A NOTE ON THE PERTURBED LAX-MILGRAM PROBLEM

Abstract

We deal with an error estimate of the
perturbed Lax-Milgram problem with respect to the perturbation in
the right-hand side. Two examples of perturbed problems in Sobolev
spaces and their detailed analysis are given. The problem
considered here is related to the effect of quadrature errors on
the finite element solution, analyzed in Ciarlet \cite{ciarlet: 91},
Strang et al \cite{strang: 73}, Janik \cite{janik: 86} and
Ko\l odziejczyk \cite{kolo: 89}. However,
in contrast to these papers, where (very often) tedious analysis
of the effect of quadrature errors of a product of two functions,
is given, a different approach is used. We consider the error by
{\it the exact integration} of the product of a projection of
function $f$ and an element from subspace where  we seek the
approximate solutions. This simplifies analysis and, as indicated
examples show, also gives  not complicated formulae. The main
characteristic of it is that numerical quadrature of a product of
two functions can be interpreted as a quadrature with respect to
only one function.