Geodesic
On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow
Mathematica Bohemica, Tome 127 (2002) no. 2, pp. 163-179
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The paper is concerned with the discontinuous Galerkin finite element method for the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems with emphasis on applications to the simulation of compressible flows. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin method, which is a generalization of the combined finite volume—finite element method. Its advantage is the use of only one mesh (in contrast to the combined finite volume—finite element schemes). However, it is of the first order only. (b) Pure discontinuous Galerkin finite element method of higher order combined with a technique avoiding spurious oscillations in the vicinity of shock waves.
The paper is concerned with the discontinuous Galerkin finite element method for the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems with emphasis on applications to the simulation of compressible flows. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin method, which is a generalization of the combined finite volume—finite element method. Its advantage is the use of only one mesh (in contrast to the combined finite volume—finite element schemes). However, it is of the first order only. (b) Pure discontinuous Galerkin finite element method of higher order combined with a technique avoiding spurious oscillations in the vicinity of shock waves.
DOI : 10.21136/MB.2002.134171
Classification : 65M15, 65M60, 76M10, 76M12, 76N15
Keywords: discontinuous Galerkin finite element method; numerical flux; conservation laws; convection-diffusion problems; limiting of order of accuracy; numerical solution of compressible Euler equations 
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Dolejší, V.; Feistauer, M.; Schwab, C. On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow. Mathematica Bohemica, Tome 127 (2002) no. 2, pp. 163-179. doi: 10.21136/MB.2002.134171

[1] Ph. Angot, V. Dolejší, M. Feistauer, J. Felcman: Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems. Appl. Math. 43 (1998), 263–310. | DOI | MR

[2] F. Bassi, S. Rebay: A high order discontinuous Galerkin method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997), 267–279. | DOI | MR

[3] B. Cockburn: Discontinuous Galerkin methods for convection dominated problems. High-Order Methods for Computational Physics, T. J. Barth, H. Deconinck (eds.), Lecture Notes in Computational Science and Engineering 9, Springer, Berlin, 1999, pp. 69–224. | MR | Zbl

[4] B. Cockburn, G. E. Karniadakis, C.-W. Shu: Discontinuous Galerkin Methods. Lect. Notes Comput. Sci. Eng. 11., Springer, Berlin, 2000. | MR

[5] V. Dolejší: Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput. Vis. Sci. 1 (1998), 165–178. | DOI

[6] V. Dolejší, M. Feistauer, C. Schwab: A finite volume discontinuous Galerkin scheme for nonlinear convection-diffusion problems. Calcolo. Preprint, Forschungsinstitut für Mathematik ETH Zürich, Januar 2001 (to appear). | MR

[7] V. Dolejší, M. Feistauer, C. Schwab: On some aspects of the discontinuous Galerkin finite element method for conservation laws. Mathematics and Computers in Simulation. The Preprint Series of the School of Mathematics, Charles University Prague, No. MATH-KNM-2001/5, 2001 (to appear). | MR

[8] M. Feistauer: Mathematical Methods in Fluid Dynamics. Longman Scientific & Technical, Harlow, 1993. | Zbl

[9] M. Feistauer: Numerical methods for compressible flow. Mathematical Fluid Mechanics. Recent Results and Open Questions, J. Neustupa, P. Penel (eds.), Birkhäuser, Basel, 2001, pp. 105–142. | MR | Zbl

[10] M. Feistauer, J. Felcman: Theory and applications of numerical schemes for nonlinear convection-diffusion problems and compressible Navier-Stokes equations. The Mathematics of Finite Elements and Applications, J. R. Whiteman (ed.), Highlights, 1996, Wiley, Chichester, 1997, pp. 175–194.

[11] M. Feistauer, J. Felcman, V. Dolejší: Numerical solution of compressible flow through cascades of profiles. Z. Angew. Math. Mech. 76 (1996), 297–300.

[12] M. Feistauer, J. Felcman, M. Lukáčová: On the convergence of a combined finite volume—finite element method for nonlinear convection-diffusion problems. Numer. Methods Partial Differ. Equations 13 (1997), 163–190. | DOI | MR

[13] M. Feistauer, J. Felcman, M. Lukáčová, G. Warnecke: Error estimates for a combined finite volume—finite element method for nonlinear convection-diffusion problems. SIAM J. Numer. Anal. 36 (1999), 1528–1548. | DOI | MR

[14] M. Feistauer, J. Slavík, P. Stupka: On the convergence of a combined finite volume—finite element method for nonlinear convection-diffusion problems. Explicit schemes. Numer. Methods Partial Differ. Equations 15 (1999), 215–235. | DOI | MR

[15] J. Felcman: On a 3D adaptation for compressible flow. Proceedings of the Conf. Finite Element Methods for Three-Dimensional Problems, Jyväskylä, June 27–July 1, 2000 (to appear). | Zbl

[16] J. Fořt, J. Halama, A. Jirásek, M. Kladrubský, K. Kozel: Numerical solution of several 2D and 3D internal and external flow problems. Numerical Modelling in Continuum Mechanics, M. Feistauer, R. Rannacher, K. Kozel (eds.), Matfyzpress, Praha, 1997, pp. 283–291.

[17] R. Hartmann, P. Houston: Adaptive discontinuous Galerkin finite element methods for nonlinear conservation laws. Preprint 2001–20, Mai 2001, SFB 359, Universität Heidelberg.

[18] J. T. Oden, I. Babuška, C. E. Baumann: A discontinuous $hp$ finite element method for diffusion problems. J. Comput. Phys. 146 (1998), 491–519. | DOI | MR

[19] S. P. Spekreijse: Multigrid Solution of the Steady Euler Equations. Centrum voor Wiskunde en Informatica, Amsterdam, 1987. | MR

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