On the Use of the Steklov Analysis to Verify the Stability Condition of Mixed Finite Element Formulations

Abstract

In finite element approximations of conservation laws using mixed formulations the stability depends strongly on the compatibility of the approximation space of the primary variable and dual variable. This compatibility is also known as the Ladyzenskaya-Babuska-Brezzi (LBB)-condition.
This work is dedicated to the development of a numerical procedure to verify the compatibility of the approximation spaces based on the analysis of the quality of approximations of a Steklov eigenvalue problem. It is observed that in the occurence of poorly configured approximation spaces, the numerical approximation of the eigenvalues of the Steklov problem either presents artificial (spurious) low energy eigenvalues or a reduced number of correctly approximated values. The first case is an indication of a poor constraint space and the latter is an indication of a too rich constraint space.
The advantage of using the Steklov eigenvalue problem is that the numerically obtained eigenvalues can be compared to either analytical values.