Preconditioning a mixed discontinuous finite element method for radiation diffusion

We propose a multilevel preconditioning strategy for the iterative solution of large sparse linear systems arising from a finite element discretization of the radiation diffusion equations. In particular, these equations are solved using a mixed finite element scheme in order to make the discretization discontinuous, which is imposed by the application in which the diffusion equation will be embedded. The essence of the preconditioner is to use a continuous finite element discretization of the original, elliptic diffusion equation for preconditioning the discontinuous equations. We have found that this preconditioner is very effective and makes the iterative solution of the discontinuous diffusion equations practical for large problems. This approach should be applicable to discontinuous discretizations of other elliptic equations. We show how our preconditioner is developed and applied to radiation diffusion problems on unstructured, tetrahedral meshes and show numerical results that illustrate its effectiveness. Published in 2004 by John Wiley & Sons, Ltd.

[1]  Valeria Simoncini,et al.  Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing , 2003, SIAM J. Sci. Comput..

[2]  Ragnar Winther,et al.  A Preconditioned Iterative Method for Saddlepoint Problems , 1992, SIAM J. Matrix Anal. Appl..

[3]  V. Simoncini,et al.  Block--diagonal and indefinite symmetric preconditioners for mixed finite element formulations , 1999 .

[4]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[5]  Brian Guthrie,et al.  GMRES as a multi-step transport sweep accelerator , 1999 .

[6]  Yvan Notay.  Flexible Conjugate Gradients , 2000, SIAM J. Sci. Comput..

[7]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[8]  T. A. Manteuffel,et al.  A look at transport theory from the point of view of linear algebra , 1988 .

[9]  J. S. WARSA,et al.  Solution ofthe Discontinuous P1 Equations in Two-Dimensional Cartesian Geometry with Two-Level Preconditioning , 2002, SIAM J. Sci. Comput..

[10]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[11]  J. Pasciak,et al.  A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems , 1988 .

[12]  Ilse C. F. Ipsen.  A Note on Preconditioning Nonsymmetric Matrices , 2001, SIAM J. Sci. Comput..

[13]  R. E. Alcouffe,et al.  Diffusion synthetic acceleration methods for the diamond-differenced discrete-ordinates equations , 1977 .

[14]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[15]  R. Bank,et al.  A class of iterative methods for solving saddle point problems , 1989 .

[16]  A. Wathen,et al.  Minimum residual methods for augmented systems , 1998 .

[17]  A. C. Hindmarsh,et al.  A linear algebraic analysis of diffusion synthetic acceleration for the Boltzmann transport equation , 1995 .

[18]  Gene H. Golub,et al.  Inexact Preconditioned Conjugate Gradient Method with Inner-Outer Iteration , 1999, SIAM J. Sci. Comput..

[19]  Shawn D. Pautz,et al.  Discontinuous Finite Element SN Methods on Three-Dimensional Unstructured Grids , 2001 .

[20]  V. Frayssé,et al.  A relaxation strategy for inexact matrix-vector products for Krylov methods , 2000 .

[21]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[22]  Jim E. Morel,et al.  Fully Consistent Diffusion Synthetic Acceleration of Linear Discontinuous SN Transport Discretizations on Unstructured Tetrahedral Meshes , 2002 .

[23]  A. Wathen,et al.  Iterative solution techniques for the stokes and Navier‐Stokes equations , 1994 .

[24]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[25]  Bernardo Cockburn.  Discontinuous Galerkin methods , 2003 .

[26]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[27]  G. Golub,et al.  Inexact and preconditioned Uzawa algorithms for saddle point problems , 1994 .

[28]  R. Freund,et al.  Software for simplified Lanczos and QMR algorithms , 1995 .

[29]  Edward W. Larsen,et al.  The Asymptotic Diffusion Limit of Discretized Transport Problems , 1992 .

[30]  H. Elman,et al.  Iterative Methods for Problems in Computational Fluid Dynamics , 1998 .

[31]  Jim E. Morel,et al.  ATTILA: A three-dimensional, unstructured tetrahedral mesh discrete ordinates transport code , 1996 .