A Petrov-Galerkin discretization with optimal test space of a mild-weak formulation of convection-diffusion equations in mixed form

Motivated by the Discontinous Petrov-Galerkin method from [Numer. Methods Partial Differential Equations, 27 (2011), 70–105] by Demkowicz and Gopalakrishnan, we study a variational formulation of second order elliptic equations in mixed form, that is obtained by piecewise integrating one of the two equations in the system w.r.t. a partition of the domain into mesh cells. We apply a Petrov-Galerkin discretization with optimal test functions, or equivalently, minimize the residual in the natural norm associated to the variational form. These optimal test functions can be found by solving local problems. Well-posedness, uniformly in the partition, and optimal error estimates are demonstrated. In the second part of the paper, the application to convection-diffusion problems is studied. The available freedom in the variational formulation and in its optimal Petrov-Galerkin discretization is used to construct a method that allows a (smooth) passing to a converging method in the convective limit, being a necessary condition to retain convergence and having a bound on the cost for a vanishing diffusion. The theoretical findings are illustrated by several numerical results.