Guaranteed stability bounds for second-order PDE problems satisfying a Garding inequality

TL;DR

This paper introduces an algorithm to verify the well-posedness of second-order PDEs satisfying a Garding inequality and provides a computable lower bound for the inf-sup constant, aiding error estimation.

Contribution

The authors develop a numerical method that estimates stability bounds for PDEs satisfying a Garding inequality, with guarantees on the approximation accuracy.

Findings

The lower bound underestimates the optimal constant by roughly a factor of two.

The method uses discrete singular value problems with finite element discretization.

The approach can be used for a posteriori error estimation.

Abstract

We propose an algorithm to numerically determined whether a second-order linear PDE problem satisfying a Garding inequality is well-posed. This algorithm further provides a lower bound to the inf-sup constant of the weak formulation, which may in turn be used for a posteriori error estimation purposes. Our numerical lower bound is based on two discrete singular value problems involving a Lagrange finite element discretization coupled with an a posteriori error estimator based on flux reconstruction techniques. We show that if the finite element discretization is sufficiently rich, our lower bound underestimates the optimal constant only by a factor roughly equal to two.