Smoothed aggregation multigrid solvers for high-order discontinuous Galerkin methods for elliptic problems

Journal of Computational Physics

Volumn 230, Issue 18, 2011, Pages 6959-6976

  Olson, Luke N ^a   Schroder, Jacob B ^b  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b UNIVERSITY OF COLORADO   (United States)

Author keywords

Algebraic multigrid; AMG; Discontinuous Galerkin; High order; Smoothed aggregation

Indexed keywords

DEGREES OF FREEDOM (MECHANICS); GALERKIN METHODS; JACOBIAN MATRICES; LINEAR SYSTEMS; NUMBER THEORY; NUMERICAL METHODS;

ALGEBRAIC MULTIGRIDS; AMG; DISCONTINUOUS GALERKIN; DOMAIN BOUNDARY; GALERKIN DISCRETIZATION; HIGH-ORDER; HIGHER-ORDER; MATRIX; MULTIGRID SOLVER; SMOOTHED AGGREGATION;

BOUNDARY CONDITIONS;

EID: 79959799151     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2011.05.009     Document Type: Article

References (32)

1. Lottes J.W., Fischer P.F. Hybrid multigrid/Schwarz algorithms for the spectral element method. J. Sci. Comput. 2005, 24:45-78.
2. Olson L. Algebraic multigrid preconditioning of high-order spectral elements for elliptic problems on a simplicial mesh. SIAM J. Sci. Comput. 2007, 29(5):2189-2209.
3. Heys J.J., Manteuffel T.A., McCormick S.F., Olson L.N. Algebraic multigrid for higher-order finite elements. J. Comput. Phys. 2005, 204(2):520-532.
4. Castillo P. Performance of discontinuous Galerkin methods for elliptic PDEs. SIAM J. Sci. Comput. 2002, 24(2):524-547.
5. Brezzi F., Manzini G., Marini D., Pietra P., Russo A. Discontinuous Galerkin approximations for elliptic problems. Numer. Meth. Partial Diff. Eq. 2000, 16:365-378.
6. Arnold D.N., Brezzi F., Cockurn B., Marini L.D. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 2002, 39:1749-1779.
7. Cockburn B., Gopalakrishnan J., Lazarov R. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 2009, 47(2):1319-1365.
8. Nguyen N.C., Peraire J., Cockburn B. A hybridizable discontinuous Galerkin method for stokes flow. Comput. Methods Appl. Mech. Eng. 2010, 199(9-12):582-597.
9. Jeon Y., Park E.. A hybrid discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 2010, 48(5):1968-1983.
10. Kanschat G. Preconditioning methods for local discontinuous Galerkin discretizations. SIAM J. Sci. Comput. 2003, 25(3):815-831.
11. Kanschat G. Robust smoothers for high-order discontinuous Galerkin discretizations of advection-diffusion problems. J. Comput. Appl. Math. 2008, 218(1):53-60. (special Issue: Finite Element Methods in Engineering and Science (FEMTEC 2006)).
12. Atkins H., Helenbrook B. Application of p-multigrid to discontinuous Galerkin formulations of the Poisson operator. AIAA J. 2005, 44(3):566-575.
13. Fidkowski K.J., Oliver T.A., Lu J., Darmofal D.L. p-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys. 2005, 207(1):92-113.
14. Dobrev V.A., Lazarov R.D., Vassilevski P.S., Zikatanov L.T. Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations. Numer. Linear Algebra Appl. 2006, 13(9):753-770.
15. Gopalakrishnan J., Kanschat G. A multilevel discontinuous Galerkin method. Numer. Math. 2003, 95(3):527-550.
16. Hemker P.W., Hoffmann W., Raalte M.H.v. Two-level Fourier analysis of a multigrid approach for discontinuous Galerkin discretizations. SIAM J. Sci. Comput. 2003, 25(3):1018-1041.
17. Prill F., Lukáčová-Medviďová M., Hartmann R. Smoothed aggregation multigrid for the discontinuous Galerkin method. SIAM J. Sci. Comput. 2009, 31(5):3503-3528.
18. Cockburn B., Shu C.-W. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 1998, 35(6):2440-2463.
19. Douglas J.J., Dupont T. Interior penalty procedures for elliptic and parabolic Galerkin methods. Lecture Notes in Physics 1976, vol. 58:207-216. Springer, Berlin.
20. F. Brezzi, G. Manzini, D. Marini, P. Pietra, A. Russo, Discontinuous finite elements for diffusion problems, in: in Atti Convegno in onore di F. Brioschi (Milan, 1997), Instituto Lombardo, Accademia di Scienze e Lettere, Italy, 1999, pp. 197-217.
21. Castillo P., Cockurn B., Perugia I., Schötzau D. An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 2000, 38:1676-1706.
22. Cockurn B., Kanschat G., Perugia I., Schötzau D. Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 2001, 39:264-285.
23. Hesthaven J., Warburton T. Nodal discontinuous Galerkin methods: algorithms, analysis, and applications. Texts in applied mathematics 2008, vol. 54. Springer, Berlin.
24. McCormick S.F., Ruge J.W. Multigrid methods for variational problems. SIAM J. Numer. Anal. 1982, 19(5):924-929.
25. Brezina M., Falgout R., MacLachlan S., Manteuffel T., McCormick S., Ruge J. Adaptive smoothed aggregation (αSA) multigrid. SIAM Rev. 2005, 47(2):317-346.
26. M. Brezina, T. Manteuffel, S. Mccormick, J. Ruge, G. Sanders, Towards adaptive smoothed aggregation (αSA) for nonsymmetric problems, SIAM J. Sci. Comput.
27. Vaněk P., Mandel J., Brezina M. Algebraic multigrid based on smoothed aggregation for second and fourth order problems. Computing 1996, 56:179-196.
28. Brannick J., Brezina M., MacLachlan S., Manteuffel T., McCormick S., Ruge J. An energy-based AMG coarsening strategy. Numer. Linear Algebra Appl. 2006, 13(2-3):133-148.
29. Olson L.N., Schroder J.B., Tuminaro R.S. A new perspective on strength measures in algebraic multigrid. Numer. Linear Algebra Appl. 2010, 17(4):713-733.
30. L.N. Olson, J.B. Schroder, R.S. Tuminaro, A general interpolation strategy for algebraic multigrid using energy-minimization, SIAM J. Sci. Comput. Submitted.
31. Warburton T., Hesthaven J., Wilcox L. Sledge++ users' guide 2006, .
32. Vaněk P., Brezina M., Mandel J. Convergence of algebraic multigrid based on smoothed aggregation. Numer. Math. 2001, 88(3):559-579.