  We design and analyze a new adaptive stabilized finite element method. We
construct a discrete approximation of the solution in a continuous trial space
by minimizing the residual measured in a dual norm of a discontinuous test
space that has inf-sup stability. We formulate this residual minimization as a
stable saddle-point problem which delivers a stabilized discrete solution and a
residual representation that drives the adaptive mesh refinement. Numerical
results on an advection-reaction model problem show competitive error reduction
rates when compared to discontinuous Galerkin methods on uniformly refined
meshes and smooth solutions. Moreover, the technique leads to optimal decay
rates for adaptive mesh refinement and solutions having sharp layers.
