Analysis of two-level method for anisotropic diffusion equations on aligned and non-aligned grids

TL;DR

This paper analyzes the convergence of two-level multigrid methods for anisotropic diffusion equations discretized with finite elements, showing that specially designed block smoothers achieve uniform convergence regardless of grid alignment or anisotropy.

Contribution

It provides a convergence analysis for two-level multigrid algorithms on anisotropic problems, introducing a block smoother that ensures uniform convergence for aligned and non-aligned grids.

Findings

Point-wise smoothers deteriorate convergence on aligned grids.

Block smoothers achieve uniform convergence across anisotropy ratios.

Numerical experiments confirm theoretical convergence results.

Abstract

This paper is devoted to the multigrid convergence analysis for the linear systems arising from the conforming linear finite element discretization of the second order elliptic equations with anisotropic diffusion. The multigrid convergence behavior is known to strongly depend on whether the discretization grid is aligned or non-aligned with the anisotropic direction and analyses in the paper will be mainly focused on two-level algorithms. For an aligned grid case, a lower bound is given for point-wise smoother which shows deterioration of convergence rate. In both aligned and non-aligned cases we show that for a specially designed block smoother the convergence is uniform with respect to both anisotropy ratio and mesh size in the energy norm. The analysis is complemented with numerical experiments which confirm the theoretical results