| In this thesis we examine a local post-processing technique and extend its applications. The post-processing technique that we use was originally developed by Bramble and Schatz for elliptic equations using continuous finite element methods. Using negative norm error estimates, Cockburn, Luskin, Shu, and Suli have shown that this highly efficient local post-processor improves the accuracy of the discontinuous Galerkin methods for linear hyperbolic equations from order k+1 to 2k+1, where k is the highest degree polynomial used in the approximation. We investigate this post-processing technique in the context of superconvergence of the derivatives of the numerical solution, two space dimensions for both tensor product local basis and the usual k-th degree polynomials basis, multi-domain problems with different mesh sizes, variable coefficient linear problems including those with discontinuous coefficients, and linearized Euler equations applied to an aeroacoustic problem. We also develop a class of one-sided post-processing techniques to enhance accuracy for the discontinuous Galerkin methods. We demonstrate through extensive numerical examples that the technique is very effective in all these situations in enhancing the accuracy of the discontinuous Galerkin solutions. Although the existing proofs for the post-processing technique are for linear hyperbolic equations, we also examine the application of this technique for the non-linear Euler equations. Additionally, we extend the applications of the TVB-limiter previously proposed by Shu to discontinuous Galerkin methods for polynomial approximations where k > 2. We do this by projecting the approximation to lower-order polynomial spaces and limiting the projection of the approximation and then limiting the approximation again in the original polynomial space. |
