Continuity properties of the inf-sup constant for the divergence

TL;DR

This paper investigates the mathematical properties of the inf-sup constant for divergence, establishing conditions for its continuity and semi-continuity with respect to domain approximation, and discusses implications for numerical methods.

Contribution

It provides new theoretical results on the continuity and semi-continuity of the LBB constant under domain approximations, with conditions applicable to numerical eigenvalue computations.

Findings

The LBB constant is an upper semi-continuous shape functional.

Sufficient conditions for the convergence of discrete LBB constants are identified.

Numerical examples support the practical relevance of the theoretical conditions.

Abstract

The inf-sup constant for the divergence, or LBB constant, is explicitly known for only few domains. For other domains, upper and lower estimates are known. If more precise values are required, one can try to compute a numerical approximation. This involves, in general, approximation of the domain and then the computation of a discrete LBB constant that can be obtained from the numerical solution of an eigenvalue problem for the Stokes system. This eigenvalue problem does not fall into a class for which standard results about numerical approximations can be applied. Indeed, many reasonable finite element methods do not yield a convergent approximation. In this article, we show that under fairly weak conditions on the approximation of the domain, the LBB constant is an upper semi-continuous shape functional, and we give more restrictive sufficient conditions for its continuity with respect to the domain. For numerical approximations based on variational formulations of the Stokes eigenvalue problem, we also show upper semi-continuity under weak approximation properties, and we give stronger conditions that are sufficient for convergence of the discrete LBB constant towards the continuous LBB constant. Numerical examples show that our conditions are, while not quite optimal, not very far from necessary.