Small data oscillation implies the saturation assumption

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 = −∆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 ∈ L 2 .

Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems

“…This led to the incorporation of the data resolution into the convergent (h − h/2)-estimator steered adaptive algorithm of [21], where η + osc is used to drive the adaptive FE algorithm. One may expect that a result similar to that of [17] should also hold for BEM or the FEM-BEM coupling. However, the non-locality of the involved boundary integral operators imposes severe difficulties, and we expect that new mathematical techniques have to be developed; (c) the results of [17,21] mentioned before in (b) provide an additional reason why one should include the resolution of the given data into the adaptive scheme and may consider discretized data U 0, of u 0 .…”

“…(a) We remark that the saturation assumption (3.13) dates back to the early work [6], but may fail to hold in general [7,17]. However, it essentially states that the numerical scheme has reached an asymptotic phase [20], Section 5.2; (b) for model problems and lowest-order FEM, the saturation assumption (3.13) can be proven, if the given data are sufficiently resolved.…”

“…Again, the main theorem on strongly monotone operators [33], Section 25.4, proves the unique solvability of (2.15) and thus of (2.14). Moreover, uniform ellipticity of S implies that the unique discrete solution U ∈ X is quasi optimal in the sense of the Céa lemma 17) where the constant C 1 > 0 depends only on Ω, see [13], Corollary 3, resp. [33], Corollary 25.7.…”

“…However, it essentially states that the numerical scheme has reached an asymptotic phase [20], Section 5.2; (b) for model problems and lowest-order FEM, the saturation assumption (3.13) can be proven, if the given data are sufficiently resolved. More precisely, [17] then states…”

Efficiency and Optimality of Some Weighted-Residual Error Estimator for Adaptive 2D Boundary Element Methods

“…Obviously, one can construct examples, for which assumption (A.1) fails to hold for an arbitrarily large number of steps by choosing φ ∈ P 0 (T (n+1) ) ⊥ , where T (n+1) := unif (n+1) (T 0 ). Then, there holds Φ ,1 = Φ = 0 for at least all meshes T with n. Up to data oscillation terms, (A.1) was proved for the finite element method and the Poisson problem [18], but still remains open for BEM. In this appendix, we attempt to prove a slightly weaker version of (A.1).…”