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

TL;DR

This paper analyzes the regularity of solutions to a Ventcel boundary value problem on polyhedral domains, deriving error estimates for finite element methods with graded meshes, supported by numerical validation.

Contribution

It provides new regularity estimates for Ventcel problems on polyhedral domains and develops an a priori error analysis for finite element approximations on anisotropic meshes.

Findings

Improved regularity estimates for the solution's trace on Ventcel boundary

Decomposition of the solution into regular and singular parts in weighted Sobolev spaces

Numerical results confirm theoretical error bounds

Abstract

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.