Optimal order a posteriori error estimates for a class of Runge-kutta and Galerkin methods

Abstract

We derive a posteriori error estimates, which exhibit optimal global order, for a class of time stepping methods of any order that include Runge-Kutta Collocation (RK-C) methods and the continuous Galerkin (cG) method for linear and nonlinear stiff ODEs and parabolic PDEs. The key ingredients in deriving these bounds are appropriate one-degree higher continuous reconstructions of the approximate solutions and pointwise error representations. The reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the time-step; they do not hinge on asymptotics. © Springer-Verlag 2009.