Potential and flux reconstructions for hp-optimal a priori and a posteriori error estimates

Given a scalar-valued discontinuous piecewise polynomial on a simplicial mesh, a potential reconstruction is a piecewise polynomial that is trace continuous, i.e., H1-conforming. It is best obtained via local homogeneous Dirichlet problems (finite element primal energy minimization) on vertex patches. Similarly, given a vector-valued discontinuous piecewise polynomial not having the target curl or divergence, a flux reconstruction is a piecewise polynomial that is tangential- or normal-trace continuous, i.e., H(curl)- or H(div)-conforming, and has the target curl or divergence. It is best obtained via local homogeneous Neumann problems (finite element dual energy constrained minimization). The theory of broken polynomial extension operators [Braess, Pillwein, Schoeberl, Equilibrated residual error estimates are p-robust, Comput. Methods Appl. Mech. Engrg. 198 (2009), 1189–1197] and [1, 2] allows to prove that such discrete, piecewise polynomial minimizations are equivalent to the continuous (Sobolev spaces) ones uniformly in the polynomial degree p. In a second step, a local p-stable decomposition such as [Schoeberl, Melenk, Pechstein, Zaglmayr, Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements, IMA J. Numer. Anal. 28 (2008), 1–24] or [Falk and Winther, Local space-preserving decompositions for the bubble transform, Found. Comput. Math. 26 (2025), 1219–1267] may be employed and some developments from algebraic topology (simplicial complexes, shellability, subdivisions) may be used.
The concepts of potential and flux reconstructions are known to lead to guaranteed, locally efficient, and polynomial-degree-robust a posteriori error estimates, which we quickly recall here.
We focus on showing that potential and flux reconstructions can also be used to obtain novel a priori error estimates. They in particular allow to devise local commuting projectors [3, 4, 5] that are uniformly stable in the polynomial degree p, with the approximation property being that of local-best unconstrained projection. This means that the precision of the commuting projector is, up to a constant only possibly depending on the mesh shape regularity, as good as that of the L2 projection over individual mesh elements, without any continuity requirement along mesh faces, and without any curl or divergence constraint. Therefrom, optimal hp approximation estimates, where the minimal Sobolev regularity is only requested separately on each mesh element, follow. The designed local commuting projectors naturally allow treatment of nontrivial domains and partial boundary conditions.

[1] A. Ern, M. Vohralík, Stable broken H1 and H(div) polynomial extensions for polynomial-degree-robust potential and flux reconstructions in three space dimensions, Math. Comp. 89 (2020), 551–594.

[2] T. Chaumont-Frelet, M. Vohralík, Constrained and unconstrained stable discrete minimizations for p-robust local reconstructions in vertex patches in the de Rham complex, Found. Comput. Math. 26 (2026), 483–524.

[3] T. Chaumont-Frelet, M. Vohralík, A stable local commuting projector and optimal hp approxi- mation estimates in H(curl), Numer. Math. 156 (2024), 2293–2342.

[4] L. Demkowicz, M. Vohralík, p-robust equivalence of global continuous constrained and local dis- continuous unconstrained approximation, a p-stable local commuting projector, and optimal ele- mentwise hp approximation estimates in H(div), Math. Comp. (2026).

[5] M. Vohralík, p-robust equivalence of global continuous and local discontinuous approximation, a p- stable local projector, and optimal elementwise hp approximation estimates in H1, SIAM J. Numer. Anal. (2026), accepted for publication.


Contatti:
paola.antonietti@polimi.it