Stopping criteria for mixed finite element problems

We study stopping criteria that are suitable in the solution by Krylov space based methods of linear and non linear systems of equations arising from the mixed and the mixed-hybrid finite-element approximation of saddle point problems. Our approach is based on the equivalence between the Babuska and Brezzi conditions of stability which allows us to apply some of the results obtained in Arioli, Loghin and Wathen [1]. Our proposed criterion involves evaluating the residual in a norm defined on the discrete dual of the space where we seek a solution. We illustrate our approach using standard iterative methods such as MINRES and GMRES. We test our criteria on Stokes and Navier-Stokes problems both in a linear and nonlinear context.