Local derivative post-processing: Challenges for a non-uniform mesh

Previous investigations into accuracy enhancement for the derivatives of a discontinuous Galerkin solution demonstrated that there are many ways to approach obtaining higher order accuracy in the derivatives, each with different advantageous properties. For the discontinuous Galerkin method, the order of accuracy without post-processing for the dth?derivative is k+1-d. For the derivative of the post-processed solution it is 2k+1-d. Additionally, it was demonstrated that not only is calculating the derivative of the post-processed solution itself unnecessary, but also that order 2k+1 can be obtained for the derivative solution for any order derivative, provided the solution is 2k+1 continuous. This is done using higher-order B-splines than used for the post-processed solution itself convolved against a finite difference derivative. This introduces higher levels of smoothness into the derivative post-processed approximation. However, this investigation was limited to a uniform mesh consideration, which is highly restrictive for practical applications. In this report, we discuss the advantages and disadvantages of extending accuracy enhancement of derivatives to non-uniform meshes in one-dimension using the ideas of local L2-projection, characteristic length as well as direct implementation as done for the post-processed solution itself