Diffusion Preconditioner for Discontinuous Galerkin Transport Problems

A simple Richardson iteration procedure converges slowly when applied to thick, diffusive problems with scattering ratios near unity. The current state of the art for overcoming this is to use a Krylov method with a diffusion preconditioner. However, the diffusion preconditioner must be tailored to the discretization of the transport operator to ensure effectiveness. We expand work from the bilinear discontinuous (BLD) finite element method (FEM) in two dimensions into a preconditioner applicable to all Discontinuous Galerkin FEMs in two and three dimensions. We demonstrate the effectiveness of our approach by applying it to the piecewise linear discontinuous (PWLD) FEM, which is notable for its flexibility with unstructured meshes. We employ a vertex-centered continuous FEM diffusion solution followed by local one-cell calculations to generate discontinuous solution corrections. Our goal is to achieve the same level of performance for PWLD and other methods, in two and three dimensions, as was previously achieved for BLD in two dimensions. We perform a Fourier analysis of this preconditioner applied to the PWLD FEM and we test the preconditioner on a variety of test problems. The preconditioned Richardson method is found to perform well in both ne and coarse mesh limits; however, it degrades for high-aspect ratio cells. These properties are typical for partially consistent diffusion synthetic acceleration (DSA) schemes, and in particular they are exactly the properties of the method that was previously developed for BLD in two dimensions. Thus, we have succeeded in our goal of generalizing the previous method to other Discontinuous Galerkin schemes. We also explore the effectiveness of our preconditioner when used within the GMRES iteration scheme. We find that with GMRES there is very little degradation for cells with high aspect ratios or for problems with strong heterogeneities. Thus we find that our preconditioned GMRES method is efficient and effective for all problems that we have tested. We have cast our diffusion operator entirely in terms of the single-cell matrices that are used by the discontinuous FEM transport method. This allows us to write our diffusion preconditioner without prior knowledge of the underlying FEM basis functions or cell shapes. As a result, a single software implementation of our preconditioner applies to a wide variety of transport options and there is no need to re-derive or re-implement a diffusion preconditioner when a new transport FEM is introduced