Scalable DPG multigrid solver with applications in high-frequency wave propagation

Badger, Jacob

doi:10.26153/tsw/55685

Wave propagation is fundamental to applications including natural resource exploration, nuclear fusion research, and military defense, among others. However, developing accurate and efficient numerical algorithms for solving time-harmonic wave propagation problems is notoriously difficult. One difficulty is that classical discretization techniques (e.g., Galerkin finite elements, finite difference, etc.) yield indefinite discrete systems that preclude the use of many scalable solution algorithms. Significant progress has been made to develop specialized preconditioners for high-frequency wave propagation problems but robust and scalable solvers for general problems, including non-homogenous media and complex geometries, remain elusive. An alternative approach is to use minimum residual discretization methods—that yield Hermitian positive-definite discrete systems—and may be amenable to more standard preconditioners. Indeed, popularization of the first-order system least-squares methodology (FOSLS) was driven by the applicability of geometric and algebraic multigrid to otherwise indefinite problems. However, for wave propagation problems, FOSLS is known to be highly dissipative and is thus less competitive in the high-frequency regime. The discontinuous Petrov–Galerkin (DPG) method of Demkowicz and Gopalakrishnan is a minimum residual finite element method with several additional attractive properties: mesh-independent stability, a built-in error indicator, and applicability to a number of variational formulations. In the context of high-frequency wave propagation, the ultraweak DPG formulation has been observed to produce pollution error roughly commensurate to Galerkin discretizations. DPG discretizations may thus deliver accuracy typical of classical discretization techniques, but result in Hermitian positive-definite discrete systems that are often more amenable to preconditioning. A multigrid preconditioner for DPG systems, developed in the dissertation work of S. Petrides, was shown to scale efficiently in a shared-memory implementation. The primary objective of this dissertation is development of an efficient, distributed implementation of the DPG multigrid solver (DPG-MG). The distributed DPG-MG solver developed in this work will be demonstrated to be massively scalable, enabling solution of three-dimensional problems with O(10¹²) degrees of freedom on up to 460 000 CPU cores, an unprecedented scale for high-frequency wave propagation. The scalability of the DPG-MG solver will be further combined with hp-adaptivity to enable efficient solution of challenging real-world high-frequency wave propagation problems including optical fiber modeling, simulation of RF heating in tokamak devices, and seismic simulation. These applications include complex three-dimensional geometries, heterogeneous and anisotropic media, and localized features; demonstrating the robustness and versatility of the solver and tools developed in this dissertation.

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