Small data oscillation implies the saturation assumption
Willy Dörfler and Ricardo H. Nochetto
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 $f=-\Delta u$, 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 $f\in L^2$.