5,234 research outputs found
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
Discrete Fourier analysis of multigrid algorithms
The main topic of this report is a detailed discussion of the discrete Fourier multilevel analysis of multigrid algorithms. First, a brief overview of multigrid methods is given for discretizations of both linear and nonlinear partial differential equations. Special attention is given to the hp-Multigrid as Smoother algorithm, which is a new algorithm suitable for higher order accurate discontinuous Galerkin discretizations of advection dominated flows. In order to analyze the performance of the multigrid algorithms the error transformation operator for several linear multigrid algorithms are derived. The operator norm and spectral radius of the multigrid error transformation are then computed using discrete Fourier analysis. First, the main operations in the discrete Fourier analysis are defined, including the aliasing of modes. Next, the Fourier symbol of the multigrid operators is computed and used to obtain the Fourier symbol of the multigrid error transformation operator. In the multilevel analysis, two and three level h-multigrid, both for uniformly and semi-coarsened meshes, are considered, and also the analysis of the hp-Multigrid as Smoother algorithm for three polynomial levels and three uniformly and semi-coarsened meshes. The report concludes with a discussion of the multigrid operator norm and spectral radius. In the appendix some useful auxiliary results are summarized
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
On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients
In this paper, we propose a novel non-standard Local Fourier Analysis (LFA)
variant for accurately predicting the multigrid convergence of problems with
random and jumping coefficients. This LFA method is based on a specific basis
of the Fourier space rather than the commonly used Fourier modes. To show the
utility of this analysis, we consider, as an example, a simple cell-centered
multigrid method for solving a steady-state single phase flow problem in a
random porous medium. We successfully demonstrate the prediction capability of
the proposed LFA using a number of challenging benchmark problems. The
information provided by this analysis helps us to estimate a-priori the time
needed for solving certain uncertainty quantification problems by means of a
multigrid multilevel Monte Carlo method
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
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
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