Generalized Filtering Decomposition

TL;DR

This paper presents a novel preconditioning method for PDE discretization matrices on unstructured grids, using a filtering property to improve convergence by mitigating low frequency mode effects.

Contribution

It introduces a generalized filtering decomposition that ensures the preconditioner matches the matrix on a filtering vector, enhancing iterative solver performance.

Findings

Improves convergence by reducing low frequency mode effects.

Applicable to matrices with arbitrary sparse structures.

Supports parallel computation through reordering techniques.

Abstract

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.