Stable broken H1 and H(div) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions

Alexandre Ern (1, 2) , Martin Vohralík (2, 1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms

We study extensions of piecewise polynomial data prescribed on faces and possibly in elements of a patch of simplices sharing a vertex. In the H1 setting, we look for functions whose jumps across the faces are prescribed, whereas in the H(div) setting, the normal component jumps and the piecewise divergence are prescribed. We show stability in the sense that the minimizers over piecewise polynomial spaces of the same degree as the data are subordinate in the broken energy norm to the minimizers over the whole broken H and H(div) spaces. Our proofs are constructive and yield constants independent of the polynomial degree. One particular application of these results is in a posteriori error analysis, where the present results justify polynomial-degree-robust efficiency of potential and flux reconstructions.

Mots clés: Robustness; A posteriori error estimate; Flux reconstruction; Best approximation; Patch of elements; Polynomial degree; Polynomial extension operator; Broken Sobolev space; Potential reconstruction

Domaines: Analyse numérique [math.NA]

Alexandre Ern, Martin Vohralík. Stable broken H1 and H(div) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions. Mathematics of Computation, 2020, 89 (322), pp.551-594. ⟨10.1090/mcom/3482⟩. ⟨hal-01422204v4⟩