Extended local fourier analysis for multigrid optimal smoothing, coarse grid correction, and preconditioning

Multigrid methods are fast iterative solvers for partial di erential equations. Especially for elliptic equations they have been proven to be highly e cient. For problems with nonelliptic and nonsymmetric features--as they often occur in typical real life applications--a rigorous mathematical theory is generally not available. For such situations Fourier smoothing and two-grid analysis can be considered as the main analysis tools to obtain quantitative convergence estimates and to optimize different multigrid components like smoothers or inter-grid transfer operators. In general, it is difficult to choose the correct multigrid components for large classes of problems. A popular alternative to construct a robust solver is the use of multigrid as a preconditioner for a Krylov subspace acceleration method like GMRES. Our contributions to the Fourier analysis for multigrid are two-fold. Firstly we extend the range of situations for which the Fourier analysis can be applied. More precisely, the Fourier analysis is generalized to k-grid cycles and to multigrid as a preconditioner. With a k-grid analysis it is possible to investigate real multigrid effects which cannot be captured by the classical two-grid analysis. Moreover, the k-grid analysis allows for a more detailed investigation of possible coarse grid correction difficulties. Additional valuable insight is obtained by evaluating multigrid as a preconditioner for GMRES. Secondly we extend the range of discretizations and multigrid components for which detailed Fourier analysis results exist. We consider four well-known singularly perturbed model problems to demonstrate the usefulness of the above generalizations: The anisotropic Poisson equation, the rotated anisotropic diffusion equation, the convection diffusion equation with dominant convection, and the driven cavity problem governed by the incompressible Navier Stokes equations. Each of these equations represents a larger class of problems with similar features and complications which are of practical relevance. With the help of the newly developed Fourier analysis methods, a comprehensive study of characteristic difficulties for singular perturbation problems can be performed. Based on the insights from this analysis it is possible to identify remedies resulting in an improved multigrid convergence. The theoretical considerations are validated by numerical test calculations

Local Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems

In this paper we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of Shifted Laplacian preconditioners are known to significantly speed-up Krylov convergence. However, these preconditioners have a parameter beta, a measure of the complex shift. Due to contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal complex shift parameter which is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed complex shift is both the minimal requirement for a multigrid V-cycle to converge, as well as being near-optimal in terms of Krylov iteration count.Comment: 20 page

HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I. Multilevel Analysis

The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space-(time) discontinuous Galerkin discretizations of advection dominated flows is presented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge-Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space-time discontinuous Galerkin discretization of the advection-diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for low and high cell Reynolds numbers and on highly stretched meshes

Multigrid optimization for space-time discontinuous Galerkin discretizations of advection dominated flows

The goal of this research is to optimize multigrid methods for higher order accurate space-time discontinuous Galerkin discretizations. The main analysis tool is discrete Fourier analysis of two- and three-level multigrid algorithms. This gives the spectral radius of the error transformation operator which predicts the asymptotic rate of convergence of the multigrid algorithm. In the optimization process we therefore choose to minimize the spectral radius of the error transformation operator. We specifically consider optimizing h-multigrid methods with explicit Runge-Kutta type smoothers for second and third order accurate space-time discontinuous Galerkin finite element discretizations of the 2D advection-diffusion equation. The optimized schemes are compared with current h-multigrid techniques employing Runge-Kutta type smoothers. Also, the efficiency of h-, p- and hp-multigrid methods for solving the Euler equations of gas dynamics with a higher order accurate space-time DG method is investigated

HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II. Optimization of the Runge-Kutta smoother

Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge-Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge-Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space-time discontinuous Galerkin finite element discretization of the advection-diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained