interface tracking
zbMATH Keywords: finite element method
zbMATH Keywords: discontinuous material coefficients
zbMATH Keywords: singular forces
zbMATH Keywords: numerical examples
zbMATH Keywords: error bounds
reviewed by: J. Illán
MaRDI profile type: Publication
full work available at URL: a:1021988001059
OpenAlex ID: W1524712200
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description: scientific article; zbMATH DE number 1886395