Multilevel Methods for Least-Squares Discretizations of Neutron Transport Equations

Gerhard Starke^{∗}

We study least-squares finite element approximations of neutron transport equations in three dimensions and the efficient solution of the resulting discretized problems by multilevel methods. This approach is based on a leastsquares variational principle in combination with a scaling transformation. An important practical feature of the least-squares framework is that it automatically provides a posteriori bounds on the discretization error.

For simplicity, we consider the single group, steady state, isotropic form of the neutron transport equation given by

Ω · ∇ψ + σ_{t}ψ − σ_{s}P ψ = q for (x, Ω) ∈ R × S^{2} ,
ψ = 0 for x ∈ ∂R with n · Ω < 0 ,

(1)

where σ_{t} is the total cross section, σ_{s} is the scattering cross section and ψ(x, Ω) is the angular flux to be determined for all points x ∈ R ⊂ IR^{3} and all possible travel directions Ω on the unit sphere S^{2}, and

(P ψ)(x) = ∫_{S^{2}} ψ(x, Ω) dΩ .

The angular dependence in the above equation is approximated by spherical harmonics up to order N, so-called P_{N} discretization. The boundary conditions will be imposed by adding a boundary functional. The P_{N} approximation to the neutron transport equation leads to a system of partial differential equations with special structure. In the diffusive limit, i.e., for σ_{t} → ∞ and σ_{a} → 0, this system essentially becomes a Poisson equation for the zeroth moment and has a divergence structure for the set of moments of order 1. This makes the design of efficient multilevel strategies a challenging task. We report on numerical results obtained with multilevel methods for P_{1} and P_{2} approximations of the least-squares formulation.

(This is joint work with Tom Manteuffel, University of Colorado at Boulder and Klaus Ressel, ETH Zürich.)

∗Institut für Praktische Mathematik, Universität Karlsruhe, Englerstrasse 2, 76128 Karlsruhe, Germany (starke@math.uni-karlsruhe.de)