Generalized Filtering Decomposition
For computational scientists solving PDEs on unstructured grids, this provides a preconditioner that can eliminate convergence plateaus in iterative solvers.
This paper introduces a new preconditioning technique for matrices from PDE discretizations on unstructured grids, which satisfies a filtering property to reduce low-frequency mode effects and eliminate convergence plateaus. The method works for arbitrary sparse structures and supports parallel computation via nested dissection.
This paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low frequency modes on convergence and so decrease or eliminate the plateau which is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process.