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

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.