Algebraic Multigrid for 3D Magnetic Field Problems

Stefan Reitzinger and Joachim Schöberl

Abstract:

In this talk we present a new algebraic multigrid method for the efficient solution of the linear system arising from a finite element discretization of variational equations in H₀(rot,Omega). The finite element discretization is done by Nédélec-elements (Whitney-1-forms or further referenced to as edge elements).

An appropriate coarsening technique is presented in order to construct suitable coarse spaces and according grid transfer operators. The prolongation operator is designed such that coarse grid kernel functions of the rot-operator are mapped to fine grid kernel functions. Furthermore, coarse grid rot-free functions are discrete gradients.

The smoothers by Hiptmair and Arnold/Falk/Winther for H₀(rot,Omega) variational problems can be used directly in the algebraic framework.

Collecting the ingredients (coarsening strategy, grid transfer operators, smoother) we end up with an algebraic multigrid method for the considered problem class. Numerical studies are presented in order to show the efficiency of the proposed technique.