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

A splitting iterative method for solving the neutron transport equation

This paper presents an iterative method based on a self‐adjoint and m‐accretive splitting for the numerical treatment of the steady state neutron transport equation. Theoretical analysis shows that this method converges unconditionally to the unique solution of the transport equation. The convergence of the method is numerically illustrated and compared with the standard Source Iteration method and multigrid method on sample problems in slab geometry and in two dimensional space. First published online: 14 Oct 201

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

Preconditioning a Mixed Discontinuous Finite Element Method for Radiation Diffusion

this paper we describe a preconditioning method for a mixed, discontinuous nite element discretization of the equations describing radiation diusion. We are interested in solving these equations in order to accelerate the convergence of an outer radiation transport iteration. The discretized transport equation results in linear systems that are far too large to be solved by direct methods. A widely used solution approach in the transport community is source iteration, or transport sweep method, which is a stationary iterative scheme based on a simple splitting of the transport operator; see for example [1], [2], [3]. This iteration can converge quite slowly for certain important classes of problems. In such cases it is impractical to solve the transport problem without some kind of acceleration. The most eective and well{known technique is based on a solution of the radiation diusion equations. This is known in the neutron transport community as Diusion Synthetic Acceleration, or DSA; see [4]. Therefore, an ecient and robust solution of the radiation diusion equations can make solution to the radiation transport equation feasible in those types of problem