A discrete Saint-Venant principle for finite element discretizations of elliptic problems

TL;DR

This paper establishes a discrete Saint-Venant principle for finite element solutions of elliptic problems, showing exponential decay of boundary influence and aiding in domain decomposition convergence analysis.

Contribution

It introduces a discrete analog of the Saint-Venant principle with decay estimates, supporting localization in finite element methods and domain decomposition analysis.

Findings

Exponential decay of boundary influence in finite element solutions.

Application of decay estimates to convergence analysis of domain decomposition methods.

Numerical examples demonstrating the theoretical results.

Abstract

The present paper studies finite element discretizations of second-order elliptic boundary value problems with homogeneous right-hand side and inhomogeneous boundary conditions. We establish discrete spatial decay estimates on element patches for the energy norm of the discrete solution, showing that the influence of boundary data decays exponentially away from the boundary. The resulting estimates are a discrete analog of Saint-Venant-type principles and provide a rigorous foundation for localization arguments in finite element methods. As an application, we present how these results can be employed in the convergence analysis of domain decomposition methods, on the example of the discrete parallel Schwarz method. Finally, the findings are thoroughly demonstrated on several numerical examples.