  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.
