Adaptive Multilevel Discontinuous Galerkin Methods on GPUs

Abstract

I present a matrix-free symmetric interior penalty discontinuous Galerkin method for adaptively refined Cartesian meshes on GPUs. At non-matching interfaces, auxiliary \emph{shadow cells} represent the adjacent coarse polynomial on the fine level. This approach converts non-matching interfaces into matching faces, permitting uniform face evaluation throughout the mesh. I prove that this construction is equivalent to the standard non-matching formulation. Furthermore, this representation yields a local geometric multigrid method in which frozen shadows provide inter-level boundary data and carry residual contributions to coarser levels. A primal--dual pairing eliminates shadow assembly from the Krylov iteration. Numerical experiments with cubic elements ($p=3$) on an NVIDIA A100 show stable multigrid convergence under increasing refinement depth and efficient GPU execution.