We introduce a novel adaptive-stabilized finite element method for unsteady advection dominant problems. We built a discrete scheme by performing a residual minimization at every time step, to an ad-hoc modification of a discrete formulation obtained from the coupling of an implicit time-marching scheme and a DG formulation in space. As a result, we obtain a stable solution and a residual estimative at every discrete time level. This residual estimative can be employed to guide mesh refinement, implying a considerable reduction in the computational effort required for implicit schemes.
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