Discontinuous Galerkin scheme for elliptic equations on extremely stretched grids

TL;DR

This paper introduces a primal Discontinuous Galerkin scheme capable of handling elliptic equations on extremely stretched grids, enabling boundary conditions at large distances, with exponential convergence and application to binary black hole initial data.

Contribution

A novel primal DG scheme that effectively manages highly stretched grids and is applicable to various elliptic problems in numerical relativity.

Findings

Scheme can stretch grids by a factor of ~10^9

Achieves exponential convergence with resolution

Enables high-quality binary black hole initial data

Abstract

Discontinuous Galerkin (DG) methods for solving elliptic equations are gaining popularity in the computational physics community for their high-order spectral convergence and their potential for parallelization on computing clusters. However, problems in numerical relativity with extremely stretched grids, such as initial data problems for binary black holes that impose boundary conditions at large distances from the black holes, have proven challenging for DG methods. To alleviate this problem we have developed a primal DG scheme that is generically applicable to a large class of elliptic equations, including problems on curved and extremely stretched grids. The DG scheme accommodates two widely used initial data formulations in numerical relativity, namely the puncture formulation and the extended conformal thin-sandwich (XCTS) formulation. We find that our DG scheme is able to stretch the grid by a factor of $\sim 10^9$ and hence allows to impose boundary conditions at large distances. The scheme converges exponentially with resolution both for the smooth XCTS problem and for the nonsmooth puncture problem. With this method we are able to generate high-quality initial data for binary black hole problems using a parallelizable DG scheme. The code is publicly available in the open-source SpECTRE numerical relativity code.