Guaranteed stability bounds for second-order PDE problems satisfying a Garding inequality
It offers a practical numerical verification method for PDE well-posedness, which is incremental as it builds on existing finite element and flux reconstruction techniques.
The paper proposes an algorithm to determine well-posedness of second-order linear PDE problems satisfying a Garding inequality, providing a lower bound for the inf-sup constant that is within a factor of two of the optimal constant for sufficiently rich discretizations.
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.