Simultaneous polynomial-degree-robust and coefficient-contrast-robust local efficiency of equilibrated flux error estimators for the diffusion problem

The equilibrated residual error estimator of Braess, Pillwein and Schöberl (2009) gives a guaranteed, constant-free upper bound on the finite element energy error and was proven efficient with a constant independent of the polynomial degree $p$ ($p$-robustness) — but only under the assumption that the diffusion coefficient $a$ in $-\nabla\cdot(a\nabla u)=f$ is piecewise constant and approximately equal to one. The paper itself left open the extension to highly varying coefficients (indicating quasi-monotonicity as the intended route), a $p$-uniform efficiency analysis of the computationally cheaper $RT^{p}$ correction variant of its Remark 2, and the gap between the constructive proof and the mixed-method implementation used in computations. Later literature resolved the simplicial-mesh conjecture underpinning the original proof (Costabel–McIntosh; Demkowicz–Gopalakrishnan–Schöberl; Ern–Vohralík) and built an extensive $p$-robust estimator theory, but essentially exclusively for the pure Poisson setting, while a separate line of work achieved robustness with respect to coefficient jumps under quasi-monotonicity without polynomial-degree-uniform constants. No published result provides local efficiency with a constant simultaneously independent of $p$ and of the coefficient contrast. The problem asks for such a simultaneous robustness theory — or a counterexample family showing it is impossible — including the degree-reduced reconstruction variants used in actual codes.

Background

A posteriori error estimation provides computable bounds on the error of a finite element (FEM) solution and drives adaptive refinement. In the $p$- and $hp$-versions of the FEM, where the local polynomial degree $p$ is raised instead of (or together with) mesh refinement, classical residual-based estimators degrade with $p$: Melenk and Wohlmuth showed that their efficiency constants decay like $p^{-1}$. Equilibrated flux (equilibrated residual) estimators avoid this. Starting from the hypercircle identity of Prager and Synge, one postprocesses the discrete solution $u_N$ into a vector field $\sigma\in H(\mathrm{div})$ (the space of $L^2$ vector fields with square-integrable divergence) satisfying the equilibrium condition $-\mathrm{div}\,\sigma=f$ exactly; then $\|a^{1/2}\nabla u_N-a^{-1/2}\sigma\|$ is a guaranteed upper bound on the energy error with constant one, and the estimator is called efficient if this quantity is also a local lower bound of the error. Braess, Pillwein and Schöberl proved that the efficiency constant of this estimator is independent of $p$ (and of the mesh size $h$). Their construction splits the residual by the hat-function partition of unity (the piecewise-linear nodal basis functions $\varphi_i$) and lifts the element and edge residuals into the broken Raviart–Thomas space $RT^{p+1}$ (locally $(\mathcal{P}^{p})^2+x\,\mathcal{P}^{p}$ on simplices; the Brezzi–Douglas–Marini space $BDM^{p}$, whose components lie in $\mathcal{P}^{p}$, is a common implementation alternative) by solving a small mixed problem on each vertex patch (the set of mesh elements sharing one vertex). The proof is complete for affine tensor-product elements; on simplices it rested on a then-unproven conjecture, the existence of a $p$-robust right inverse of the divergence operator on a single simplex (a lifting $r_T\mapsto\sigma_T$ with $\mathrm{div}\,\sigma_T=r_T$ and $\|\sigma_T\|\le C\|r_T\|_{(H^1_0)^*}$, $C$ independent of $p$). Two features limit the result in practice. First, the analysis assumes $a$ piecewise constant with $a\approx 1$; the authors state only that it could be extended to a highly varying coefficient $a$ satisfying the quasi-monotonicity condition. Quasi-monotonicity, the classical framework of Bernardi and Verfürth for jump robustness, requires that on each vertex patch the distribution of the piecewise constant coefficient admits a monotone path from each element to a maximal neighbor; it is known to be sufficient — and its absence essentially necessary — for estimator constants that do not blow up with the coefficient contrast. Second, the constructive proof differs from the simple local mixed-method implementation used in computations, and the numerically attractive variant of Remark 2 (reconstructing the correction in $RT^{p}$ with residuals orthogonally projected onto $\mathcal{P}^{p}$) was supported only by numerical experiments. Subsequent work resolved the simplicial conjecture and generalized the $p$-robust theory (Ern–Vohralík and coauthors, using the polynomial-preserving integral operators of Costabel–McIntosh and the polynomial extension operators of Demkowicz–Gopalakrishnan–Schöberl), and separately developed coefficient-jump-robust equilibrated estimators at fixed or non-uniform degree — but these two robustness properties have not been combined.

Problem Statement

Prove — or disprove by an explicit family of examples drawn from the same admissible class (shape-regular meshes of the stated types and piecewise constant diffusion coefficients satisfying the local quasi-monotonicity condition) — that the equilibrated residual (equilibrated flux) estimator of Braess–Pillwein–Schöberl and its degree-reduced variant of Remark 2 (the correction flux reconstructed in $RT^{p}$ with the local residuals orthogonally projected onto $\mathcal{P}^{p}$, as used in actual implementations) are locally efficient with a constant simultaneously independent of the mesh size $h$, the polynomial degree $p$, and the contrast (ratio of largest to smallest value) of a piecewise constant diffusion coefficient $a$ in $-\nabla\cdot(a\nabla u)=f$ on shape-regular simplicial meshes and affine tensor-product (quadrilateral or hexahedral) meshes in two and three space dimensions — at least under the local quasi-monotonicity condition that the source itself indicated as the route from its $a\approx 1$ assumption to highly varying coefficients. Local efficiency here means an estimate of the form $\eta_K \le C\,\|\nabla(u-u_N)\|_{a,\omega_K}$ (up to data oscillation, the standard additive term accounting for the piecewise-polynomial approximation of the load $f$) on element patches $\omega_K$, where $\eta_K$ is the local estimator contribution, $u_N$ the conforming degree-$(p+1)$ finite element solution, and $\|\cdot\|_{a}$ the energy norm induced by $a$. The single scientific objective subsumes the remaining components left open in the source, since they are steps of the same theory: (i) a $p$-uniform and contrast-robust efficiency analysis of the computationally cheaper correction variant of Remark 2, which reconstructs the correction flux in $RT^{p}$ with the local residuals orthogonally projected onto $\mathcal{P}^{p}$, as used in actual implementations; and (ii) a rigorous account of the gap, noted in the source, between the constructive proof and the mixed-method construction used in actual computations (in particular the choice $BDM^{p}$ versus $RT^{p+1}$ in the local mixed solves), so that the proven bounds apply to the estimator as actually computed.

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