Multigrid approaches to non‐linear diffusion problems on unstructured meshes

By Dimitri J. Mavriplis. Numerical Linear Algebra with Applications, 2001

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LOCAL DISCONTINUOUS GALERKIN METHODS FOR THREE CLASSES OF NONLINEAR WAVE EQUATIONS (2004, methodology): “The schemes we present extend the previous work of Yan and Shu [30, 31] and of Levy, Shu and Yan [24] on local discontinuous Galerkin method solving partial differential equations with higher spatial derivatives.”
Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Differential Equations (2009, methodology): “Error estimates The sub-optimal O(hk+ 1 2 ) error estimates in L2 norm can be obtained for the smooth solution of the equation [100, 104] ut+ f (u)x+uxxx =0.”
Local discontinuous Galerkin methods for nonlinear Schrödinger equations (2005, methodology): “Shu / Journal of Computational Physics 205 ( 2005 ) 72 – 97 83 to ±10%Dx .”
Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems (2016, methodology): “Namely, F̃nK ,K(u, u) = F (u) · nK , and F̃nK ,K(a, b)+ F̃nK′ ,K′(b, a) = 0 for arbitrary a, b, where K and K ′ share the same edge; see [22] for more details.”
Discontinuous Galerkin Method for Time-Dependent Problems: Survey and Recent Developments (2014, background): “A sub-optimal L(2) error estimate of order O(h) is also proved in [103].”