Quasi-optimal nonconforming approximation of elliptic PDES with contrasted coefficients and $H^{1+r}$, $r>0$, regularity - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Foundations of Computational Mathematics Year : 2021

Quasi-optimal nonconforming approximation of elliptic PDES with contrasted coefficients and $H^{1+r}$, $r>0$, regularity

Abstract

In this paper, we investigate the approximation of a diffusion model problem with contrasted diffusivity for various nonconforming approximation methods. The essential difficulty is that the Sobolev smoothness index of the exact solution may be just barely larger than 1. The lack of smoothness is handled by giving a weak meaning to the normal derivative of the exact solution at the mesh faces. We derive robust and quasi-optimal error estimates. Quasi-optimality means that the approximation error is bounded, up to a generic constant, by the best-approximation error in the discrete trial space, and robustness means that the generic constant is independent of the diffusivity contrast. The error estimates use a mesh-dependent norm that is equivalent, at the discrete level, to the energy norm and that remains bounded as long as the exact solution has a Sobolev index strictly larger than 1. Finally, we briefly show how the analysis can be extended to the Maxwell's equations.
Fichier principal
Vignette du fichier
contrasted_diffusivity.pdf (438.23 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-01964299 , version 1 (21-12-2018)
hal-01964299 , version 2 (03-05-2021)
hal-01964299 , version 3 (24-11-2021)

Identifiers

Cite

Alexandre Ern, Jean-Luc Guermond. Quasi-optimal nonconforming approximation of elliptic PDES with contrasted coefficients and $H^{1+r}$, $r>0$, regularity. Foundations of Computational Mathematics, 2021, 22 (5), pp.1273--1308. ⟨10.1007/s10208-021-09527-7⟩. ⟨hal-01964299v3⟩
284 View
205 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More