A discontinuous hp finite element method for convection-diffusion problems

Computer Methods in Applied Mechanics and Engineering

Volumn 175, Issue 3-4, 1999, Pages 311-341

  Baumann, Carlos Erik ^a,b   Tinsley Oden, J ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b COMCO ^*  (United States)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; HEAT CONVECTION; MATHEMATICAL MODELS; POLYNOMIALS; PROBLEM SOLVING; THERMAL DIFFUSION;

ARTIFICIAL DIFFUSION; DISCONTINUOUS BASIS FUNCTIONS;

FINITE ELEMENT METHOD;

CONVECTION; DIFFUSION; FINITE ELEMENT METHOD;

EID: 0345062335     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7825(98)00359-4     Document Type: Article

References (41)

1. [1] S.R. Allmaras, A coupled Euler/navier-Stokes algorithm for 2-D unsteady transonic shock/boundary-layer interaction, Ph.D. Dissertation, Massachusetts Institute of Technology, Feb. 1989.
2. [2] T. Arbogast and M.F. Wheeler, A characteristic-mixed finite element method for convection-dominated transport problems, SIAM J. Numer. Anal. 32 (1995) 404-424.
3. [3] D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19(4) (1982) 742-760.
4. [4] H.L. Atkins and C.-W. Shu, Quadrature-free implementation of discontinuous Galerkin methods for hyperbolic equations, ICASE Report 96-51, 1996.
5. [5] A.K. Aziz and I. Babuška, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Academic Press, 1972).
6. [6] I. Babuška and M. Suri, The hp-version of the finite element method with quasiuniform meshes, Math. Model. Numer. Anal. 21 (1987) 199-238.
7. [7] I. Babuška, J. Tinsley Oden and C.E. Baumann, A discontinuous hp finite element method for diffusion problems: 1-D Analysis, Comput. Math. Applic., also TICAM Report 97-22, to appear.
8. [8] F. Bassi and R. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., submitted.
9. [9] F. Bassi, R. Rebay, M. Savini and S, Pedinotti, The Discontinuous Galerkin method applied to CFD problems, in: Second European Conf. on Turbomachinery, Fluid Dynamics and Thermodynamics, ASME, 1995.
10. [10] C.E. Baumann, An hp-adaptive discontinuous finite element method for computational fluid dynamics, Ph.D. Dissertation, The University of Texas at Austin, Aug. 1997.
11. [11] C.E. Baumann and J. Tinsley Oden, A discontinuous hp finite element method for the solution of the Euler equations of Gas Dynamics, in: Tenth Int. Conf. on Finite Elements in Fluids, Jan. 5-8, 1998.
12. [12] C.E. Baumann and J. Tinsley Oden, A discontinuous hp finite element method for the solution of the Navier-Stokes equations, in: Tenth Int. Conf. on Finite Elements in Fluids, Jan. 5-8, 1998.
13. [13] K.S. Bey and J.T. Oden, A Runge-Kutta discontinuous finite element method for high speed flows, in: AIAA 10th Computational Fluid Dynamics Conference, June 1991.
14. [14] K.S. Bey, J.T. Oden and A. Patra, hp-Version discontinuous Galerkin methods for hyperbolic conservation laws, Comput. Methods Appl. Mech. Engrg. 133(3-4) (1996) 259-286.
15. [15] K. S. Bey, An hp-adaptive discontinuous Galerkin method for hyperbolic conservation laws, Ph.D. dissertation, The University of Texas at Austin, May 1994.
16. [16] B. Cockburn, An introduction to the discontinuous Galerkin method for convection-dominated problems, School of Mathematics, University of Minnesota, 1997.
17. [17] B. Cockburn, S. Hou and C.W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element for conservation laws IV: The multi-dimensional case, Math. Comput. 54 (1990) 545.
18. [18] B. Cockburn, S. Hou and C.W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, ICASE Report 97-43, 1997.
19. [19] B. Cockburn, S.Y. Lin and C.W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element for conservation laws III: One-dimensional systems, J. Comput. Phys. 84 (1989) 90-113.
20. [20] B. Cockburn and C.W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element for conservation laws II: General framework, Math. Comput. 52 (1989) 411-435.
21. [21] B. Cockburn and C.W. Shu, The local discontinuous Galerkin method for time dependent convection-diffusion systems, SIAM J. Numer. Anal., submitted.
22. [22] C.N. Dawson, Godunov-mixed methods for advection-diffusion equations, SIAM J. Numer. Anal. 30 (1993) 1315-1332.
23. [23] L.M. Delves and C. A. Hall, An implicit matching principle for global element calculations, J. Inst. Math. Appl. 23 (1979) 223-234.
24. [24] U. Ghia, K.N. Ghia and C.T. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982) 387-411.
25. [25] J.A. Hendry and L.M. Delves, The global element method applied to a harmonic mixed boundary value problem, J. Comput. Phys. 33 (1979) 33-44.
26. [26] C. Hu and C.-W. Shu, A discontinuous Galerkin finite element method for the Hamilton-Jacobi equations, Division of Applied Mathematics, Brown University, 1997.
27. [27] C. Johnson and J.Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comput. 46 (1986) 1-26.
28. [28] J. Lang and A. Walter, An adaptive discontinuous finite element method for the transport equation, J. Comput. Phys. 117 (1995) 28-34.
29. [29] P. Lesaint and P.A. Raviart, On a finite element method for solving the neutron transport, in: C. de Boor, ed., Mathematical Aspects of Finite Elements in Partial Differential Equations (Academic Press, 1974) 89-123.
30. [30] P. Lesaint and P.A. Raviart, Finite element collocation methods for first-order systems, Math. Comput. 33(147) (1979) 891-918.
31. [31] I. Lomtev and G.E. Karniadakis, A discontinuous Galerkin method for the Navier-Stokes equations, Int. J. Numer. Methods Fluids, submitted.
32. [32] I. Lomtev and G.E. Karniadakis, Simulations of viscous supersonic flows on unstructured meshes, AIAA-97-0754, 1997.
33. [33] I. Lomtev, C.B. Quillen and G.E. Karniadakis, Spectral/hp methods for viscous compressible flows on unstructured 2d meshes, J. Comput. Phys. (1998) to appear.
34. [34] I. Lomtev, C.W. Quillen and G. Karniadakis, Spectral/hp methods for viscous compressible flows on unstructured 2d meshes, Technical report, Center for Fluid Mechanics Turbulence and Computation, Brown University, Box 1966, Providence RI 02912, Dec. 1996.
35. [35] R.B. Lowrie, Compact higher-order numerical methods for hyperbolic conservation laws, Ph.D. Dissertation, University of Michigan, 1996.
36. [36] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971) 9-15.
37. [37] J.T. Oden and G.F. Carey, Texas Finite Elements Series, Vol. IV - Mathematical Aspects (Prentice-Hall, 1983).
38. [38] J. Tinsley Oden, I. Babuska and C.E. Baumann, A discontinuous hp finite element method for diffusion problems, J. Comput. Phys., also TICAM Report 97-21, to appear.
39. [39] P. Percell and M.F. Wheeler, A local residual finite element procedure for elliptic equations, SIAM J. Numer. Anal. 15(4) (1978) 705-714.
40. [40] G.R. Richter, An optimal-order error estimate for the discontinuous Galerkin method, Math. Comput. 50 (1988) 75-88.
41. [41] T.C. Warburton, I. Lomtev, R.M. Kirby and G.E. Karniadakis, A discontinuous Galerkin method for the Navier-Stokes equations on hybrid grids, Center for Fluid Mechanics 97-14, Division of Applied Mathematics, Brown University, 1997.