Preconditioning a mixed discontinuous finite element method for radiation diffusion

Numerical Linear Algebra with Applications

Volumn 11, Issue 8-9, 2004, Pages 795-811

  Warsa, James S ^a   Benzi, Michele ^b   Wareing, Todd A ^a   Morel, Jim E ^a  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

^b EMORY UNIVERSITY   (United States)

Author keywords

Indefinite matrices; Inner outer iteration; Mixed discontinuous finite element method; Preconditioned Krylov subspace methods; Radiation diffusion; Two level preconditioning

Indexed keywords

EID: 10044251393     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/nla.347     Document Type: Article

References (35)

1. Faber V, Manteuffel TA. A look at transport theory from the point of view of linear algebra. In Transport Theory, Invariant Imbedding and Integral Equations, Nelson P, Faber V, Manteuffel TA, Seth DL, White AB (eds). Lecture Notes in Pure and Applied Mathematics, vol. 115. Marcel Dekker: New York, NY, 1989; 31-61.
2. Ashby SF, Brown PN, Dorr MR, Hindmarsh AC. A linear algebraic analysis of diffusion synthetic acceleration for the Boltzmann transport equation. SIAM Journal on Numerical Analysis 1995; 32:128-178.
3. Guthrie B, Holloway JP, Patton BW. GMRES as a multi-step transport sweep accelerator. Transport Theory and Statistical Physics 1999; 28:83-102.
4. Alcouffe RE. Diffusion synthetic acceleration methods for diamond-differenced discrete-ordinates equations. Nuclear Science and Engineering 1977; 64:344-355.
5. Larsen EW. The asymptotic diffusion limit of discretized transport problems. Nuclear Science and Engineering 1992; 112:336-346.
6. n methods on three-dimensional unstructured grids. Nuclear Science and Engineering 2001; 138:1-13.
7. Cockburn B, Shu CW. The local discontinuous Galerkin finite element method for convection-diffusion systems. SIAM Journal on Numerical Analysis 1998; 35:2440-2463.
8. Cockburn B, Karniadakis GE, Shu CW (eds). Discontinuous Galerkin Methods. Springer: Berlin, Heidelberg, 2000.
9. 1 equations in two-dimensional Cartesian geometry with two-level preconditioning. SIAM Journal on Scientific Computing, 2003; 24:2093-2124.
10. Warsa JS, Wareing TA, Morel JE. Fully consistent diffusion synthetic acceleration of linear discontinuous transport discretizations on three-dimensional unstructured meshes. Nuclear Science and Engineering 2002; 141:1-16.
11. Perugia I, Schötzau D, Warsa JS. A note on a discontinuous Galerkin method for radiation diffusion. Los Alamos National Laboratory Technical Report LA-UR-02-4754. Los Alamos, NM, May 2002.
12. Arnold DM, Brezzi F, Cockburn B, Marini D. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis 2002; 39:1749-1779.
13. Greenbaum A. Iterative Methods for Solving Linear Systems. SIAM: Philadelphia, PA, 1997.
14. Fischer B, Ramage A, Silvester DJ, Wathen AJ. Minimal residual methods for augmented systems. BIT 1998; 38:527-543.
15. Paige CC, Saunders MA. Solution of sparse indefinite systems of linear equations. SIAM Journal on Numerical Analysis 1975; 12:617-629.
16. Freund RW, Nachtigal NM. Software for simplified Lanczos and QMR algorithms. Applied Numerical Mathematics 1995; 19:319-341.
17. Saad Y, Schultz MH. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 1986; 7:856-869.
18. van der Vorst HA. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 1992; 13:631-644.
19. Freund RW. A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM Journal on Scientific Computing 1993; 14:425-448.
20. Bank RE, Welfert BD, Yserentant HY. A class of iterative methods for solving saddle point problems. Numerische Mathematik 1990; 56:645-666.
21. Elman HC, Golub GH. Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM Journal on Numerical Analysis 1994; 31:1645-1661.
22. Bramble JH, Pasciak JE. A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Mathematics of Computation 1988; 50:1-17.
23. Rusten T, Winther R. A preconditioned iterative method for saddlepoint problems. SIAM Journal on Matrix Analysis und Applications 1992; 13:887-904.
24. Ramage A, Wathen AJ. Iterative solution techniques for the Stokes and Navier-Stokes equations. International Journal for Numerical Methods in Fluids 1994; 19:67-83.
25. Elman HC, Silvester DJ, Wathen AJ. Iterative methods for problems in computational fluid dynamics. In Iterative Methods in Scientific Computing, Chan R, Chan T, Golub GH (eds). Springer: Singapore, 1997; 271-327.
26. Murphy MF, Golub GH, Wathen AJ. A note on preconditioning for indefinite linear systems. SIAM Journal on Scientific Computing 2000; 21:1969-1972.
27. Perugia I, Simoncini V. Block-diagonal and symmetric indefinite preconditioners for mixed finite element formulations. Numerical Linear Algebra with Applications 2000; 7:585-616.
28. Ipsen ICF. A note on preconditioning non-symmetric matrices. SIAM Journal on Scientific Computing 2001; 23:1050-1051.
29. Bouras A, Frayssé V. A relaxation strategy for inexact matrix-vector products for Krylov methods. Technical Report CERFACS TR/PA/00/15, Toulouse, France, September 2000.
30. Bouras A, Frayssé V, Giraud L. A relaxation strategy for inner-outer linear solvers in domain decomposition methods. Technical Report CERFACS TR/PA/00/17, Toulouse, France, September 2000.
31. Golub GH, Ye Q. Inexact preconditioned conjugate gradient method with inner-outer iteration. SIAM Journal on Scientific Computing 1999; 21:1305-1320.
32. Simoncini V, Szyld DB. Theory of inexact Krylov subspace methods and applications to scientific computing. SIAM Journal on Scientific Computing, 2003; 25:454-477.
33. Saad Y. A flexible inner-outer preconditioned GMRES algorithm. SIAM Journal on Scientific Computing 1993; 14:461-469.
34. Notay Y. Flexible conjugate gradients. SIAM Journal on Scientific Computing 2000; 22:1444-1460.
35. Wareing TA, McGhee JM, Morel JE. ATTILA: A three-dimensional, unstructured tetrahedral mesh discrete ordinates transport code. Transactions of the American Nuclear Society 1996; 75:146-147.