Justification of the saturation assumption

Abstract

The saturation assumption is widely used in computational science and engineering, usually without any rigorous theoretical justification and even despite of counterexamples for some coarse meshes known in the mathematical literature. On the other hand, there is overwhelming numerical evidence at least in an asymptotic regime for the validity of the saturation. In the generalized form, the assumption states, for any $0<\varepsilon \le 1$, that

$$\begin{aligned} ||| u - {\hat{U}} |||^2 \le (1- \varepsilon /C) ||| u - U |||^2+ \varepsilon \mathrm{osc}^2(f,\mathcal {N}) \end{aligned}$$

(SA)

for the exact solution u and the first-order conforming finite element solution U (resp. ${\hat{U}}$) of the Poisson model problem with respect to a regular triangulation $\mathcal {T}$ (resp. ${\hat{\mathcal {T}}}$) and its uniform refinement ${\hat{\mathcal {T}}}$ within the class $\mathbb {T}$ of admissible triangulations. The point is that the patch-oriented oscillations $\mathrm{osc}(f,\mathcal {N})$ vanish for constant right-hand sides $f\equiv 1$ and may be of higher order for smooth f, while the strong reduction factor $(1- \varepsilon /C)<1$ involves some universal constant C which exclusively depends on the set of admissible triangulations and so on the initial triangulation only. This paper proves the inequality (SA) for the energy norms of the errors for any admissible triangulation $\mathcal {T}$ in $\mathbb {T}$ up to computable pathological situations characterized by failing the weak saturation test (WS). This computational test (WS) for some triangulation $\mathcal {T}$ states that the solutions U and ${\hat{U}}$ do not coincide for the constant right-hand side $f\equiv 1$. The set of possible counterexamples is characterized as $\mathcal {T}$ with no interior node or exactly one interior node which is the vertex of all triangles and ${\hat{\mathcal {T}}}$ is a particular uniform bisec3 refinement. In particular, the strong saturation assumption holds for all triangulations with more than one degree of freedom. The weak saturation test (WS) is only required for zero or one degree of freedom and gives a definite outcome with O(1) operations. The only counterexamples known so far are regular n-polygons. The paper also discusses a generalization to linear elliptic second-order PDEs with small convection to prove that saturation is somehow generic and fails only in very particular situations characterised by (WS).