Cubature of integrands containing derivatives
type: Item
instance of: scholarly article
title: Cubature of integrands containing derivatives
zbMATH Open document ID: 0897.65012
author: James N. Lyness
author: Ulrich Rüde
published in: Numerische Mathematik
publication date: 1998-02-01T00:00:00Z
review text: The authors present an accurate quadrature method for approximation to planar integrals \[ \iint_A(\nabla u)^Ta(\nabla v)dx dy\tag{i} \] useful in finite element approximations. \(A\) is either a parallelogram or triangle bounded by vectors \(I_1\), \(I_2\). It is sufficient to approximate integrals \[ \iint_A(\partial u/\partial n_i)a(\partial v/\partial n_j)dx dy,\quad i,j= 1,2,\tag{ii} \] \(n_1= I_1/| I_1|\), \(n_2= I_2/| I_2|\). For the parallelogram, \(A\) is partitioned into \((m+ 1)^2\) subdivisions similar to \(A\) and (ii) is approximated by a Riemann sum \(J^{(m)}\), where \(u\), \(v\) are evaluated at vertex points and \(a\) at center points of the grid. For \(u,v,a\in C^{p+ 1}\) in an open region about \(A\) it is proved that the error in applying \(J^{(m)}\) has the form \[ \sum^{p- 1}_{q= 1} D_q/m^q+ O(1/m^p).\tag{iii} \] The integrals (ii) are approximated by extrapolating values of \(J^{(m)}\), \(m= 1,2,4,8,\dots\), as in Romberg quadrature. The authors present a similar discretization when \(A\) is a triangle. Results are presented for a specific example where \(A\) is the triangle bounded by \((0,0)\), \((1,0)\), \((1,1)\). 15-significant figure accuracy was obtained.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 1111584
zbMATH Keywords: cubature
zbMATH Keywords: integrands containing derivatives
zbMATH Keywords: quadrature method
zbMATH Keywords: finite element
zbMATH Keywords: Riemann sum
zbMATH Keywords: Romberg quadrature
reviewed by: John B. Butler jun.
MaRDI profile type: Publication
full work available at URL: s002110050320
OpenAlex ID: W2045075499
DOI: 10.1007/S002110050320
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description: scientific article; zbMATH DE number 1111584