Abstract
In this paper we present numerical investigations of four different formulations of the discontinuous Galerkin method for diffusion problems. Our focus is to determine, through numerical experimentation, practical guidelines as to which numerical flux choice should be used when applying discontinuous Galerkin methods to such problems. We examine first an inconsistent and weakly unstable scheme analyzed in Zhang and Shu, Math. Models Meth. Appl. Sci. (M 3 AS) 13, 395–413 (2003), and then proceed to examine three consistent and stable schemes: the Bassi–Rebay scheme (J. Comput. Phys. 131, 267 (1997)), the local discontinuous Galerkin scheme (SIAM J. Numer. Anal. 35, 2440–2463 (1998)) and the Baumann–Oden scheme (Comput. Math. Appl. Mech. Eng. 175, 311–341 (1999)). For an one-dimensional model problem, we examine the stencil width, h-convergence properties, p-convergence properties, eigenspectra and system conditioning when different flux choices are applied. We also examine the ramifications of adding stabilization to these schemes. We conclude by providing the pros and cons of the different flux choices based upon our numerical experiments.
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References
D. N. Arnold F. Brezzi B. Cockburn D. Marini (2000) Discontinuous Galerkin Methods for Elliptic Problems. B. Cockburn G.E. Karniadakis C.-W. Shu (Eds) Discontinuous Galerkin Methods: Theory Computation and Applications Springer Berlin
D. N. Arnold F. Brezzi B. Cockburn L. D. Marini (2002) ArticleTitleUnified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 1749 Occurrence Handle10.1137/S0036142901384162
Atkins, H., and Shu, C.-W. (1999). Analysis of the discontinuous Galerkin method applied to the diffusion operator. In 14th AIAA Computational Fluid Dynamics Conference AIAA, pp. 99–3306.
F. Bassi S. Rebay (1997) ArticleTitleA high-order accurate discontinuous finite element method forthe numerical solution of the compressible Navier–Stokes equations. J. Comp. Phys. 131 267 Occurrence Handle10.1006/jcph.1996.5572
C. E. Baumann J. T. Oden (1999) ArticleTitleA discontinuous hp finite element method for convection-diffusion problems Comp. Meth. Appl. Mech. Eng. 175 311–341 Occurrence Handle10.1016/S0045-7825(98)00359-4
Brezzi, F., Manzini, G., Marini, D., Pietra, P., and Russo, A. (1999). Discontinuous finite elements for diffusion problems. Atti Convegno in onore di F. Brioschi (Milano 1997) Istituto Lombardo Accademia di Scienze e Lettere, pp. 197–217.
C. Canuto M. Y. Hussaini A. Quarteroni T. A. Zang (1987) Spectral Methods in Fluid Mechanics Springer-Verlag New York
P. Castillo (2002) ArticleTitlePerformance of discontinuous Galerkin Methods for elliptic problems SIAM J. Numer. Anal. 24 IssueID2 524–547
B. Cockburn C.-W. Shu (1998) ArticleTitleThe local discontinuous Galerkin for convection-diffusion systems SIAM J. Numer. Anal. 35 2440–2463 Occurrence Handle10.1137/S0036142997316712
B. Cockburn G. E. Karniadakis C.-W. Shu (2000) The development of discontinuous Galerkin methods B. Cockburn G. E. Karniadakis C.-W. Shu (Eds) Discontinuous Galerkin Methods: Theory Computation and Applications Springer Berlin
Helenbrook, B., Mavriplis, D., and Atkins, H. (2003). Analysis of p-multigrid for continuous and discontinuous finite element discretizations. In 16th AIAA Computational Fluid Dynamics Conference AIAA, pp. 99–3989.
J. S. Hesthaven D. Gottlieb (1996) ArticleTitleA stable penalty method for the compressible Navier–Stokes Equations. I. Open boundary conditions. SIAM J. Sci. Comp. 17 IssueID3 579–612 Occurrence Handle10.1137/S1064827594268488
B. Rivière M. F. Wheeler V. Girault (2001) ArticleTitleA priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems SIAM J. Numer. Anal. 39 IssueID3 902–931 Occurrence Handle10.1137/S003614290037174X
A. Romkes S. Prudhomme J. Tinsley Oden (2003) ArticleTitleA posteriori error estimation for a new stabilized discontinuous Galerkin method Appl. Math. Lett. 16 IssueID4 447–452 Occurrence Handle10.1016/S0893-9659(03)00018-1 Occurrence HandleMR1983711
C. -W. Shu (2001) Different formulations of the discontinuous Galerkin method for the viscous terms Z.-C. Shi M. Mu W. Xue J. Zou (Eds) Advances in Scientific Computing Science Press mascou 144–155
M. Zhang C.-W. Shu (2003) ArticleTitleAn analysis of three different formulations of the discontinuous Galerkin method for diffusion equations Math. Models Meth. Appl. Sci. 13 395–413 Occurrence Handle10.1142/S0218202503002568
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Kirby, R.M., Karniadakis, G.E. Selecting the Numerical Flux in Discontinuous Galerkin Methods for Diffusion Problems. J Sci Comput 22, 385–411 (2005). https://doi.org/10.1007/s10915-004-4145-5
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DOI: https://doi.org/10.1007/s10915-004-4145-5