Abstract
We apply the discontinuous Galerkin finite element method with a degree p polynomial basis to the linear advection equation and derive a PDE which the numerical solution solves exactly. We use a Fourier approach to derive polynomial solutions to this PDE and show that the polynomials are closely related to the \(\frac{p}{p+1}\) Padé approximant of the exponential function. We show that for a uniform mesh of N elements there exist \((p+1)N\) independent polynomial solutions, N of which can be viewed as physical and pN as non-physical. We show that the accumulation error of the physical mode is of order \(2p+1\). In contrast, the non-physical modes are damped out exponentially quickly. We use these results to present a simple proof of the superconvergence of the DG method on uniform grids as well as show a connection between spatial superconvergence and the superaccuracies in dissipation and dispersion errors of the scheme. Finally, we show that for a class of initial projections on a uniform mesh, the superconvergent points of the numerical error tend exponentially quickly towards the downwind based Radau points.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover, New York (1965)
Adjerid, S., Devine, K., Flaherty, J.E., Krivodonova, L.: A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191, 1097–1112 (2002)
Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198, 106–130 (2004)
Baker, G.A., Graves-Morris, P.R.: Padé approximants. Cambridge University Press (1996)
Biswas, R., Devine, K., Flaherty, J.E.: Parallel adaptive finite element methods for conservation laws. Appl. Numer. Math. 14, 255–284 (1994)
Cao, W., Zhang, Z., Zou, Q.: Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations. SIAM J. Numer. Anal. 52(5), 2555–2573 (2014)
Cheng, Y., Shu, C.-W.: Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J. Comput. Phys. 227, 9612–9627 (2008)
Cheng, Y., Shu, C.-W.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension. SIAM J. Numer. Anal. 47(6), 4044–4072 (2010)
Guo, W., Zhong, X., Qiu, J.-M.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: Eigen-structure analysis based on Fourier approach. J. Comput. Phys. 235, 458–485 (2013)
Hu, F.Q., Atkins, H.L.: Eigensolution analysis of the discontinuous Galerkin method with nonuniform grids. J. Comput. Phys. 182, 516–545 (2002)
Krivodonova, L., Qin, R.: An analysis of the spectrum of the discontinuous Galerkin method. Appl. Numer. Math. 64, 1–18 (2013)
Krivodonova, L., Qin, R.: An analysis of the spectrum of the discontinuous Galerkin method II: Nonuniform grids. Appl. Numer. Math. 71, 41–62 (2013)
Le Saint, P., Raviart, P.: On a finite element method for solving the neutron transport equation. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–145. Academic Press, New York (1974)
Yang, Y., Shu, C.-W.: Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 50(6), 3110–3133 (2012)
Zhong, X., Shu, C.-W.: Numerical resolution of discontinuous Galerkin methods for time dependent wave equations. Comput. Methods Appl. Mech. Eng. 200(41), 2814–2827 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chalmers, N., Krivodonova, L. Spatial and Modal Superconvergence of the Discontinuous Galerkin Method for Linear Equations. J Sci Comput 72, 128–146 (2017). https://doi.org/10.1007/s10915-016-0349-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-016-0349-8