Solution of the Discontinuous P1 Equations in Two-Dimensional Cartesian Geometry with Two-Level Preconditioning

  • Authors:

  • J. S. Warsa;T. A. Wareing;J. E. Morel

  • Affiliations:

  • -;-;-

  • Venue:
  • SIAM Journal on Scientific Computing
  • Year:
  • 2002

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Abstract

We present a new bilinear discontinuous (Galerkin) finite element discretization of the P1 (spherical harmonics) equations, a first order system of equations used for describing neutral particle radiation transport or modeling radiative transfer problems. The discrete equations are described for two-dimensional rectangular meshes; we solve the linear system with Krylov iterative methods. We have developed a novel, two-level preconditioner to improve convergence of the Krylov solvers that is based on a linear continuous finite element discretization of the diffusion equation, solved with a conjugate gradient iteration, preceded and followed by one of several different smoothing relaxations. A Fourier analysis shows that our approach is very effective over a wide range of problems. Numerical experiments confirm the results of the Fourier analysis. Computations for a realistic problem show that the preconditioner is effective and the solution method is efficient in practice.