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 non-smooth 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：求解椭圆方程的不连续伽辽金（DG）方法因其高阶谱收敛性及在计算集群上的并行化潜力，正逐渐在计算物理学界受到青睐。然而，涉及极端拉伸网格的数值相对论问题（例如需在距黑洞远处施加边界条件的双黑洞初始数据问题）对DG方法构成了挑战。为解决此问题，我们提出了一种通用的原始DG格式，适用于包括弯曲网格及极端拉伸网格在内的大类椭圆方程。该格式兼容数值相对论中两种广泛使用的初始数据构造方法，即穿孔公式和扩展共形薄三明治（XCTS）公式。实验表明，我们的DG格式能实现约$10^9$倍的网格拉伸系数，从而可在远距离处施加边界条件。该格式对光滑的XCTS问题与非光滑的穿孔问题均呈现指数收敛特性。借助该方法，我们能够利用可并行化的DG方案生成高质量双黑洞初始数据。该代码已开源发布于SpECTRE数值相对论代码库中。