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I wrote a number of 1D/2D FE and FD programs as a bachelor student, but the main problem I continually came into contact with was gradient shocks related to convection/diffusion problems in convection-dominated flows. From my limited research this seems to be an inherent problem in continuous Galerkin FE methods.

Now I'm interested in jumping into the DG FE world. I've been doing some reading on the various forms of DG available, but I'm not really sure where to start. I've read about hybrid CG/DG methods, various limiters, h/p adaptivity, etc. All of this is really interesting, but difficult to find a beginning to start with.

Is it worth just focusing on the first paper by Reed and Hill from '73? Is there a more recent paper/textbook/lecture notes that gives a decent intro into DG methods without any overly specific information for a particular application? I'm looking to start but don't know where..

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    $\begingroup$ Both books that Paul suggests are great. I'd just like to add one more reference that I found extremely useful to the list. It's a paper summarizing DG methods for various different problems, and it will take you step by step through the development of the methods. It doesn't have much in the way of implementation, but it supplements some of the more implementation-focused resources very well. publications.mi.fu-berlin.de/767/1/Cockburn2003DGMethods.pdf $\endgroup$
    – Tyler Olsen
    Mar 31, 2016 at 12:20
  • $\begingroup$ Howdy Tyler, thanks for the supporting document. This looks like a great place to begin before I invest in one of the books suggested by the other commenters. $\endgroup$
    – cbcoutinho
    Apr 1, 2016 at 8:16

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For parabolic/elliptic PDE's, I highly recommend Beatrice Riviere's book: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation.

For hyperbolic PDE's and general (i.e. nonlinear) conservation laws, I recommend Hesthaven & Warburton's book: Nodal discontinuous Galerkin methods: algorithms, analysis, and applications.

Both books include matlab code that you can step through in a debugger to understand how they are really implemented.

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  • $\begingroup$ Howdy Paul, thanks for the info. Since I'm interested in advection-dominated flows I will probably lean towards the hyperbolic text book on Nodal DG. The fact that both have matlab code to parse through makes bridging the gap much simpler. Thanks again for your input $\endgroup$
    – cbcoutinho
    Apr 1, 2016 at 8:28

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