### 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.

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.

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