# Small data oscillation implies the saturation assumption

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

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

## 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 $f\in L^2$.