SMALL DATA OSCILLATION IMPLIES THE SATURATION ASSUMPTION

WILLY DÖRFLER RICARDO H. NOCHETTO

Abstract. The saturation assumption asserts that the best approximation error in H_{0}^{1} 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 = −Δu, 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 ∈ L^{2}.