Automated testing of saddle point stability conditions

Abstract

Over the last five decades, there has been a substantial body of research on the theory of mixed finite element methods. Mixed finite element methods are finite element methods where two or more finite element spaces are used to approximate separate variables. These methods have often been applied to saddle point problems arising from constrained minimization problems. Examples include the Stokes equations, the equations of Darcy flow (or the mixed Laplacian) or the Hellinger–Reissner formulation for linear elasticity. For equations involving several variables, and where elimination of any of the variables is not a viable option, the usefulness of such methods is evident. For other equations, discretizations based on the introduction of additional variables may have improved properties. The goal of this chapter is to demonstrate that one may automate the examination of the stability of any given discretization.