Unified analysis of discontinuous Galerkin methods for elliptic problems

SIAM Journal on Numerical Analysis

Volumn 39, Issue 5, 2001, Pages 1749-1779

  Arnold, Douglas N ^a   Brezzi, Franco ^b   Cockburn, Bernardo ^c   Donatella Marini, L ^b  

^a UNIVERSITY OF MINNESOTA   (United States)

^b CNR   (Italy)

^c School of Mathematics University of Minnesota ^*  (United States)

Author keywords

Discontinuous Galerkin; Elliptic problems; Interior penalty

Indexed keywords

EID: 0036410391     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142901384162     Document Type: Article

References (70)

1. P. ALOTTO, A. BERTONI, I. PERUGIA, AND D. SCHÖTZAU, Discontinuous finite element methods for the simulation of rotating electrical machines, COMPEL, 20 (2001), pp. 448-462.
2. D. ARNOLD, F. BREZZI, B. COCKBURN, AND D. MARINI, Discontinuous Galerkin methods for elliptic problems, in Discontinuous Galerkin Methods. Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes in Comput. Sci. Engrg. 11, Springer-Verlag, New York, 2000, pp. 89-101.
3. D. N. ARNOLD, An Interior Penalty Finite Element Method with Discontinuous Elements, Ph.D. thesis, The University of Chicago, Chicago, IL, 1979.
4. D. N. ARNOLD, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), pp. 742-760.
5. J. P. AUBIN, Approximation des problèmes aux limites non homogènes pour des opérateurs non linéaires, J. Math. Anal. Appl., 30 (1970), pp. 510-521.
6. I. BABUŠKA, The finite element method with penalty, Math. Comp., 27 (1973), pp. 221-228.
7. I. BABUŠKA AND M. ZLÁMAL, Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal., 10 (1973), pp. 863-875.
8. G. A. BAKER, Finite element methods for elliptic equations using nonconforming elements, Math. Comp., 31 (1977), pp. 45-59.
9. G. A. BAKER, W. N. JUREIDINI, AND O. A. KARAKASHIAN, Piecewise solenoidal vector fields and the Stokes problem, SIAM J. Numer. Anal., 27 (1990), pp. 1466-1485.
10. F. BASSI AND S. REBAY, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), pp. 267-279.
11. F. BASSI AND S. REBAY, GMRES for discontinuous Galerkin solution of the compressible Navier-Stokes equations, in Discontinuous Galerkin Methods. Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes in Comput. Sci. Engrg. 11, Springer-Verlag, New York, 2000, pp. 197-208.
12. F. BASSI AND S. REBAY, Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids, to appear.
13. F. BASSI, S. REBAY, G. MARIOTTI, S. PEDINOTTI, AND M. SAVINI, A high-order accurate discontinuous finite, element method for inviscid and viscous turbomachinery flows, in Proceedings of 2nd European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics, R. Decuypere and G. Dibelius, eds., Technologisch Instituut, Antwerpen, Belgium, 1997, pp. 99-108.
14. C. E. BAUMANN AND J. T. ODEN, A Discontinuous hp finite element method for the Euler and Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 31 (1999), pp. 79-95.
15. C. E. BAUMANN AND J. T. ODEN, A discontinuous hp finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 311-341.
16. R. BECKER AND P. HANSBO, Discontinuous Galerkin methods for convection-diffusion problems with arbitrary Péclet number, in Numerical Mathematics and Advanced Applications, Proceedings of ENUMATH 99, P. Neittaanmäki, T. Tiihonen, and P. Tarvainen, eds., World Scientific, River Edge, NJ, 2000, pp. 100-109.
17. R. BECKER AND P. HANSBO, A Finite Element Method for Domain Decomposition with Non-Matching Grids, Tech. Report 3613, INRIA, Le Chesnay, France, 1999.
18. C. BERNARDI, Y. MADAY, AND A. T. PATERA, Domain decomposition by the mortar element method, in Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, H. G. Kaper and M. Garbey, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993, pp. 269-286.
19. C. BERNARDI, Y. MADAY, AND A. T. PATERA, A new nonconforming approach to domain decomposition: The mortar element method, in Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. XI, H. Brézis and J. L. Lions, eds., Pitman Res. Notes Math. Ser., 299, Longman Sci. Tech., Harlow, 1994, pp. 13-51.
20. C. BERNARDI, N. DEBIT, AND Y. MADAY, Coupling finite element and spectral methods: First results, Math. Comp., 54 (1990), pp. 21-39.
21. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
22. F. BREZZI, G. MANZINI, D. MARINI, P. PIETRA, AND A. RUSSO, Discontinuous finite elements for diffusion problems, in Atti Convegno in onore di F. Brioschi (Milan, 1997), Istituto Lombardo, Accademia di Scienze e Lettere, Milan, Italy, 1999, pp. 197-217.
23. F. BREZZI, G. MANZINI, D. MARINI, P. PIETRA, AND A. RUSSO, Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differential Equations, 16 (2000), pp. 365-378.
24. P. CASTILLO, An optimal error estimate for the local discontinuous Galerkin method, in Discontinuous Galerkin Methods. Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes in Comput. Sci. Engrg., 11, Springer-Verlag, New York, 2000, pp. 285-290.
25. P. CASTILLO, Performance of Discontinuous Galerkin Methods for Elliptic PDE's, IMA Preprint 1764, Minneapolis, MN, April 2001.
26. P. CASTILLO, B. COCKBURN, I. PERUGIA, AND D. SCHÖTZAU, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal., 38 (2000), pp. 1676-1706.
27. P. CASTILLO, B. COCKBURN, D. SCHÖTZAU, AND C. SCHWAB, Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems, Math. Comp., to appear.
28. Z. CHEN, B. COCKBURN, C. GARDNER, AND J. JEROME, Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode, J. Comput. Phys., 117 (1995), pp. 274-280.
29. Z. CHEN, B. COCKBURN, J. JEROME, AND C.-W. SHU, Mixed-RKDG finite element methods for the 2-D hydrodynamic model for semiconductor device simulation, VLSI Design, 3 (1995), pp. 145-158.
30. P. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
31. B. COCKBURN, Discontinuous Galerkin methods for convection-dominated problems, in High-Order Methods for Computational Physics, T. Barth and H. Deconink, eds., Lecture Notes in Comput. Sci. Engrg. 9, Springer-Verlag, New York, 1999, pp. 69-224.
32. B. COCKBURN, Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws, J. Comput. Appl. Math., 128 (2001), pp. 187-204.
33. B. COCKBURN AND C. DAWSON, Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions, in Proceedings of the Conference on the Mathematics of Finite Elements and Applications: MAFELAP X, J. R. Whiteman, ed., Elsevier, New York, 2000, pp. 225-238.
34. B. COCKBURN, S. HOU, AND C.-W. SHU, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Math. Comp., 54 (1990), pp. 545-581.
35. B. COCKBURN, J. JEROME, AND C.-W. SHU, The utility of modeling and simulation in determining performance and symmetry properties of semiconductors, in Discontinuous Galerkin Methods. Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes in Comput. Sci. Engrg. 11, Springer-Verlag, New York, 2000, pp. 147-156.
36. B. COCKBURN, G. KANSCHAT, I. PERUGIA, AND D. SCHÖTZAU, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39 (2001), pp. 264-285.
37. B. COCKBURN, G. E. KARNIADAKIS, AND C.-W. SHU, The development of discontinuous Galerkin methods, in Discontinuous Galerkin Methods. Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes in Comput. Sci. Engrg. 11, Springer-Verlag, New York, 2000, pp. 3-50.
38. B. COCKBURN, S. Y. LIN, AND C.-W. SHU, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems, J. Comput. Phys., 84 (1989), pp. 90-113.
39. B. COCKBURN AND C.-W. SHU, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework, Math. Comp., 52 (1989), pp. 411-435.
40. 1-discontinuous Galerkin method for scalar conservation laws, RAIRO Modél. Math. Anal. Numér., 25 (1991), pp. 337-361.
41. B. COCKBURN AND C.-W. SHU, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), pp. 2440-2463.
42. B. COCKBURN AND C.-W. SHU, The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199-224.
43. B. COCKBURN AND C.-W. SHU, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., to appear.
44. C. N. DAWSON, Godunov-mixed methods for advection flow problems equations in one space dimension, SIAM J. Numer. Anal., 28 (1991), pp. 1282-1309.
45. C. DAWSON, Godunov-mixed methods for advection-diffusion equations in multidimensions, SIAM J. Numer. Anal., 30 (1993), pp. 1315-1332.
46. C. DAWSON, Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations, SIAM J. Numer. Anal., 35 (1998), pp. 1709-1724.
47. C. N. DAWSON AND V. AIZINGER, Upwind-mixed methods for transport equations, Comp. Geosci., 3 (1999), pp. 93-110.
48. E. G. DUTRA DO CARMO AND A. V. C. DUARTE, A discontinuous finite element-based domain decomposition method, Comput. Methods Appl. Mech. Engrg., 190 (2000), pp. 825-843.
49. J. DOUGLAS, JR., B. L. DARLOW, R. P. KENDALL, AND M. F. WHEELER, Self-adaptive Galerkin methods for one-dimensional, two-phase immiscible flow, in Proceedings AIME Fifth Symposium on Reservoir Simulation (Denver, Colorado), Society of Petroleum Engineers, Dallas, TX, 1979, pp. 65-72.
50. J. DOUGLAS, JR. AND T. DUPONT, Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods, Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976.
51. X. FENG AND O. A. KARAKASHIAN, Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems, SIAM J. Numer. Anal., 39 (2001), pp. 1343-1365.
52. J. GREENSTADT, An abridged history of cell discretization, in Discontinuous Galerkin Methods. Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes in Comput. Sci. Engrg. 11, Springer-Verlag, New York, 2000, pp. 337-342.
53. J. GOPALAKRISHNAN AND G. KANSHAT, A multilevel discontinuous Galerkin method, Numer. Math., to appear.
54. P. HOUSTON, C. SCHWAB, AND E. SÜLI, Discontinuous hp Finite Element Methods for Advection-Diffusion Problems, Tech. Report NA 00-15, Oxford University Computing Laboratory, Oxford, UK, 2000.
55. C. LASSER AND A. TOSELLI, An Overlapping Domain Decomposition Preconditioner for a Class of Discontinuous Galerkin Approximations of Advection-Diffusion Problems, Tech. Report 2000-12, Seminar für Angewandte Mathematik, ETH Zürich, 2000.
56. J.-L. LIONS, Problèmes aux limites non homogènes à donées irrégulières: Une méthode d'approximation, in Numerical Analysis of Partial Differential Equations (C.I.M.E. 2 Ciclo, Ispra, 1967), Edizioni Cremonese, Rome, 1968, pp. 283-292.
57. I. LOMTEV, C. W. QUILLEN, AND G. E. KARNIADAKIS, Spectral/hp methods for viscous compressible flows on unstructured 2D meshes, J. Comput. Phys., 144 (1998), pp. 325-357.
58. J. A. NITSCHE, Über ein Variationsprinzip zur Lösung Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unteworfen sind, Abh. Math. Sem. Univ. Hamburg, 36 (1971), pp. 9-15.
59. J. T. ODEN, IVO BABUŠKA, AND C. E. BAUMANN, A discontinuous hp finite element method for diffusion problems, J. Comput. Phys., 146 (1998), pp. 491-519.
60. I. PERUGIA AND D. SCHÖTZAU, On the coupling of local discontinuous Galerkin and conforming finite element methods, J. Sci. Comput., to appear.
61. W. H. REED AND T. R. HILL, Triangular Mesh Methods for the Neutron Transport Equation, Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973.
62. G. R. RICHTER, The discontinuous Galerkin method with diffusion, Math. Comp., 58 (1992), pp. 631-643.
63. B. RIVIÈRE AND M. F. WHEELER, A discontinuous Galerkin method applied to nonlinear parabolic equations, in Discontinuous Galerkin Methods. Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes in Comput. Sci. Engrg. 11, Springer-Verlag, New York, 2000, pp. 231-244.
64. B. RIVIÈRE, M. F. WHEELER, AND V. GIRAULT, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems I, Comput. Geosci., 3 (1999), pp. 337-360.
65. T. RUSTEN, P. S. VASSILEVSKI, AND R. WINTHER, Interior penalty preconditioners for mixed finite element approximations of elliptic problems, Math. Comp., 65 (1996), pp. 447-466.
66. C.-W. SHU, Different formulations of the discontinuous Galerkin method for the viscous terms, in Proceedings of Conference in Honor of Professor H.-C. Huang on the Occasion of His Retirement, W. M. Xue, Z. C. Shi, M. Mu, and J. Zou, eds., Science Press, New York, Beijing, 2000, pp. 14-45.
67. E. SÜLI, P. HOUSTON, AND C. SCHWAB, hp-finite element methods for hyperbolic problems, in Proceedings of the Conference on the Mathematics of Finite Elements and Applications: MAFELAP X, J. R. Whiteman, ed., Elsevier, New York, 2000, pp. 143-162.
68. E. SÜLI, C. SCHWAB, AND P. HOUSTON, hp-DGFEM for partial differential equations with non-negative characteristic form, in Discontinuous Galerkin Methods. Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes in Comput. Sci. Engrg. 11, Springer-Verlag, New York, 2000, pp. 221-230.
69. H. SWANN, The cell discretization algorithm: An overview, in Discontinuous Galerkin Methods. Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes in Comput. Sci. Engrg. 11, Springer-Verlag, New York, 2000, pp. 433-438.
70. M. F. WHEELER, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal., 15 (1978), pp. 152-161.