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