Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The discontinuous Galerkin method with diffusion

Author: Gerard R. Richter
Journal: Math. Comp. 58 (1992), 631-643
MSC: Primary 65M60; Secondary 65M15, 65N30, 76M25, 76Rxx
MathSciNet review: 1122076
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We propose a way of extending the discontinuous Galerkin method from pure hyperbolic equations to convection-dominated equations with an $ O(h)$ diffusion term. The resulting method is explicit and can be applied with polynomials of degree $ n \geq 1$. The extended method satisfies the same $ O({h^{n + 1/2}})$ error estimate previously established for the discontinuous Galerkin method as applied to hyperbolic problems. Numerical results are provided.

References [Enhancements On Off] (What's this?)

  • [1] R. S. Falk and G. R. Richter, Analysis of a continuous finite element method for hyperbolic equations, SIAM J. Numer. Anal. 24 (1987), 257-278. MR 881364 (88d:65133)
  • [2] T. J. R. Hughes and A. Brooks, A multidimensional upwind scheme with no crosswind diffusion, Finite Element Methods for Convection Dominated Flows (T. J. R. Hughes, ed.), AMD (ASME) 34 (1979). MR 571681 (81f:76040)
  • [3] C. Johnson, U. Nävert, and J. Pitkäranta, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 45 (1984), 285-312. MR 759811 (86a:65103)
  • [4] C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986), 1-26. MR 815828 (88b:65109)
  • [5] P. Lesaint and P. A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical Aspects of Finite Elements in Partial Differential Equations, (C. deBoor, ed.), Academic Press, 1974, pp. 89-123. MR 0658142 (58:31918)
  • [6] T. E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, preprint. MR 1083327 (91m:65250)
  • [7] W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory Report LA-UR-73-479, 1973.
  • [8] G. R. Richter, An optimal-order error estimate for the discontinuous Galerkin method, Math. Comp. 50 (1988), 75-88. MR 917819 (88j:65197)
  • [9] -, A finite element method for time-dependent convection-diffusion equations, Math. Comp. 54 (1990), 81-106. MR 993932 (90g:65152)
  • [10] -, An explicit finite element method for convection-dominated steady state convection-diffusion equations, SIAM J. Numer. Anal. 28 (1991), 744-759. MR 1098416 (92e:65155)
  • [11] M. I. Vishik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with a small parameter, Uspekhi Mat. Nauk 12 (1957), 3-122; English transl, in Amer. Math. Soc. Transl. (2) 20 (1962), 239-364. MR 0136861 (25:322)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M60, 65M15, 65N30, 76M25, 76Rxx

Retrieve articles in all journals with MSC: 65M60, 65M15, 65N30, 76M25, 76Rxx

Additional Information

Keywords: Finite element, hyperbolic
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society