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

Alexandre Ern (1, 2) , Jean-Luc Guermond (3)

1 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
3 TAMU - Department of Mathematics [Texas]

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.

Mots clés: Finite elements; Nonconforming methods; Error estimates; Minimal regularity; Nitsche's method; discontinuous Galerkin; Elliptic equations; Maxwell's equations

Domaines: Analyse numérique [math.NA]

Alexandre Ern, Jean-Luc Guermond. Quasi-optimal nonconforming approximation of elliptic PDES with contrasted coefficients and $H^{1+r}$, $r>0$, regularity. 2021. ⟨hal-01964299v2⟩