The effect of numerical integration in finite element methods for nonlinear Sobolev equations
type: Item
instance of: scholarly article
zbMATH Open document ID: 0985.65119
author: Germain N'Guimbi
publication date: 2002-05-08T00:00:00Z
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1616517
zbMATH Keywords: finite element method
zbMATH Keywords: nonlinear Sobolev equations
zbMATH Keywords: quadrature rules
zbMATH Keywords: error estimates
MaRDI profile type: Publication
title: The effect of numerical integration in finite element methods for nonlinear Sobolev equations
published in: Numerical Mathematics
review text: This paper deals with the numerical approximation of solutions of nonlinear initial boundary value Sobolev (evolution) equations. Their weak formulation looks for a solution \(u\) defined on the time interval and with values in \(H^1_0(\Omega)\), where \(\Omega \) is a polygonal domain in Euclidean space. By means of some quadrature rule on \(\Omega \), one can substitute the integrals in the weak formulation and look for a solution \(U\) of this discretized weak formulation, whose values are in a finite element subspace of functions in \(H^1_0(\Omega)\), which are continuous on the closure of \(\Omega \). NEWLINENEWLINENEWLINEA typical result proved in the paper is that if the quadrature rule is exact in \(P_{2k}\), i.e. the space of polynomials of degree smaller or equal than \(2k\), and the set of points in \(\Omega \) used in the rule contains a unisolvent subset for \(P_k(\Omega)\), and if \(u\) and \(u_t\) are in \(L_{\infty }(W^{k+1,\infty }(\Omega))\), then for the finite element method mesh size \(h\) sufficiently small, there exists a constant \(C=C(u)\), such that NEWLINE\[NEWLINE\|u-U\|_{L_{\infty }(L_2)}+\|u_t-U_t\|_{L_{\infty }(L_2)}\leq Ch^{k+1},NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\|u-U\|_{L_{\infty }(H^1)}+\|u_t-U_t\|_{L_{\infty }(H^1)}\leq Ch^k.NEWLINE\]
reviewed by: Juan Pedro Milaszewicz
description: scientific article; zbMATH DE number 1616517