Even-parity finite-element transport methods in the diffusion limit
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We use an asymptotic analysis to investigate the behavior of continuous finite-element-method (CFEM) discretizations of the even-parity transport equation, in problems containing optically thick diffusive regions. Our first interesting result is that we can analyze the entire family of even-parity CFEMs, and can do so in three dimensions on an arbitrarily-connected grid. (Previous asymptotic analyses have been restricted to specific discretizations, either in slab geometry or in XY geometry on a rectangular grid.) We show that every even-parity CFEM transport solution satisfies a corresponding CFEM discretization of the correct diffusion equation in the diffusion limit, which is a highly desirable property. We further show that this solution is subject to a Dirichlet boundary condition given by a cosine (|n|) weighting of the incident intensity. We show that this boundary condition, which is less accurate than we would like, means that in certain problems the transport solution in a diffusive region can be more than a factor of two greater than the correct solution. We also show that the CFEM transport solution can be incorrect in non-diffusive regions that are adjacent to diffusive regions, no matter how fine the spatial grid is in the non-diffusive region. We give numerical results from slab geometry verifying the predictions of our analysis. 1991.
