With an emphasis on achieving ideal multigrid solver performance, this paper explores
the design of local discontinuous Galerkin schemes for multiphase elliptic interface
problems. In particular, for cases exhibiting coefficient discontinuities several orders
in magnitude, the role of viscosity-weighted numerical fluxes on interfacial mesh faces
is examined: findings support a known strategy of harmonic weighting, but also show
that further improvements can be made via a stronger kind of biasing, denoted herein
as viscosity-upwinded weighting. Applying this strategy, multigrid performance is
assessed for a variety of elliptic interface problems in 1D, 2D, and 3D, across 16 orders of
viscosity ratio. These include constant- and variable-coefficient problems, multiphase
checkerboard patterns, implicitly defined interfaces, and 3D problems with intricate
geometry. With the exception of a challenging case involving a lattice of vanishingly
small droplets, in all demonstrated examples the condition number of the multigrid
V-cycle preconditioned system has unit order magnitude, independent of the mesh
size
.
Keywords
elliptic interface problems, multigrid methods, local
discontinuous Galerkin methods, implicitly defined meshes,
harmonic weights, viscosity-upwinded weighting, operator
coarsening