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...version of the finite element method. As in [10], the analysis is based on rewriting the method in a non-consistent manner using lifting operators in the spirit of Arnold, Brezzi, Cockburn and Marini =-=[1]-=- (see also Perugia and Schötzau [18] and Schötzau, Schwab and Toselli [20]), and employing a new hp-version decomposition result discontinuous finite element spaces; the proof of this result will be r...

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...ively. However, these are not the only choices for numerical traces we could use to generate stabilization through the jumps of the approximate solution. Indeed, in the unified analysis of DG methods =-=[5]-=-, we see that we can also take �q h = q h + αr((uh − λh)n), for the BMMPR method [13], and �q h = −agrad uh + αr((uh − λh)), for the BRMPS method [7]. Here, for any ϕ ∈ L 2 (∂K) the vector αr(ϕ) is th...

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...he RB-literature. In particular, our framework covers finite volume (FV) schemes [4, 8, 12, 13] for parabolic and hyperbolic transport equations. Similarly, local discontinuous Galerkin (LDG) methods =-=[2, 6, 5]-=- can be cast in this framework. Discrete evolution schemes are based on an approximating high-dimensional discrete space WH ⊂ L2 (Ω) (or subset of some other Hilbert space) and by approximating the ex...

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...he LDG approach requires that we rewrite the above equations as a system of first order equations by introducing additional variables for the viscous fluxes ∂U ∂t +∇ · F −∇ · q = 0 (11) q − ε∇U = 0 . =-=(12)-=- In order to eliminate the effect of the added viscosity away from discontinuities we use the discontinuity sensor described above for the scalar case and apply it to a characteristic quantity such as...

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...ltaneously, the jumps of u and ∂u/∂n vanish on the interelement boundaries. To express this in a more precise way we need to introduce suitable notational conventions. We use the notation employed in =-=[2]-=-. Let e be an interior edge shared by elements T1 and T2. Define the unit normal vectors n1 and n2 on e pointing exterior to T1 and T2, respectively. For a function ϕ, piecewise smooth on Th, with ϕi ...

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... σ, τ by σh, τ h ∈ Σh and u, v by uh, vh ∈ Vh. Then, approximate the traces of u and σ across interelement boundaries by the so-called numerical fluxes denoted by ûh and ̂σh, respectively (see, e.g., =-=[2]-=- for details) and obtain ∫ K K ∫ iω σh · τ h dV + K K ∫ uh ∇ · τ h dV − ∫ ∫ ∫ iω uh vh dV + σh · ∇vh dV − ∂K ∂K ûh τ h · ndS = 0 ∀τ h ∈ Σh(K) ̂σh · nvh dS = 1 ∫ f vh dV ∀vh ∈ Vh(K) . iω K (3.3) At thi...