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Superconvergent error estimates for position-dependent smoothness-increasing accuracy-conserving (SIAC) post-processing of discontinuous Galerkin solutions


Authors: Liangyue Ji, Paulien van Slingerland, Jennifer K. Ryan and Kees Vuik
Journal: Math. Comp. 83 (2014), 2239-2262
MSC (2010): Primary 65M60; Secondary 35L02
DOI: https://doi.org/10.1090/S0025-5718-2014-02835-4
Published electronically: April 9, 2014
MathSciNet review: 3223331
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Abstract: Superconvergence of discontinuous Galerkin methods is an area of increasing interest due to the ease with which higher order information can be extracted from the approximation. Cockburn, Luskin, Shu, and Süli showed that by applying a B-spline filter to the approximation at the final time, the order of accuracy can be improved from $ \mathcal {O}(h^{k+1})$ to $ \mathcal {O}(h^{2k + 1})$ in the $ \mathcal {L}^{2}$-norm for linear hyperbolic equations with periodic boundary conditions (where $ k$ is the polynomial degree and $ h$ is the mesh element diameter) [Math. Comp. (2003)]. The applicability of this filter for linear hyperbolic problems with non-periodic boundary conditions was computationally extended and renamed a position-dependent smoothness-increasing accuracy-conserving (SIAC) filter by van Slingerland, Ryan, Vuik [SISC (2011)]. However, error estimates in the $ \mathcal {L}^2$-norm for this new position-dependent SIAC filter were never given. Furthermore, error estimates in the $ \mathcal {L}^\infty $-norm have not been established for the original kernel nor the position-dependent kernel. In this paper, for the first time we establish that it is possible to obtain $ \mathcal {O}(h^{\min \{2k+1,2k + 2-\frac {d}{2}\}})$ accuracy in the $ \mathcal {L}^{\infty }$-norm for the position-dependent SIAC filter, where $ d$ is the dimension. Furthermore, we extend the error estimates given by Cockburn et al. so that they are applicable to the entire domain when implementing the position-dependent SIAC filter. We also computationally demonstrate the applicability of this filter for visualization of streamlines.


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Additional Information

Liangyue Ji
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China.
Address at time of publication: University of Minnesota, School of Mathematics, 206 Church St. SE Minneapolis, Minnesota 55455

Paulien van Slingerland
Affiliation: Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlands

Jennifer K. Ryan
Affiliation: Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlands
Address at time of publication: University of East Anglia, School of Mathematics, Norwich Research Park, Norwich NR4 7TJ, United Kingdom
Email: Jennifer.Ryan@uea.ac.uk

Kees Vuik
Affiliation: Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlands

DOI: https://doi.org/10.1090/S0025-5718-2014-02835-4
Keywords: High-order methods, discontinuous Galerkin, SIAC filtering, accuracy enhancement, B-splines, post-processing, superconvergence, error estimates
Received by editor(s): May 25, 2011
Received by editor(s) in revised form: February 1, 2012, and November 8, 2012
Published electronically: April 9, 2014
Additional Notes: The second author was supported by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-09-1-3055.
The third author was supported by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-09-1-3055.
The U.S Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright notation thereon