A ghost-point smoothing strategy for geometric multigrid on curved boundaries

TL;DR

This paper introduces a boundary local Fourier analysis to optimize relaxation parameters in multigrid methods for elliptic equations on curved domains, improving smoothing near boundaries with ghost-point techniques.

Contribution

It proposes a novel optimization of relaxation parameters based on ghost-point distances, enhancing multigrid smoothing on curved boundaries.

Findings

Optimized relaxation parameters improve convergence rates.

Boundary effects do not degrade internal smoothing.

Numerical tests confirm theoretical predictions in 1D, 2D, and 3D.

Abstract

We present a Boundary Local Fourier Analysis (BLFA) to optimize the relaxation parameters of boundary conditions in a multigrid framework. The method is implemented to solve elliptic equations on curved domains embedded in a uniform Cartesian mesh, although it is designed to be extended for general PDEs in curved domains, wherever a multigrid technique can be implemented. The boundary is implicitly defined by a level-set function and a ghost-point technique is employed to treat the boundary conditions. Existing strategies in literature adopt a constant relaxation parameter on the whole boundary. In this paper, the relaxation parameters are optimized in terms of the distance between ghost points and boundary, with the goal of smoothing the residual along the tangential direction. Theoretical results are confirmed by several numerical tests in 1D, 2D and 3D, showing that the convergence factor associated with the smoothing on internal equations is not degraded by boundary effects.