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An efficient $hp$-multigrid scheme is presented for local discontinuousGalerkin (LDG) discretizations of elliptic problems, formulated around the ideaof separately coarsening the underlying discrete gradient and divergenceoperators. We show that traditional multigrid coarsening of the primalformulation leads to poor and suboptimal multigrid performance, whereascoarsening of the flux formulation leads to optimal convergence and isequivalent to a purely geometric multigrid method. The resultingoperator-coarsening schemes do not require the entire mesh hierarchy to beexplicitly built, thereby obviating the need to compute quadrature rules,lifting operators, and other mesh-related quantities on coarse meshes. We showthat good multigrid convergence rates are achieved in a variety of numericaltests on 2D and 3D uniform and adaptive Cartesian grids, as well as for curveddomains using implicitly defined meshes and for multi-phase elliptic interfaceproblems with complex geometry. Extension to non-LDG discretizations is brieflydiscussed.

Efficient operator-coarsening multigrid schemes for local discontinuous Galerkin methods