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.

翻译：求解椭圆方程的间断伽辽金（DG）方法因其高阶谱收敛特性以及在计算集群上的并行化潜力，正日益受到计算物理学界的青睐。然而，在数值相对论中涉及极端拉伸网格的问题——例如需要在距离黑洞极远处施加边界条件的双黑洞初始数据问题——已被证明对DG方法构成严峻挑战。为缓解此问题，我们开发了一种原始DG格式，该格式普遍适用于包括弯曲及极端拉伸网格问题在内的一大类椭圆方程。该DG格式兼容数值相对论中两种广泛使用的初始数据构建框架：穿刺表述与扩展共形薄三明治（XCTS）表述。我们发现，所提出的DG格式能够实现约$10^9$倍的网格拉伸，从而允许在极远距离处施加边界条件。该格式在光滑的XCTS问题与非光滑的穿刺问题上均表现出随分辨率指数收敛的特性。借助这一方法，我们能够利用可并行化的DG格式为双黑洞问题生成高质量的初始数据。相关代码已在开源数值相对论软件SpECTRE中公开发布。