A generalized inf–sup test for multi-field mixed-variational methods

Abstract

We present a variational approach towards identifying conditions for the stability of Galerkin methods for multi-field saddle point problems in continuum mechanics and continuum thermodynamics with free energy functions that have positive second variations at critical points. The framework aims to generalize the discrete inf–sup theory and its numerical verification in the context of general problems in an arbitrary number of fields for both linear and nonlinear problems. In utilizing a linearized second derivative test the proposed scheme is purely based on uniqueness properties of a mixed Lagrangian around the solution, thus providing a purely variational requirement for the well-posedness of Galerkin approximations. In maintaining the dependence on parameters and in incorporating the notion of multifold inf–sup conditions the scheme provides a generalization of the classical LBB theory that allows the study of multi-field saddle point problems in the limit of vanishing parameters and their connection to generalized numerical eigenvalue tests. The proposed generalization is trusted to provide a helpful tool for the development of mixed methods that arise in many novel engineering problems due to the coupling of multiple irreversible phenomena or the incorporation of Lagrange multipliers in advanced methods. An emphasis is given with regard to time-dependent problems in irreversible thermodynamics that arise from Biot-type variational formulations if conjugate variables are employed by means of a Legendre–Fenchel transformation. Examples are given for a two-field variational principle in finite deformation poroelasticity in the undrained limit, a three-field variational principle for elasticity in the incompressible limit, and a recent four-field variational principle in gradient-extended plasticity.