Tips for implementing multigrid methods on domains containing holes

TL;DR

This paper presents effective strategies for implementing multigrid methods on domains with holes, ensuring second-order accuracy in solving elliptic equations relevant to numerical relativity, with practical guidance and open-source code.

Contribution

It introduces specific techniques for handling holes in multigrid methods, including restriction operator choices and boundary definitions, validated through 3D elliptic problem solutions.

Findings

Achieved globally second-order-accurate solutions in 2D and 3D domains with holes.

Demonstrated the importance of smoothing concentration near holes for convergence.

Provided publicly available code as a semi-pedagogical resource.

Abstract

As part of our development of a computer code to perform 3D `constrained evolution' of Einstein's equations in 3+1 form, we discuss issues regarding the efficient solution of elliptic equations on domains containing holes (i.e., excised regions), via the multigrid method. We consider as a test case the Poisson equation with a nonlinear term added, as a means of illustrating the principles involved, and move to a "real world" 3-dimensional problem which is the solution of the conformally flat Hamiltonian constraint with Dirichlet and Robin boundary conditions. Using our vertex-centered multigrid code, we demonstrate globally second-order-accurate solutions of elliptic equations over domains containing holes, in two and three spatial dimensions. Keys to the success of this method are the choice of the restriction operator near the holes and definition of the location of the inner boundary. In some cases (e.g. two holes in two dimensions), more and more smoothing may be required as the mesh spacing decreases to zero; however for the resolutions currently of interest to many numerical relativists, it is feasible to maintain second order convergence by concentrating smoothing (spatially) where it is needed most. This paper, and our publicly available source code, are intended to serve as semi-pedagogical guides for those who may wish to implement similar schemes.