Algebraic tailoring of discontinuous Galerkin p-multigrid for convection

• p-Multigrid is a simple but imperfect solver for high-order DG.
• Effective p-multigrid smoothers do not scale well.
• Coarse level spaces can be tuned to complement simple smoothers.
• An element-local singular value analysis provides an efficient coarse space.

This work presents an element-local algebraic approach to constructing coarse spaces for p-multigrid solvers and preconditioners of high-order discontinuous Galerkin discretizations. The target class of problems is convective systems on unstructured meshes, a class for which traditional p-multigrid typically fails to reach textbook multigrid efficiency due to a mismatch between smoothers and coarse spaces. Smoothers that attempt to alleviate this mismatch, such as line-implicit, incomplete LU, or Gauss–Seidel, deteriorate on grids that are not aligned with the flow, and they rely on sequential operations that do not scale well to distributed-memory architectures. In this work we shift attention from the smoothers to the coarse spaces, and we present an algebraic definition of the coarse spaces within each element based on a singular-value decomposition of the neighbor influence matrix. On each multigrid level, we employ a block-Jacobi smoother, which maintains algorithmic scalability as all elements can be updated in parallel. We demonstrate the performance of our solver on discretizations of advection and the linearized compressible Euler equations.