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We design and analyze a new adaptive stabilized finite element method. Weconstruct a discrete approximation of the solution in a continuous trial spaceby minimizing the residual measured in a dual norm of a discontinuous testspace that has inf-sup stability. We formulate this residual minimization as astable saddle-point problem which delivers a stabilized discrete solution and aresidual representation that drives the adaptive mesh refinement. Numericalresults on an advection-reaction model problem show competitive error reductionrates when compared to discontinuous Galerkin methods on uniformly refinedmeshes and smooth solutions. Moreover, the technique leads to optimal decayrates for adaptive mesh refinement and solutions having sharp layers.

An adaptive stabilized conforming finite element method via residual minimization on dual discontinuous Galerkin norms