The double adaptivity paradigm: (How to circumvent the discrete inf–sup conditions of Babuška and Brezzi)

Abstract

We present an efficient implementation of the double adaptivity algorithm of Cohen et al. (2012) within the setting of the Petrov–Galerkin method with optimal test functions. We apply this method to the ultraweak variational formulation of a general linear variational problem discretized with the standard Galerkin finite element method. As an example, we demonstrate the feasibility of the method in the context of the convection-dominated diffusion problem. The presented ideas, however, apply to virtually any well-posed system of first-order partial differential equations, including singular perturbation problems.