Fast and accurate multigrid solution of Poissons equation using diagonally oriented grids

TL;DR

The paper develops diagonally oriented multigrid hierarchies for the Poisson equation $\nabla^2u=f$, introducing 2D grids with spacings $2^{-\ abla/2}$ aligned at $45^\circ$ and a more complex 3D scheme with $2^{-\ abla/3}$ spacings across multiple intermediate grids. It replaces conventional interpolation with red-black Jacobi-based prolongation and uses simple smoothing restriction, achieving rapid convergence in a basic V-cycle; with optimized over-relaxation parameters, the method can be about twice as fast as simple conventional multigrid in 2D, and yields strong convergence in 3D as well (e.g., $\bar{\rho}\approx0.043$ on a $17^3$ grid after optimization). Second-order diagonals yield substantial speedups, while fourth-order residuals are less beneficial and may require staged approaches; the work points to the potential of W-cycles to further enhance performance on diagonal grids. Overall, the diagonal-grid multigrid framework offers a simple yet effective path to faster Poisson solvers with potential broad applicability.

Abstract

We solve Poisson's equation using new multigrid algorithms that converge rapidly. The novel feature of the 2D and 3D algorithms are the use of extra diagonal grids in the multigrid hierarchy for a much richer and effective communication between the levels of the multigrid. Numerical experiments solving Poisson's equation in the unit square and unit cube show simple versions of the proposed algorithms are up to twice as fast as correspondingly simple multigrid iterations on a conventional hierarchy of grids.

Figures (4)

• Figure 1: three levels of grids in the 2D multigrid hierarchy: the dotted green grid is the finest, spacing $h$ say; the dashed red grid is the next finest diagonal grid with spacing $\sqrt2h$; the solid blue grid is the coarsest shown grid with spacing $2h$. Coarser levels of the multigrid follow the same pattern.
• Figure 2: restriction stencils are simple weighted averages of neighbouring grid points on all levels of the grid.
• Figure 3: the interpolation in a prolongation step is replaced by simply a "red-black" Jacobi iteration: (a) compute the new values at the red grid points, then refine the values at the blue points; (b) compute the new values at the green points, then refine those at the red points.
• Figure 4: one cell of an amalgam of four levels of the hierarchy of grids used to form the multigrid V-cycle in 3D: green is the finest grid shown; red is the next level coarser grid; magenta shows the next coarser grid; and the blue cube is the coarsest to be shown. This stereoscopic view is to be viewed cross-eyed as this seems to be more robust to changes of viewing scale.