Domain decomposition preconditioners for higher-order discontinuous Galerkin discretizations

Abstract

Aerodynamic flows involve features with a wide range of spatial and temporal scales which need to be resolved in order to accurately predict desired engineering quantities. While computational fluid dynamics (CFD) has advanced considerably in the past 30 years, the desire to perform more complex, higher-fidelity simulations remains. Present day CFD simulations are limited by the lack of an efficient high-fidelity solver able to take advantage of the massively parallel architectures of modern day supercomputers. A higher-order hybridizable discontinuous Galerkin (HDG) discretization combined with an implicit solution method is proposed as a means to attain engineering accuracy at lower computational cost. Domain decomposition methods are studied for the parallel solution of the linear system arising at each iteration of the implicit scheme. A minimum overlapping additive Schwarz (ASM) preconditioner and a Balancing Domain Decomposition by Constraints (BDDC) preconditioner are developed for the HDG discretization. An algebraic coarse space for the ASM preconditioner is developed based on the solution of local harmonic problems. The BDDC preconditioner is proven to converge at a rate independent of the number of subdomains and only weakly dependent on the solution order or the number of elements per subdomain for a second-order elliptic problem. The BDDC preconditioner is extended to the solution of convection-dominated problems using a Robin-Robin interface condition. An inexact BDDC preconditioner is developed based on incomplete factorizations and a p-multigrid type coarse grid correction. It is shown that the incomplete factorization of the singular linear systems corresponding to local Neumann problems results in a nonsingular preconditioner. The inexact BDDC preconditioner converges in a similar number of iterations as the exact BDDC method, with significantly reduced CPU time. The domain decomposition preconditioners are extended to solve the Euler and Navier- Stokes systems of equations. An analysis is performed to determine the effect of boundary conditions on the convergence of domain decomposition methods. Optimized Robin-Robin interface conditions are derived for the BDDC preconditioner which significantly improve the performance relative to the standard Robin-Robin interface conditions. Both ASM and BDDC preconditioners are applied to solve several fundamental aerodynamic flows. Numerical results demonstrate that for high-Reynolds number flows, solved on anisotropic meshes, a coarse space is necessary in order to obtain good performance on more than 100 processors.