Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods

Abstract

An efficient hp-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discrete gradient and divergence operators. We show that traditional multigrid coarsening of the primal formulation leads to poor and suboptimal multigrid performance, whereas coarsening of the flux formulation leads to optimal convergence and is equivalent to a purely geometric multigrid method. The resulting operator-coarsening schemes do not require the entire mesh hierarchy to be explicitly built, thereby obviating the need to compute quadrature rules, lifting operators, and other mesh-related quantities on coarse meshes. We show that good multigrid convergence rates are achieved in a variety of numerical tests on 2D and 3D uniform and adaptive Cartesian grids, as well as for curved domains using implicitly defined meshes and for multi-phase elliptic interface problems with complex geometry. Extension to non-LDG discretizations is briefly discussed.

Efficient multigrid solution of elliptic interface problems using viscosity-upwinded local discontinuous Galerkin methods

“…Using similar methods as was shown in [29], one can show that this coarsemesh operator is equivalent to that which would be obtained if the coarse-mesh problem was explicitly discretized with LDG. In particular, the coarsened µweighted identity operator Ꮿ( µ ) effectively coarsens the influence of µ on the fine mesh to larger and larger elements throughout the hierarchy, consistently with performing an L 2 projection of µ multiplied by piecewise polynomial functions on the coarse meshes.…”

“…The multigrid algorithms used in this work follow the operator-coarsening schemes presented by Fortunato et al [29], except with two important modifications: (i) the methods are generalized to handle variable viscosity, and (ii) penalty parameters are halved in strength each level down the mesh hierarchy. (Further details on these modifications are provided shortly.)…”

“…Operator-coarsening multigrid. Here, a brief description of the multigrid algorithms is given; for further details and motivation, the reader is referred to [29]. The essential components of the multigrid methods are as follows:…”

“…Importantly, element agglomeration is only permitted between elements of the same phase -as such, the interface of an elliptic interface problem is sharply preserved throughout the entire multigrid hierarchy. (An example is shown in Figure 4.8 of [29]. )…”

“…In [29], operator-coarsening multigrid methods are derived for single-phase Poisson problems −∇ 2 u = f as follows. On the finest mesh, the LDG discretization results in the linear system (−DG + τ E)u = ‫ސ‬ V h f , where G is the discrete gradient operator, D = − adj(G) = −M −1 G T M is the discrete divergence operator, and E is a penalty stabilization operator.…”