# A hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity

## TL;DR

This paper introduces a novel hp-adaptive discontinuous Galerkin method for elliptic equations in numerical relativity, featuring a scalable multigrid solver and boundary compactification, demonstrated on neutron star and black hole initial data.

## Contribution

It presents a new hp-adaptive discontinuous Galerkin scheme with multigrid preconditioning for elliptic problems in numerical relativity, capable of handling complex geometries and boundary conditions.

## Key findings

- Effective on test problems including neutron star phase transitions
- Successfully constructs initial data for multiple black holes
- Demonstrates scalability and flexibility of the solver

## Abstract

A considerable amount of attention has been given to discontinuous Galerkin methods for hyperbolic problems in numerical relativity, showing potential advantages of the methods in dealing with hydrodynamical shocks and other discontinuities. This paper investigates discontinuous Galerkin methods for the solution of elliptic problems in numerical relativity. We present a novel hp-adaptive numerical scheme for curvilinear and non-conforming meshes. It uses a multigrid preconditioner with a Chebyshev or Schwarz smoother to create a very scalable discontinuous Galerkin code on generic domains. The code employs compactification to move the outer boundary near spatial infinity. We explore the properties of the code on some test problems, including one mimicking Neutron stars with phase transitions. We also apply it to construct initial data for two or three black holes.