We present a discontinuous Galerkin internal-penalty scheme that is applicable to a large class of linear and nonlinear elliptic partial differential equations. The unified scheme can accommodate all second-order elliptic equations that can be formulated in first-order flux form, encompassing problems in linear elasticity, general relativity, and hydrodynamics, including problems formulated on a curved manifold. It allows for a wide range of linear and nonlinear boundary conditions, and accommodates curved and nonconforming meshes. Our generalized internal-penalty numerical flux and our Schur-complement strategy of eliminating auxiliary degrees of freedom make the scheme compact without requiring equation-specific modifications. We demonstrate the accuracy of the scheme for a suite of numerical test problems. The scheme is implemented in the open-source SpECTRE numerical relativity code.

翻译：我们提出了一个不连续的Galerkin内部刑罚制度,适用于一大批线性和非线性椭圆部分偏差方程式。统一办法可以容纳以一阶通量形式制定的所有二级椭圆方程式,包括线性弹性、一般相对性和流体动力学方面的问题,包括曲线形数上的问题。它允许一系列广泛的线性和非线性边界条件,并包含曲线和不相容的线性。我们普遍的内线性数值通量和我们消除辅助自由度的舒尔补充战略,使该计划的契约不要求具体方程式的修改。我们展示了一套数字测试问题的准确性。这个办法在开放源码SPECTRE数字相对性代码中实施。