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On the order of convergence of the discontinuous Galerkin method for hyperbolic equations

Author: Gerard R. Richter
Journal: Math. Comp. 77 (2008), 1871-1885
MSC (2000): Primary 65M60, 65M15
Published electronically: May 8, 2008
MathSciNet review: 2429867
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Abstract: The basic error estimate for the discontinuous Galerkin method for hyperbolic equations indicates an $ O(h^{n+\frac{1}{2}})$ convergence rate for $ n\textrm{th}$ degree polynomial approximation over a triangular mesh of size $ h$. However, the optimal $ O(h^{n+1})$ rate is frequently seen in practice. Here we extend the class of meshes for which sharpness of the $ O(h^{n+\frac{1}{2}})$ estimate can be demonstrated, using as an example a problem with a ``nonaligned'' mesh in which all triangle sides are bounded away from the characteristic direction. The key to realizing $ h^{n+\frac{1}{2}}$ convergence is a mesh which, to the extent possible, directs the error to lower frequency modes which are approximated, not damped, as $ h\rightarrow 0$.

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  • 1. B. Cockburn, B. Dong, and J. Guzmán, Optimal convergence of the original DG method for the transport-reaction equation on special meshes, Institute for Mathematics and its Applications, University of Minnesota, IMA Preprint Series # 2147, December, 2006.
  • 2. C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comput. 47 (1986), pp. 285-312.
  • 3. P. Lesaint and P. A. Raviart On a finite element for solving the neutron transport equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, ed., Academic Press, New York, 1974, pp. 89-123. MR 0658142 (58:31918)
  • 4. T. E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal. 28 (1991), pp. 133-140. MR 1083327 (91m:65250)
  • 5. W. H. Reed and T. R. Hill Triangular mesh methods for the neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos National Laboratory, Los Alamos NM, 1973.
  • 6. G. R. Richter An optimal-order error estimate for the discontinuous Galerkin method, Math.Comp. 50 (1988), pp. 75-88. MR 917819 (88j:65197)

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Additional Information

Gerard R. Richter
Affiliation: Department of Computer Science, Rutgers University, Busch Campus, Piscataway New Jersey 08854-8019

Keywords: Finite element, hyperbolic
Received by editor(s): February 15, 2007
Received by editor(s) in revised form: November 20, 2007
Published electronically: May 8, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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