An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations

Journal of Computational Physics

Volumn 227, Issue 22, 2008, Pages 9670-9685

  Hartmann, Ralf ^a   Houston, Paul ^b  

^a GERMAN AEROSPACE CENTER DLR   (Germany)

^b UNIVERSITY OF NOTTINGHAM   (United Kingdom)

Author keywords

Adjoint consistency; Compressible Navier Stokes equations; Discontinuous Galerkin methods; Finite element methods

Indexed keywords

DRAG; GALERKIN METHODS; NAVIER STOKES EQUATIONS; NUMERICAL METHODS; VISCOUS FLOW;

ADJOINT CONSISTENCY; ADJOINTS; COMPRESSIBLE NAVIER-STOKES EQUATIONS; DISCONTINOUS GALERKIN METHODS; DISCONTINUOUS GALERKIN; GALERKIN DISCRETIZATION; INTERIOR PENALTIES; OPTIMAL ORDERING; SYMMETRICS; TARGET FUNCTIONALS;

FINITE ELEMENT METHOD;

EID: 52749088289     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2008.07.015     Document Type: Article

References (38)

1. Arnold D., Brezzi F., Cockburn B., and Marini D. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 5 (2002) 1749-1779
2. Babuška I., and Miller A. The post processing approach in the finite element method. Part 1: Calculation of displacements, stresses and other higher derivatives of the displacements. Int. J. Numer. Methods Eng. 34 (1984) 1085-1109
3. Babuška I., and Miller A. The post processing approach in the finite element method. Part 2: The calculation of stress intensity factors. Int. J. Numer. Methods Eng. 34 (1984) 1111-1129
4. Babuška I., and Miller A. The post processing approach in the finite element method. Part 3: A posteriori estimates and adaptive mesh selection. Int. J. Numer. Methods Eng. 34 (1984) 1131-1151
5. W. Bangerth, R. Hartmann, G. Kanschat, deal.II Differential Equations Analysis Library, Technical Reference. http://www.dealii.org/, 6.0 edition, September 2007. First edition 1999.
6. Bangerth W., Hartmann R., and Kanschat G. deal.II - A general purpose object oriented finite element library. ACM Trans. Math. Softw. 33 4 (2007)
7. T.J. Barth, An introduction to upwind finite volume and finite element methods: some unifying and contrasting themes, VKI Lecture Series on High Order Discretization Methods in CFD, von Karman Institute for Fluid Dynamics, 2006, pp. 1-64.
8. Baumann C.E., and Oden J.T. A discontinuous hp finite element method for the solution of the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 31 (1999) 79-95
9. Bassi F., Crivellini A., Rebay S., and Savini M. Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k-ω turbulence model equations. Comput. Fluids 34 (2005) 507-540
10. Bassi F., and Rebay S. High-order accurate discontinuous finite element solution of the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267-279
11. Bassi F., and Rebay S. High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138 (1997) 251-285
12. Bassi F., and Rebay S. GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations. In: Cockburn B., Karniadakis G., and Shu C.-W. (Eds). Discontinuous Galerkin Methods. Lecture Notes in Comput. Sci. Engrg. vol. 11 (2000), Springer, Berlin 197-208
13. Bassi F., and Rebay S. Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 40 (2002) 197-207
14. F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, M. Savini, A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, in: R. Decuypere, G. Dibelius (Eds.), Second European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Antwerpen, Belgium, March 5-7, 1997, Technologisch Instituut, 1997, pp. 99-108.
15. P.J. Capon, Adaptive Stable Finite Element Methods for the Compressible Navier-Stokes Equations, Ph.D. Thesis, University of Leeds, 1995.
16. Cockburn B., and Shu C.-W. The Runge-Kutta discontinuous Galerkin finite element method for conservation laws. V: Multidimensional systems. J. Comput. Phys. 141 (1998) 199-224
17. Dolejší V. On the discontinuous Galerkin method for the numerical solution of the Navier-Stokes equations. Int. J. Numer. Methods Fluids 45 (2004) 1083-1106
18. Dolejší V., Feistauer M., and Schwab Ch. On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow. Math. Bohem. 127 2 (2002) 163-179
19. Feistauer M., Felcman J., and Straškraba I. Mathematical and Computational Methods for Compressible Flow (2003), Clarendon Press, Oxford
20. Georgoulis E.H., Hall E., and Houston P. Discontinuous Galerkin methods for advection-diffusion-reaction problems on anisotropically refined meshes. SIAM J. Sci. Comput. 30 1 (2007) 246-271
21. S. Giani, E. Hall, P. Houston, AptoFEM Users Manual Version 1.0. Technical Report, University of Nottingham, in preparation.
22. K. Harriman, D.J. Gavaghan, E. Süli, The importance of adjoint consistency in the approximation of linear functionals using the discontinuous Galerkin finite element method. Technical Report NA04/18, Oxford University Computing Laboratory, 2004.
23. K. Harriman, P. Houston, B. Senior, E. Süli, hp-Version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form, in: C.-W. Shu, T. Tang, S.-Y. Cheng (Eds.), Proceedings of the International Conference on Scientific Computing and Partial Differential Equations. AMS Contemporary Mathematics 330 (2003) 89-119.
24. R. Hartmann, Adaptive Finite Element Methods for the Compressible Euler Equations, PhD Thesis, University of Heidelberg, 2002.
25. Hartmann R. Adjoint consistency analysis of discontinuous Galerkin discretizations. SIAM J. Numer. Anal. 45 6 (2007) 2671-2696
26. R. Hartmann, J. Held, T. Leicht, F. Prill. PADGE Parallel Adaptive Discontinuous Galerkin Environment. Technical Reference. DLR Braunschweig, 2008, in preparation.
27. Hartmann R., and Houston P. Symmetric interior penalty DG methods for the compressible Navier-Stokes equations. I: Method formulation. Int. J. Numer. Anal. Model. 3 1 (2006) 1-20
28. Hartmann R., and Houston P. Symmetric interior penalty DG methods for the compressible Navier-Stokes equations. II: Goal-oriented a posteriori error estimation. Int. J. Numer. Anal. Model. 3 2 (2006) 141-162
29. Houston P., Rannacher R., and Süli E. A posteriori error analysis for stabilised finite element approximations of transport problems. Comput. Methods Appl. Mech. Eng. 190 (2000) 1483-1508
30. Houston P., Robson J., and Süli E. Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems. I: The scalar case. IMA J. Numer. Anal. 25 (2005) 726-749
31. Karniadakis G.E., and Sherwin S. Spectral/hp Finite Element Methods in CFD (1999), Oxford University Press
32. Kröner D. Numerical Schemes for Conservation Laws (1997), Wiley-Teubner, Stüttgart
33. Larson M.G., and Barth T.J. A posteriori error estimation for discontinuous Galerkin approximations of hyperbolic systems. In: Cockburn B., Karniadakis G., and Shu C.-W. (Eds). Discontinuous Galerkin Methods. Lecture Notes in Comput. Sci. Engrg. vol. 11 (2000), Springer, Berlin 363-368
34. J. Lu, D.L. Darmofal, Dual-consistency analysis and error estimation for discontinuous Galerkin discretization: application to first-order conservation laws, IMA J. Numer. Anal., submitted for publication.
35. Gassner G., Lörcher F., and Munz C.-D. A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes. J. Comput. Phys. 224 2 (2007) 1049-1063
36. Gassner G., Lörcher F., and Munz C.-D. A discontinuous Galerkin scheme based on a space-time expansion. II. Viscous flow equations in multi-dimensions. J. Sci. Comput. 34 3 (2008) 260-286
37. Šolín P., Segeth K., and Doležel I. Higher-Order Finite Element Methods (2004), Chapman & Hall/CRC
38. Toro E. Riemann Solvers and Numerical Methods for Fluid Dynamics (1997), Springer, Berlin