A posteriori error analysis of twofold saddle point variational formulations for nonlinear boundary value problems

Abstract

In this paper we recast the analysis of twofold saddle point variational formulations for several nonlinear boundary value problems arising in continuum mechanics, and derive reliable and efficient residual-based a posteriori error estimators for the associated Galerkin schemes. We illustrate the main results with nonlinear elliptic equations modelling heat conduction and hyperelasticity. The main tools of our analysis include a global inf–sup condition for a linearization of the problem, Helmholtz's decompositions, local approximation properties of the Raviart–Thomas and Clément interpolation operators, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions. Finally, several numerical results confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithms are provided.

twofold saddle point equation, mixed finite element method, a posteriori error estimator