High order convergent multigrid methods on domains containing holes for black hole initial data

TL;DR

This paper extends multigrid methods to higher order accuracy for solving elliptic equations in domains with holes, crucial for initial data in numerical relativity, demonstrating high-order convergence in 2D and 3D problems.

Contribution

It introduces high-order (fourth, sixth, eighth) multigrid methods for domains with holes, enabling rapid, accurate solutions aligned with evolution code requirements in general relativity.

Findings

Achieved high-order convergence in 2D and 3D elliptic problems with holes.

Demonstrated solution of the conformally flat Hamiltonian constraint in 3D.

Validated the method's effectiveness for realistic general relativity initial data.

Abstract

It is well known that multigrid methods are optimally efficient for solution of elliptic equations (O(N)), which means that effort is proportional to the number of points at which the solution is evaluated). Thus this is an ideal method to solve the initial data/constraint equations in General Relativity for (for instance) black hole interactions, or for other strong-field gravitational configurations. Recent efforts have produced finite difference multigrid solvers for domains with holes (excised regions). We present here the extension of these concepts to higher order (fourth-, sixth- and eigth-order). The high order convergence allows rapid solution on relatively small computational grids. Also, general relativity evolution codes are moving to typically fourth-order; data have to be computed at least as accurately as this same order for straightfoward demonstration of the proper order of convergence in the evolution. Our vertex-centered multigrid code demonstrates globally high-order-accurate solutions of elliptic equations over domains containing holes, in two spatial dimensions with fixed (Dirichlet) outer boundary conditions, and in three spatial dimensions with {\it Robin} outer boundary conditions. We demonstrate a ``real world'' 3-dimensional problem which is the solution of the conformally flat Hamiltonian constraint of General Relativity. The success of this method depends on: a) the choice of the discretization near the holes; b) the definition of the location of the inner boundary, which allows resolution of the hole even on the coarsest grids; and on maintaining the same order of convergence at the boundaries as in the interior of the computational domain.