A New Approach to Recovery of Discontinuous Galerkin (Journal of Computational Mathematics, 2021)
A new recovery operator $P : Q^{disc}_n (\mathcal{T}) → Q^{disc}_{n+1}(\mathcal{M})$ for discontinuous Galerkin is derived. It is based on the idea of projecting a discontinuous, piecewise polynomial solution on a given mesh $\mathcal{T}$ into a higher order polynomial space on a macro mesh $\mathcal{M}$. In order to do so, we define local degrees of freedom using polynomial moments and provide global degrees of freedom on the macro mesh. We prove consistency with respect to the local $L_2$-projection, stability results in several norms and optimal anisotropic error estimates. As an example, we apply this new recovery technique to a stabilized solution of a singularly perturbed convection-diffusion problem using bilinear elements.
Keywords: Discontinuous Galerkin; Postprocessing; Recovery.