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Minimal stabilization for discontinuous Galerkin finite element methods for hyperbolic problems.

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We consider a discontinuous Galerkin finite element method for the advection-reaction equation in two space-dimensions. For polynomial approximation spaces of degree greater than or equal to two on triangles we propose a method where stability is obtained by a penalization of only the upper portion of the polynomial spectrum of the jump of the solution over element edges. We prove stability in the standard $h$-weighted graph norm and obtain optimal order error estimates with respect to mesh-size.

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