Aspects of discontinuous Galerkin methods for hyperbolic conservation laws

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Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Differential Equations (2009, background): “As mentioned in [53], ideal a posteriori error estimates should be • inexpensive relative to the cost of the solution; • accurate in the sense that they converge to the true error under h- and p-refinement; • robust in the sense that they provide error bounds over a wide range of mesh spacings, poly”
Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws (2004, methodology): “A basis for Pp(Ωj) is chosen to be orthogonal in L(2) [30, 14, 26] on Ωj and this leads to the Dubiner basis commonly used with spectral methods [24].”
Slope limiting for discontinuous Galerkin approximations with a possibly non‐orthogonal Taylor basis (2013, background): “A major bottleneck in the design of high-order discontinuous Galerkin (DG) methods [5, 6, 8, 11] for convection-dominated transport problems is the lack of reliable mechanisms that ensure nonlinear stability and prevent the onset of spurious oscillations.”