Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

Serge Nicaise (1) , Hengguang Li (1) , Anna L. Mazzucato (1)

1 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015

We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face, and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an a priori error analysis for the Finite Element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis.

Mots clés: Ventcel boundary conditions; Elliptic boundary value problems; finite element; anistropic meshes; weighted Sobolev spaces; polyhedral domains

Domaines: Mathématiques [math]

Serge Nicaise, Hengguang Li, Anna L. Mazzucato. Regularity and a priori error analysis of a Ventcel problem in polyhedral domains. Mathematical Methods in the Applied Sciences, 2017, 40 (5), ⟨10.1002/mma.4083⟩. ⟨hal-01332720v3⟩