Fourier analysis for discontinuous Galerkin and related methods

In this paper we review a series of recent work on using a Fourier analysis technique to study the stability and error estimates for the discontinuous Galerkin method and other related schemes. The advantage of this approach is that it can reveal instability of certain “bad”’ schemes; it can verify stability for certain good schemes which are not easily amendable to standard finite element stability analysis techniques; it can provide quantitative error comparisons among different schemes; and it can be used to study superconvergence and time evolution of errors for the discontinuous Galerkin method. We will briefly describe this Fourier analysis technique, summarize its usage in stability and error estimates for various schemes, and indicate the advantages and disadvantages of this technique in comparison with other finite element techniques. Supported by the National Natural Science Foundation of China (Grant Nos. 10671190, 10671190), NSF Grant DMS-0809086 and DOE Grant DE-FG02-08ER25863