Stopping criteria for adaptive finite element solvers

We consider a family of practical stopping criteria for linear solvers for adaptive finite element methods for symmetric elliptic problems. A contraction property between two consecutive levels of refinement of the adaptive algorithm is shown when a family of smallness criteria for the corresponding linear solver residuals are assumed on each level or refinement. More importantly, based on known and new results for the estimation of the residuals of the conjugate gradient method, we show that the smallness criteria give rise to practical stopping criteria for the iterations of the linear solver, which guarantees that the (inexact) adaptive algorithm converges. A series of numerical experiments highlights the practicality of the theoretical developments.