A wide class of discontinuous Galerkin (DG) methods, the so called interior penalty methods, arise from the idea that inter-element continuity could be attained by mimicking the techniques previously developed for weakly enforcing suitable boundary conditions for PDE's, see [7]. Although the DG methods are usually defined by means of the so called numerical fluxes between neighboring mesh cells, see [1], for most of the interior penalty methods for second order elliptic problems it is possible to correlate the expression of the numerical fluxes with a corresponding set of local interface conditions that are weakly enforced on each inter-element boundary. Such conditions are suitable to couple elliptic PDE's with smooth coefficients and it seems that a little attention is paid to the case of problems with discontinuous data or to the limit case where the viscosity vanishes in some parts of the computational domain.
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Zunino, P. (2008). Mortar and Discontinuous Galerkin Methods Based on Weighted Interior Penalties. In: Langer, U., Discacciati, M., Keyes, D.E., Widlund, O.B., Zulehner, W. (eds) Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science and Engineering, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75199-1_38
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