Approaches for Adjoint-Based A Posteriori Analysis of Stabilized Finite Element Methods

Abstract

This study derives a posteriori error estimates for linear functionals of the solution of systems of partial differential equations discretized using stabilized continuous Galerkin methods. We investigate a convection-diffusion equation, the Stokes equations, and incompressible low Reynolds number flow governed by the Navier--Stokes equations. We consider three well-known stabilization methods and show that only one of the three is adjoint consistent and that even this case is contingent upon a proper treatment of the adjoint data. A standard approach for a posteriori error analysis uses the adjoint of the stabilized formulation, which inherits the difficulties induced by the lack of adjoint consistency. We introduce and analyze two alternative approaches. The first is based on the addition of stabilization terms to the adjoint data, while the second is based on a stabilized formulation of the formal adjoint problem. We show that any of the three approaches can be used to derive a fully computable error representation. However, numerical results show that these new alternative approaches result in more accurate error estimates for a variety of problems.