Title :

A stable local commuting projector and optimal hp approximation estimates in H(curl)

Author(s) :

Chaumont-Frelet, Théophile [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Reliable numerical approximations of dissipative systems [RAPSODI]
Vohralík, Martin [Auteur correspondant]
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique [CERMICS]
Simulation for the Environment: Reliable and Efficient Numerical Algorithms [SERENA]

English keyword(s) :

Sobolev space H(curl)
Minimal regularity
Commuting projector
Piecewise polynomial
Nédélec space
hp approximation
Error bound
Constrained-unconstrained equivalence

HAL domain(s) :

Mathématiques [math]/Analyse numérique [math.NA]

English abstract : [en]

We design an operator from the infinite-dimensional Sobolev space H(curl) to its finite-dimensional subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh that has the following properties: 1) it is ...
Show more >We design an operator from the infinite-dimensional Sobolev space H(curl) to its finite-dimensional subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh that has the following properties: 1) it is defined over the entire H(curl), including boundary conditions imposed on a part of the boundary; 2) it is defined locally in a neighborhood of each mesh element; 3) it is based on simple piecewise polynomial projections; 4) it is stable in the L2-norm, up to data oscillation; 5) it has optimal (local-best) approximation properties; 6) it satisfies the commuting property with its sibling operator on H(div); 7) it is a projector, i.e., it leaves intact objects that are already in the Nédélec piecewise polynomial space. This operator can be used in various parts of numerical analysis related to the H(curl) space. We in particular employ it here to establish the two following results: i) equivalence of global-best, tangential-trace-and curl-constrained, and local-best, unconstrained approximations in H(curl) including data oscillation terms; and ii) fully h- and p- (mesh-size- and polynomial-degree-) optimal approximation bounds valid under the minimal Sobolev regularity only requested elementwise. As a result of independent interest, we also prove a p-robust equivalence of curl-constrained and unconstrained best-approximations on a single tetrahedron in the H(curl)setting, including hp data oscillation terms.Show less >

Language :

Anglais

European Project :

Guaranteed fully adaptive algorithms with tailored inexact solvers for complex porous media flows

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL