Domain decomposition preconditioners for higher-order discontinuous Galerkin discretizations

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, February 2012."September 2011." Cataloged from PDF version of thesis.Includes bibliographical references (p. 147-155).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.by Laslo Tibor Diosady.Ph.D

Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations

Hybridizable Discontinuous Galerkin (HDG) is an important family of methods, which combine the advantages of both Discontinuous Galerkin in terms of flexibility and standard finite elements in terms of accuracy and efficiency. The impact of this method is partly evidenced by the prolificacy of research work in this area. Weak Galerkin (WG) is a relatively newly proposed method by introducing weak functions and generalizing the differential operator for them. This method has also drawn remarkable interests from both numerical practitioners and analysts recently. HDG and WG are different but closely related. BDDC algorithms are developed for numerical solution of elliptic problems with both methods. We prove that the optimal condition number estimate for BDDC operators with standard finite element methods can be extended to the counterparts arising from the HDG and WG methods, which are nonconforming finite element methods. Numerical experiments are conducted to verify the theoretical analysis. Further, we propose BDDC algorithms for the saddle point system arising from the Stokes equations using both HDG and WG methods. By design of the preconditioner, the iterations are restricted to a benign subspace, which makes the BDDC operator effectively positive definite thus solvable by the conjugate gradient method. We prove that the algorithm is scalable in the number of subdomains with convergence rate only dependent on subdomain problem size. The condition number bound for the BDDC preconditioned Stokes system is the same as the optimal bound for the elliptic case. Numerical results confirm the theoretical analysis

Segregated Runge–Kutta time integration of convection-stabilized mixed finite element schemes for wall-unresolved LES of incompressible flows

In this work, we develop a high-performance numerical framework for the large eddy simulation (LES) of incompressible flows. The spatial discretization of the nonlinear system is carried out using mixed finite element (FE) schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint. These additional terms introduced at the discrete level have been proved to act as implicit LES models. In order to perform meaningful wall-unresolved simulations, we consider a weak imposition of the boundary conditions using a Nitsche’s-type scheme, where the tangential component penalty term is designed to act as a wall law. Next, segregated Runge–Kutta (SRK) schemes (recently proposed by the authors for laminar flow problems) are applied to the LES simulation of turbulent flows. By the introduction of a penalty term on the trace of the acceleration, these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms. SRK schemes are excellent for large-scale simulations, since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level, leading to two uncoupled systems. The pressure system is a Darcy-type problem that can easily be preconditioned using a traditional block-preconditioning scheme that only requires a Poisson solver. At the end, only coercive systems have to be solved, which can be effectively preconditioned by multilevel domain decomposition schemes, which are both optimal and scalable. The framework is applied to the Taylor–Green and turbulent channel flow benchmarks in order to prove the accuracy of the convection-stabilized mixed FEs as LES models and SRK time integrators. The scalability of the preconditioning techniques (in space only) has also been proven for one step of the SRK scheme for the Taylor–Green flow using uniform meshes. Moreover, a turbulent flow around a NACA profile is solved to show the applicability of the proposed algorithms for a realistic problem.Peer ReviewedPostprint (author's final draft

Local Fourier Analysis of Domain Decomposition and Multigrid Methods for High-Order Matrix-Free Finite Elements

High-order matrix-free finite element operators offer superior performance on modern high performance computing hardware when compared to assembled sparse matrices, both with respect to floating point operations needed for operator evaluation and the memory transfer needed for a matrix-vector product. However, high-order matrix-free operators require iterative solvers, such as Krylov subspace methods, and these methods converge slowly for ill-conditioned operators. Preconditioning techniques are needed to improve the convergence of these iterative solvers independent of problem size and resolution. P-multigrid and domain decomposition methods are particularly well suited for problems on unstructured meshes, but these methods can involve parameters that require careful tuning to ensure proper convergence. Local Fourier Analysis (LFA) of these preconditioners describes how these methods affect frequency modes in the error for the operator defined on an infinite or periodic domain and can provide sharp convergence estimates and parameter tuning while only requiring computation on a single representative element or macro-element patch. We develop LFA of high-order finite element operators, focusing on multigrid and domain decomposition preconditioning techniques. The LFA of p-multigrid is validated with numerical experiments, and we extend this LFA to reproduce previous work with h-multigrid by using macro-elements consisting of multiple low-order finite elements. We also develop LFA of the lumped and Dirichlet versions of Balancing Domain Decomposition by Constraints (BDDC) preconditioners for high-order finite elements. By using Fast Diagonalization Method approximate subdomain solvers, the increased setup costs for the Dirichlet BDDC preconditioner, relative to the lumped variant, can be substantially reduced, making Dirichlet BDDC an attractive preconditioner. We validate this work against previous numerical experiments and exactly reproduce previous work on the LFA of BDDC for subdomains with multiple low-order finite elements. Aggressive coarsening in p-multigrid is not supported by traditional polynomial smoothers, such as Chebyshev.Dirichlet BDDC can be used as a smoother for p-multigrid to target a wider range of frequency modes, which facilitates more aggressive coarsening. We provide LFA of p-multigrid with Dirichlet BDDC smoothing to demonstrate the suitability of this approach for preconditioning high-order matrix-free finite element operators.</p

Local Fourier analysis for saddle-point problems

The numerical solution of saddle-point problems has attracted considerable interest in recent years, due to their indefiniteness and often poor spectral properties that make efficient solution difficult. While much research already exists, developing efficient algorithms remains challenging. Researchers have applied finite-difference, finite element, and finite-volume approaches successfully to discretize saddle-point problems, and block preconditioners and monolithic multigrid methods have been proposed for the resulting systems. However, there is still much to understand. Magnetohydrodynamics (MHD) models the flow of a charged fluid, or plasma, in the presence of electromagnetic fields. Often, the discretization and linearization of MHD leads to a saddle-point system. We present vector-potential formulations of MHD and a theoretical analysis of the existence and uniqueness of solutions of both the continuum two-dimensional resistive MHD model and its discretization. Local Fourier analysis (LFA) is a commonly used tool for the analysis of multigrid and other multilevel algorithms. We first adapt LFA to analyse the properties of multigrid methods for both finite-difference and finite-element discretizations of the Stokes equations, leading to saddle-point systems. Monolithic multigrid methods, based on distributive, Braess-Sarazin, and Uzawa relaxation are discussed. From this LFA, optimal parameters are proposed for these multigrid solvers. Numerical experiments are presented to validate our theoretical results. A modified two-level LFA is proposed for high-order finite-element methods for the Lapalce problem, curing the failure of classical LFA smoothing analysis in this setting and providing a reliable way to estimate actual multigrid performance. Finally, we extend LFA to analyze the balancing domain decomposition by constraints (BDDC) algorithm, using a new choice of basis for the space of Fourier harmonics that greatly simplifies the application of LFA. Improved performance is obtained for some two- and three-level variants

Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds