Ana I. Garralda Guillem, Gabriel N. Gatica, Antonio Marquez, Manuel Ruiz Galan:
A posteriori error analysis of twofold saddle point variational formulations for nonlinear boundary value problems
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 decompositions, local approximation properties of the Raviart-Thomas and Clement 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.
This preprint gave rise to the following definitive publication(s):
Ana I. GARRALDA GUILLEM, Gabriel N. GATICA, Antonio MARQUEZ, Manuel RUIZ GALAN: A posteriori error analysis of twofold saddle point variational formulations for nonlinear boundary value problems. IMA Journal of Numerical Analysis, vol. 34, 1, pp. 326-361, (2014).