Nodal auxiliary space preconditioning in H(curl) and H(div) spaces

SIAM Journal on Numerical Analysis

Volumn 45, Issue 6, 2007, Pages 2483-2509

  Hiptmair, Ralf ^a   Xu, Jinchao ^b  

^a ETH ZURICH   (Switzerland)

^b PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

Algebraic multigrid; Auxiliary space preconditioning; Edge and face finite elements; Fictitious space preconditioning; H(curl) and H(div)

Indexed keywords

ALGEBRAIC MULTIGRID; AUXILIARY SPACE PRECONDITIONING; EDGE AND FACE FINITE ELEMENTS; FICTITIOUS SPACE PRECONDITIONING; H(CURL) AND H(DIV);

LINEAR SYSTEMS; POISSON EQUATION;

LINEAR EQUATIONS;

EID: 44049108809     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/060660588     Document Type: Article

References (38)

1. C. AMROUCHE, C. BERNARDI, M. DAUGE, AND V. GIRAULT, Vector potentials in three-dimensional nonsmooth domains, Math. Methods Appl. Sci., 21 (1998), pp. 823-864.
2. D. ARNOLD, R. FALK, AND R. WINTHER, Multigrid Preconditioning in H(div) on Non-convex Polygons, Tech. rep., Penn State University, College Park, PA, 1997.
3. D. ARNOLD, R. FALK, AND R. WINTHER, Preconditioning in H(div) and applications, Math. Comp., 66 (1997), pp. 957-984.
4. D. ARNOLD, R. FALK, AND R. WINTHER, Multigrid in H(div) and H(curl), Numer. Math., 85 (2000), pp. 175-195.
5. R. BECK, Algebraic Multigrid by Component Splitting for Edge Elements on Simplicial Triangulations, Tech. rep. SC 99-40, ZIB, Berlin, Germany, 1999.
6. 2- theory of the Maxwell operator in arbitrary domains, Russian Math. Surveys, 42 (1987), pp. 75-96.
7. P. B. BOCHEV, C. J. GARASI, J. J. HU, A. C. ROBINSON, AND R. S. TUMINARO, An improved algebraic multigrid method for solving Maxwell's equations, SIAM J. Sci. Comput., 25 (2003), pp. 623-642.
8. F. BORNEMANN, A Sharpened Condition Number Estimate for the BPX-Preconditioner of Elliptic Finite Element Problems on Highly Non-uniform Triangulations, Tech. rep. SC 91-9, ZIB, Berlin, Germany, 1991.
9. A. BOSSAVIT, Computational Electromagnetism. Variational Formulation, Complementarity, Edge Elements, Electromagnetism 2, Academic Press, San Diego, CA, 1998.
10. W. BOYSE, D. LYNCH, K. PAULSEN, AND G. MINERBO, Nodal-based finite-element modeling of Maxwell's equations, IEEE Trans. Antennas and Propagation, 40 (1992), pp. 642-651.
11. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
12. A. BUFFA, M. COSTABEL, AND D. SHEEN, On traces for H(curl, Ω) in Lipschitz domains, J. Math. Anal. Appl., 276 (2002), pp. 845-867.
13. P. CIARLET, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4, North-Holland, Amsterdam, 1978.
14. M. COSTABEL, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), pp. 365-368.
15. M. COSTABEL, A coercive bilinear form for Maxwell's equations, J. Math. Anal. Appl., 157 (1991), pp. 527-541.
16. M. COSTABEL AND M. DAUGE, Maxwell and Lamé eigenvalues on polyhedra, Math. Methods Appl. Sci., 22 (1999), pp. 243-258.
17. V. GIRAULT AND P. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986.
18. M. GRIEBEL AND P. OSWALD, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math., 70 (1995), pp. 163-180.
19. R. HIPTMAIR, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal., 36 (1998), pp. 204-225.
20. R. HIPTMAIR, Discrete Hodge operators, Numer. Math., 90 (2001), pp. 265-289.
21. R. HIPTMAIR, Finite elements in computational electromagnetism, Acta Numer., 11 (2002), pp. 237-339.
22. R. HIPTMAIR, Analysis of multilevel methods for eddy current problems, Math. Comp., 72 (2003), pp. 1281-1303.
23. R. HIPTMAIR, Coupling of finite elements and boundary elements in electromagnetic scattering, SIAM J. Numer. Anal., 41 (2003), pp. 919-944.
24. 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 Vol. Math. Appl. 120, Springer-Verlag, Berlin, 1999, pp. 181-202.
25. 0(curl, Ω), Numer. Math., 103 (2006), pp. 435-459.
26. R. HIPTMAIR AND W.-Y. ZHENG, Local multigrid in H(curl), Tech. rep. 2007-03, SAM, ETH Zürich, Zürich, Switzerland, 2007.
27. 1-based Auxiliary Space AMG Solver for H(curl) Problems, Report UCRL-TR-2222763, LLNL, Livermore, CA, 2006.
28. 1-based Auxiliary Space AMG for H(curl) Problems, Report UCRL-TR-221841, LLNL, Livermore, CA, 2006.
29. 3, Numer. Math., 35 (1980), pp. 315-341.
30. 3, Numer. Math., 50 (1986), pp. 57-81.
31. S. NEPOMNYASCHIKH, Decomposition and fictitious domain methods for elliptic boundary value problems, in Proceedings of the Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, D. Keyes, T. Chan, G. Meurant, J. Scroggs, and R. Voigt, eds., SIAM, Philadelphia, 1992, pp. 62-72.
32. J. PASCIAK AND J. ZHAO, Overlapping Schwarz methods in H(curl) on polyhedral domains, J. Numer. Math., 10 (2002), pp. 221-234.
33. S. REITZINGER AND J. SCHÖBERL, Algebraic multigrid for edge elements, Numer. Linear Algebra Appl., 9 (2002), pp. 223-238.
34. L. R. SCOTT AND Z. ZHANG, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483-493.
35. K. STÜBEN, An Introduction to Algebraic Multigrid, Academic Press, London, 2001, pp. 413-528.
36. U. TROTTENBERG, C. OOSTERLEE, AND A. SCHüller, Multigrid, Academic Press, London, 2000.
37. J. XU, Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), pp. 581-613.
38. J. XU, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215-235.