Abstract.
A variable V-cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented. An analysis under a mild regularity assumption shows that the preconditioner is uniform. The interior penalty method is then combined with a discontinuous Galerkin scheme to arrive at a discretization scheme for an advection-diffusion problem, for which an error estimate is proved. A multigrid algorithm for this method is presented, and numerical experiments indicating its robustness with respect to diffusion coefficient are reported.
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D. Arnold, F. Brezzi, B. Cockburn, D. Marini. Discontinuous Galerkin methods for elliptic problems. In: B. Cockburn, G. Karniadakis, C.-W. Shu (eds.) First International Symposium on Discontinuous Galerkin Methods, volume 11 of Lecture Notes in Computational Science and Engineering, pages 89-101. Springer Verlag (2000)
D.N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4): 742-760 (1982)
I. Babuška. The finite element method with penalty. Math. Comp. 27: 221-228 (1973)
W. Bangerth, G. Kanschat. Concepts for object-oriented finite element software - the deal.ii library. Preprint 99-43, SFB 359, Universität Heidelberg, October (1999)
W. Bangerth, G. Kanschat. deal.II Differential Equations Analysis Library, Technical Reference, October (1999) +http://gaia.iwr.uni-heidelberg.de/deal/+
J. H. Bramble. Multigrid Methods. Number 294 in Pitman research notes in mathematics series. Longman Scientific & Technical, Harlow, UK (1993)
J.H. Bramble, R.E. Ewing, J.E. Pasciak, J. Shen. The analysis of multigrid algorithms for cell centered finite difference methods. Adv. Comput. Math. 5(1): 15-29 (1996)
J.H. Bramble, J.E. Pasciak. The analysis of smoothers for multigrid algorithms. Math. Comp. 58(198): 467-488 (1992)
J.H. Bramble, J.E. Pasciak, J. Xu. The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math. Comp. 56(193): 1-34 (1991)
A. Brandt. Multigrid techniques: 1984 guide with applications to fluid dynamics. Gesellschaft für Mathematik und Datenverarbeitung m.b.H., Bonn, St. Augustin (1984)
P. Castillo, B. Cockburn, I. Perugia, D. Schötzau. An a priori error analysis of the LDG method for elliptic problems. SIAM J. Numer. Anal. (2000) To appear
B. Cockburn. Discontinuous Galerkin methods for convection-dominated problems. In: High-order methods for computational physics, pp. 69-224. Springer, Berlin (1999)
J. Douglas, Jr., T. Dupont. Interior penalty procedures for elliptic and parabolic Galerkin methods. In: R. Glowinski, J. L. Lions (eds.) Computing methods in applied sciences, pp. 207-216. (Lecture Notes in Phys) Vol. 58, Springer, Berlin (1976) Second International Symposium on Computing Methods in Applied Sciences and Engineering, held at Versailles (Dec. 1975)
S.C. Eisenstat, H.C. Elman, M.H. Schultz. Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20(2): 345-357 (1983)
X. Feng, O. Karakashian. Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. Preprint (2001)
P. Grisvard. Elliptic Problems in Nonsmooth Domains. Number 24 in Monographs and Studies in Mathematics. Pitman Advanced Publishing Program, Marshfield, Massachusetts (1985)
W. Hackbusch. Multi-Grid Methods and Applications. Number 4 in Springer series in Computational Mathematics. Berlin, Springer (1985)
W. Hackbusch, T. Probst. Downwind Gauss-Seidel smoothing for convection dominated problems. Numer. Linear Algebra Appl. 4(2): 85-102 (1997)
P. Houston, C. Schwab, E. Süli. Discontinuous hp-finite element methods for advection-diffusion problems. Technical Report 00/15, Oxford University Computing Lab (2000)
P. Houston, C. Schwab, E. Süli. Stabilized hp-finite element methods for first-order hyperbolic problems. SIAM J. Numer. Anal. 37(5): 1618-1643 ((electronic) 2000)
C. Johnson. Numerical solution of partial differential equations by the finite element method. Cambridge, Cambridge University Press (1995)
C. Johnson, J. Pitkäranta. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46(173): 1-26 (1986)
R.B. Kellog. Interpolation between subspaces of a Hilbert space. Technical Note BN-719, University of Maryland, College Park, Maryland, November (1971)
P. Lasaint, P.-A. Raviart. On a finite element method for solving the neutron transport equation. In: C. de Boor (ed.) Mathematical aspects of finite elements in partial differential equations, pp. 89-123. Academic Press New York (1974). Proceedings of a symposium conducted by Math. Res. Center, Univ. of Wisconsin-Madison (in April, 1974)
C. Lasser, A. Toselli. An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems. Technical Report 2000-12 Eidgenössische Technische Hochschule, CH-8092 Zürich, November (2000)
I. Persson, K. Samuelsson, A. Szepessy. On the convergence of multigrid methods for flow problems. Electron. Trans. Numer. Anal. 8: 46-87 ((electronic) 1999)
C. Pflaum. Robust convergence of multilevel algorithms for convection-diffusion equations. SIAM J. Numer. Anal. 37(2): 443-469 ((electronic) 2000)
B. Riviére, M.F. Wheeler, V. Girault. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I. Comput. Geosci. 3(3-4): 337-360 ((2000) 1999)
T. Rusten, P. S. Vassilevski, R. Winther. Interior penalty preconditioners for mixed finite element approximations of elliptic problems. Math. Comp. 65(214): 447-466 (1996)
L.R. Scott, S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190): 483-493 (1990)
M.F. Wheeler. An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15(1): 152-161 (1978)
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Received: 5 June 2001, Revised: 12 December 2001, Published online: 4 April 2003
Mathematics Subject Classification (1991):
65F10, 65N55, 65N30
This research was supported in part by Institute for Mathematics and its Applications, Supercomputing Institute of University of Minnesota, and Deutsche Forschungsgemeinschaft.
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Gopalakrishnan, J., Kanschat, G. A multilevel discontinuous Galerkin method. Num. Math. 95, 527–550 (2003). https://doi.org/10.1007/s002110200392
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DOI: https://doi.org/10.1007/s002110200392