Jan Van lent

Multigrid Methods for Implicit Runge-Kutta and Boundary Value Method Discretizations of Parabolic PDEs

Advanced time discretization schemes for stiff systems of ordinary differential equations (ODEs), such as implicit Runge--Kutta and boundary value methods, have many appealing properties. However, the resulting systems of equations can be quite large and expensive to solve. Many techniques, exploiting the structure of these systems, have been developed for general ODEs. For spatial discretizations of time-dependent partial differential equations (PDEs) these techniques are in general not sufficient and also the structure arising from spatial discretization has to be taken into consideration. We show here that for time-dependent parabolic problems, this can be done by multigrid methods, as in the stationary elliptic case. The key to this approach is the use of a smoother that updates several unknowns at a spatial grid point simultaneously. The overall cost is essentially proportional to the cost of integrating a scalar ODE for each grid point. Combination of the multigrid principle with both time stepping and waveform relaxation techniques is described, together with a convergence analysis. Numerical results are presented for the isotropic heat equation and a general diffusion equation with variable coefficients.

Local Fourier Analysis of Multigrid for the Curl-Curl Equation

We present a local Fourier analysis of multigrid methods for the two-dimensional curl-curl formulation of Maxwells equations. Both the hybrid smoother proposed by Hiptmair and the overlapping block smoother proposed by Arnold, Falk, and Winther are considered. The key to our approach is the identification of two-dimensional eigenspaces of the discrete curl-curl problem by decoupling the Fourier modes for edges with different orientations. This procedure is used to quantify the smoothing properties of the considered smoothers and the convergence behavior of the multigrid methods. Additionally, we identify the Helmholtz splitting in Fourier space. This allows several well known properties to be recovered in Fourier space, such as the commutation properties of the classical Nedelec prolongator and the equivalence of the curl-curl operator and the vector Laplacian for divergence-free vectors. We show how the approach used in this paper can be generalized to two- and three-dimensional problems in

Multigrid Waveform Relaxation for Anisotropic Partial Differential Equations

H

Waveform relaxation using spectral collocation in time

(curl) and

Multigrid waveform relaxation for delay partial differential equations

H

A unified framework based on operational calculus for the convergence analysis of waveform relaxation methods

(div) and to other types of regular meshes.

Algebraic multigrid for fully implicit Runge-Kutta discretizations of the time-dependent divgrad and curlcurl equations

Multigrid waveform relaxation provides fast iterative methods for the solution of time-dependent partial differential equations. In this paper we consider anisotropic problems and extend multigrid methods developed for the stationary elliptic case to waveform relaxation methods for the time-dependent parabolic case. We study line-relaxation, semicoarsening and multiple semicoarsening multilevel methods. A two-grid Fourier–Laplace analysis is used to estimate the convergence of these methods for the rotated anisotropic diffusion equation. We treat both continuous time and discrete time algorithms. The results of the analysis are confirmed by numerical experiments.