Tweed and wireframe: accelerated relaxation algorithms for multigrid solution of elliptic PDEs on stretched structured grids

TL;DR

This paper introduces two novel relaxation schemes, Tweed and wireframe, that significantly improve multigrid solutions of elliptic PDEs on stretched structured grids by optimizing line relaxation based on grid clustering.

Contribution

The paper presents two new relaxation schemes, Tweed and wireframe, tailored for efficient smoothing in multigrid methods on stretched grids, outperforming existing relaxation techniques.

Findings

Tweed relaxation excels near boundary grid clustering.

Wireframe relaxation is optimal near central grid clustering.

Both schemes outperform checkerboard and zebra relaxations on stretched grids.

Abstract

Two new relaxation schemes are proposed for the smoothing step in the geometric multigrid solution of PDEs on 2D and 3D stretched structured grids. The new schemes are characterized by efficient line relaxation on branched sets of lines of alternating colour, where the lines are constructed to be everywhere orthogonal to the local direction of maximum grid clustering. Tweed relaxation is best suited for grid clustering near the boundaries of the computational domain, whereas wireframe relaxation is best suited for grid clustering near the centre of the computational domain. On strongly stretched grids of these types, multigrid leveraging these new smoothing schemes significantly outperforms multigrid based on other leading relaxation schemes, such as checkerboard and alternating-direction zebra relaxation, for the numerical solution of large linear systems arising from the discretization of elliptic PDEs.