  Certain Petrov-Galerkin schemes are inherently stable formulations of
variational problems on a given mesh. This stability is primarily obtained by
computing an optimal test basis for a given approximation space. Furthermore,
these Petrov-Galerkin schemes are equipped with a robust a posteriori error
estimate which makes them an ideal candidate for mesh adaptation. One could
extend these Petrov-Galerkin schemes not only to have optimal test spaces but
also optimal approximation spaces with respect to current estimates of the
solution. These extensions are the main focus of this article. In this article,
we provide a methodology to drive mesh adaptation to produce optimal meshes for
resolving solution features and output functionals which we demonstrate through
numerical experiments.
