Solving the Einstein constraints in periodic spaces with a multigrid approach

TL;DR

This paper introduces a multigrid solver for elliptic PDEs with periodic boundary conditions, enabling efficient initial data generation for cosmological models like black-hole lattices in Numerical Relativity.

Contribution

It develops and tests a new multigrid algorithm integrated with the Einstein Toolkit for solving Einstein constraints on periodic spaces, a novel approach for cosmological simulations.

Findings

Successfully implemented the solver within the Einstein Toolkit.

Validated the solver against existing methods with favorable results.

Generated initial data for a periodic black-hole lattice.

Abstract

Novel applications of Numerical Relativity demand for more flexible algorithms and tools. In this paper, I develop and test a multigrid solver, based on the infrastructure provided by the Einstein Toolkit, for elliptic partial differential equations on spaces with periodic boundary conditions. This type of boundary often characterizes the numerical representation of cosmological models, where space is assumed to be made up of identical copies of a single fiducial domain, so that only a finite volume (with periodic boundary conditions at its edges) needs to be simulated. After a few tests and comparisons with existing codes, I use the solver to generate initial data for an infinite, periodic, cubic black-hole lattice.