Timotheus Boonen

Algebraic multigrid for stationary and time‐dependent partial differential equations with stochastic coefficients

We consider the numerical solution of time-dependent partial differential equations with random coefficients. A spectral approach, called stochastic finite element method, is used to compute the statistical characteristics of the solution. This method transforms a stochastic partial differential equation into a coupled system of deterministic equations by means of a Galerkin projection onto a generalized polynomial chaos. An algebraic multigrid method is presented to solve the algebraic systems that result after discretization of this coupled system. High-order time integration schemes of an implicit Runge-Kutta type, and spatial discretization on unstructured finite element meshes are considered. The convergence properties of the algebraic multigrid method are demonstrated by a convergence analysis and by numerical tests.

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

On algebraic multigrid methods derived from partition of unity nodal prolongators

H

Algebraic multigrid for implicit Runge-Kutta discretizations of the eddy current problem

(curl) and

Optimised electromagnetic 3d field solver for frequencies below the first resonance

H

An algebraic multigrid method for high order time-discretizations of the div-grad and the curl-curl equations

(div) and to other types of regular meshes.

An Optimized Electromagnetic 3D Field Solver for Frequencies Below the First Resonance

This paper is concerned with algebraic multigrid for finite element discretizations of the divgrad, curlcurl and graddiv equations on tetrahedral meshes with piecewise linear shape functions. First, an edge, face and volume prolongator are derived from an arbitrary partition of unity nodal prolongator for a tetrahedral fine mesh, using the formulas for edge, face and volume elements. This procedure can be repeated recursively. The implied coarse topology and the normalization of the prolongators are analysed. It is proved that the range spaces of the nodal prolongator and of the derived edge, face and volume prolongators form a discrete de Rham complex if these prolongators have full rank. It is shown that on simplicial meshes, the constructed edge prolongator is a generalization of the Reitzinger–Schoberl prolongator. The derived edge and face prolongators are applied in an algebraic multigrid method for the curlcurl and graddiv equations, and numerical results are presented. Copyright

A hybrid Picard-Newton Acceleration Scheme for Non-Linear Time-Harmonic Problems

In this paper, we present an algebraic multigrid algorithm for fully coupled implicit Runge-Kutta (IRK) time discretizations of the eddy current problem. The algorithm uses a blocksmoother. By a theoretical analysis and numerical experiments, we show that the convergence is similar to the convergence of the scalar algebraic multigrid algorithm on which it is based