H(div) preconditioning for a mixed finite element formulation of the diffusion problem with random data

Mathematics of Computation

Volumn 79, Issue 270, 2010, Pages 733-760

  Elman, Howard C ^a   Furnival, Darran G ^b,c   Powell, Catherine E ^d  

^a University of Maryland ^*  (United States)

^b UNIVERSITY OF MARYLAND   (United States)

^c NATIONAL OCEANOGRAPHY CENTRE   (United Kingdom)

^d UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77956054384     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-09-02274-1     Document Type: Article

References (25)

1. Douglas N. Arnold, Richard S. Falk, and R. Winther. Preconditioning in H(div) and applications. Math. Comp., 66:957-984, 1997. MR1401938 (97i:65177)
2. Ivo Babuška, Raúl Tempone, and Georgios E. Zouraris. Galerkin finite element approximation of stochastic elliptic partial differential equations. SIAM J. Numer. Anal., 42:800-825, 2004. MR2084236 (2005h:65012)
3. Dietrich Braess. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, second edition, 2001. MR1827293 (2001k:65002)
4. Franco Brezzi and Michel Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, 1991. MR1115205 (92d:65187)
5. Zdzislaw Brzeźniak and Tomasz Zastawniak. Basic Stochastic Processes. Springer-Verlag, 1999. MR1713252 (2000f:60002)
6. Claudio Canuto and Tomas Kozubek. A fictitious domain approach to the numerical solution of PDES in stochastic domains. Numer. Math., 107:257-293, 2007. MR2328847 (2008h:60277)
7. Manas K. Deb, Ivo M. Babuška, and J. Tinsley Oden. Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Engrg., 190:6359-6372, 2001. MR1870425 (2003g:65009)
8. Howard Elman and Darran Furnival. Solving the stochastic steady-state diffusion problem using multigrid. IMA. J. Numer. Anal., 27:675-688, 2007. MR2371827 (2009b:60248)
9. O. G. Ernst, C. E. Powell, D. J. Silvester, and E. Ullman. Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data. SIAM Journal Sci. Comp., 31:1424-1447, 2000. MR2486837
10. P. Frauenfelder, C. Schwab, and R. A. Todor. Finite elements for elliptic problems with stochastic coefficients. Comput. Meth. Appl. Mech. Engrg., 194:205-228, 2005. MR2105161 (2005i:65186)
11. Avner Friedman. Foundations of Modern Analysis. Dover, 1982. MR663003 (83h:46001)
12. Roger G. Ghanem and Pol D. Spanos. Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, 1991. MR1083354 (91k:73102)
13. R. Hiptmair. Multigrid method for H(div) in three dimensions. Electron. Trans. Numer. Anal., 6:133-152, 1997. MR1615161 (99c:65232)
14. O. P. Le Maître, O. M. Knio, B. J. Debusschere, H. N. Najm, and R. G. Ghanem. A multi-grid solver for two-dimensional stochastic diffusion equations. Comput. Methods Appl. Mech. Engrg., 192:4723-4744, 2003. MR2012487 (2004h:65012)
15. P. Vassilevski and J. Wang. Multilevel iterative methods for mixed finite element discretizations of elliptic problems. Numer. Math., 63:503-520, 1992. MR1189534 (93j:65187)
16. C. C. Paige and M. A. Saunders. Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal., 12:617-629, 1975. MR0383715 (52:4595)
17. Catherine E. Powell. Parameter-free H(div) preconditioning for mixed finite element formulation of diffusion problems. IMA J. Numer. Anal., 25:783-796, 2005. MR2170523 (2007b:65113)
18. Catherine Elizabeth Powell and David Silvester. Optimal preconditioning for Raviart-Thomas mixed formulation of second-order problems. SIAM J. Matrix Anal. Appl., 25:718-738, 2003. MR2081231 (2005h:65054)
19. P. A. Raviart and J. M. Thomas. A mixed finite elment method for second order elliptic problems. In I. Galligani and E. Magenes, editors, Mathematical Aspects of the Finite Element Method, no. 606 in Lecture Notes in Math., pages 292-315. Springer-Verlag, New York, 1977. MR0483555 (58:3547)
20. Wim Schoutens. Stochastic Processes and Orthogonal Polynomials. Springer-Verlag, 2000. MR1761401 (2001f:60095)
21. A. Toselli and 0. Widlund. Domain Decomposition Methods - Algorithms and Theory. Springer-Verlag, 2005. MR2104179 (2005g:65006)
22. Fraņcois Treves. Topological Vector Spaces, Distributions and Kernels. Academic Press, 1967. MR0225131 (37:726)
23. Dongbin Xiu. Generalized (Wiener-Askey) Polynomial Chaos. Ph.D. thesis, Brown University, 2004.
24. Dongbin Xiu and George Em Karniadakis. Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Engrg., 191:4927-4948, 2002. MR1932024 (2003h:65008)
25. C. I. Goldstein, Z. Cai and J. E. Pasciak. Multilevel iteration for mixed finite element systems with penalty. SIAM J. Sci. Comput., 14(5):1072-1088, 1993. MR1232176 (94h:65116)-