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.
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