Vol. 14, No. 2, 2019

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

Robert I. Saye

Vol. 14 (2019), No. 2, 247–283
Abstract

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 h.

Keywords
elliptic interface problems, multigrid methods, local discontinuous Galerkin methods, implicitly defined meshes, harmonic weights, viscosity-upwinded weighting, operator coarsening
Mathematical Subject Classification 2010
Primary: 65F08, 65N30, 65N55
Milestones
Received: 11 July 2019
Accepted: 5 November 2019
Published: 31 December 2019
Authors
Robert I. Saye
Mathematics Group
Lawrence Berkeley National Laboratory
Berkeley, CA
United States