Quasi-optimal nonconforming approximation of elliptic PDES with contrasted coefficients and minimal 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 and the error analysis of various nonconforming approximation methods. The essential difficulty is that the Sobolev smoothness index of the exact solution may be just barely larger than one. The lack of smoothness is handled by giving a weak meaning to the normal derivative of the exact solution at the mesh faces. The error estimates are robust with respect to the diffusivity contrast. We briefly show how the analysis can be extended to the Maxwell's equations.

Mots clés: Finite elements; Nonconforming methods; Error estimates; Minimal regularity; 11 Nitsche method; Boundary penalty; 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 minimal regularity. 2018. ⟨hal-01964299v1⟩