Exact de Rham sequences of spaces defined on macro-elements in two and three spatial dimensions

SIAM Journal on Scientific Computing

Volumn 30, Issue 5, 2007, Pages 2427-2446

  Pasciak, Joseph E ^a   Vassilevsk, Panayot S ^b  

^a TEXAS A AND M UNIVERSITY   (United States)

^b LAWRENCE LIVERMORE NATIONAL LABORATORY   (United States)

Author keywords

Agglomeration; Algebraic multigrid; Composite elements; Exact sequences; Mixed finite elements

Indexed keywords

CONJUGATE GRADIENT METHOD; PIECEWISE LINEAR TECHNIQUES;

ALGEBRAIC MULTIGRIDS; COMPOSITE ELEMENT; EXACT SEQUENCE; FINITE ELEMENT SPACE; LOW ORDER; MACRO ELEMENT; MIXED FINITE ELEMENTS; REGULAR STRUCTURE; SPATIAL DIMENSION; STANDARD ELEMENTS;

AGGLOMERATION;

EID: 55349083637     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/070698178     Document Type: Article

References (20)

1. D. N. ARNOLD, R. S. FALK, AND R. WINTHER, Finite element differential forms, Proceedings in Applied Mathematics and Mechanics, 2008, to appear.
2. D. N. ARNOLD, R. S. FALK, AND R. WINTHER, Multigrid in H(div) and H(curl), Numer. Math., 85 (2000), pp. 197-217.
3. D. N. ARNOLD, R. S. FALK, AND R. WINTHER, Preconditioning in H(div) and applications, Math. Comp., 66 (1997), pp. 957-984.
4. D. N. ARNOLD, R. S. FALK, AND R. WINTHER, Differential complexes and stability of finite element methods 1: The de Rham complex, in Compatible Spatial Discretizations, IMA Volumes in Mathematics and its Applications 142, Springer, Berlin, pp. 23-46.
5. P. BOCHEV, C. GARASI, J. HU, A. ROBINSON, AND R. TUMINARO, An improved algebraic multigrid method for solving Maxwell's equations, SIAM J. Sci. Comput., 25 (2003), pp. 623-642.
6. M. BREZINA, A. J. CLEARY, R. D. FALGOUT, V. E HENSON, J. E. JONES, T. A. MANTEUFFEL, S. F. MCCORMIOK, AND J. W. RUGE, Algebraic multigrid based on element interpolation (AMGe), SIAM J. Sci. Comput., 22 (2000), pp. 1570-1592.
7. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
8. T. CHARTIER, R. FALGOUT, V. E. HENSON, J. JONES, T. MANTEUFFEL, S. MCCORMICK, J. RUGE, AND P. S. VASSILEVSKI, Spectral element agglomerate AMGe, in Domain Decomposition Methods in Science and Engineering XVI, Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin-Heidelberg, 55 (2007), pp. 515-524.
9. P. W. GROSS AND P. R. KOTIUGA, Electromagnetic Theory and Computation: A Topological Approach, MSRI Monograph series, vol. 48 (2004), Cambridge University Press.
10. R. HIPTMAIR, Multigrid method for H(div) in three dimensions, Electronic Trans. Numer. Anal., 6 (1997) pp. 133-152.
11. R. HIPTMAIR, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal., 36 (1999), pp. 204-225.
12. R. HIPTMAIR AND A. TOSELLI, Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions, in Parallel Solution of Partial Differential Equations, P. Bjorstad and M. Luskin, eds., IMA Volumes in Mathematics and its Applications 120, Springer, Berlin, 1999, pp. 181-202.
13. R. HIPTMAIR, Finite elements in computational electromagnetism, Acta Numer., 11 (2002), pp. 237-339.
14. J. E. JONES AND P. S. VASSILEVSKI, AMGe based on element agglomerations, SIAM J. Sci. Comput., 23 (2001), pp. 109-133.
15. G. KARYPIS AND V. KUMAR, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Sci. Comput., 20 (1998), pp. 359-392.
16. T. V. KOLEV AND P. S. VASSILEVSKI, AMG by element agglomeration and constrained energy minimization interpolation, Numer. Linear Algebra. Appl., 13 (2006), pp. 771-788.
17. F. MUSY, L. NICOLAS, AND R. PERRUSSEL, Compatible coarse nodal and edge elements through energy functionals, SIAM J. Sci. Comput., 29 (2007), pp. 1315-1337.
18. S. REITZINGER AND J. SCHÖBERL, An algebraic multigrid method for finite element discretization with edge elements, Numer. Linear Algebra. Appl., 9 (2002), pp. 215-235.
19. P. S. VASSILEVSKI, Sparse matrix element topology with application to AMG and preconditioning, Numer. Linear Algebra Appl., 9 (2002), pp. 429-444.
20. P. S. VASSILEVSKI AND L. T. ZIKATANOV, Multiple vector preserving interpolation mappings in algebraic multigrid, SIAM J. Matrix Anal. Appl., 27 (2005-2006), pp. 1040-1055.