# Small data oscillation implies the saturation assumption

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Numer. Math., 91 (2002), 1-12.
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Willy Doerfler

Institut fuer Angewandte Mathematik

Universitaet Karlsruhe

Kaiserstrasse 12

D-76128 Karlsruhe, Germany

doerfler@math.uni-karlsruhe.de
Ricardo H. Nochetto

Department of Mathematics

University of Maryland, College Park

College Park, MD 20742, USA

rhn@math.umd.edu
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Abstract

The saturation assumption asserts that the best approximation error
in $H^1_0$ with piecewise quadratic finite elements is strictly
smaller than that of piecewise linear finite elements. We establish a
link between this assumption and the oscillation of $f=-\De u$, and
prove that small oscillation relative to the best error with
piecewise linears implies the saturation assumption. We also show that
this condition is necessary, and asymptotically valid provided
$f\in L^2$.