Small data oscillation implies the saturation assumption
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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 \(f\in L^2\).
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