Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Enhanced accuracy by post-processing for finite element methods for hyperbolic equations
Mathematics of Computation
Superconvergence in the generalized finite element method
Numerische Mathematik
Postprocessing for the Discontinuous Galerkin Method over Nonuniform Meshes
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
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Smoothness-increasing accuracy-conserving (SIAC) filtering has demonstrated its effectiveness in raising the convergence rate of discontinuous Galerkin solutions from order $k+\frac{1}{2}$ to order 2k+1 for specific types of translation invariant meshes (Cockburn et al. in Math. Comput. 72:577---606, 2003; Curtis et al. in SIAM J. Sci. Comput. 30(1):272---289, 2007; Mirzaee et al. in SIAM J. Numer. Anal. 49:1899---1920, 2011). Additionally, it improves the weak continuity in the discontinuous Galerkin method to k驴1 continuity. Typically this improvement has a positive impact on the error quantity in the sense that it also reduces the absolute errors. However, not enough emphasis has been placed on the difference between superconvergent accuracy and improved errors. This distinction is particularly important when it comes to understanding the interplay introduced through meshing, between geometry and filtering. The underlying mesh over which the DG solution is built is important because the tool used in SIAC filtering--convolution--is scaled by the geometric mesh size. This heavily contributes to the effectiveness of the post-processor. In this paper, we present a study of this mesh scaling and how it factors into the theoretical errors. To accomplish the large volume of post-processing necessary for this study, commodity streaming multiprocessors were used; we demonstrate for structured meshes up to a 50脳 speed up in the computational time over traditional CPU implementations of the SIAC filter.