Nonlinear three-dimensional magnetic field computations using Lagrange finite-element functions and algebraic multigrid

Abstract

The paper proposes an efficient solution strategy for nonlinear three-dimensional (3-D) magnetic field problems. The spatial discretization of Maxwell's equations uses Lagrange finite-element functions. The paper shows that this discretization is appropriate for the problem class. The nonlinear equation is linearized by the standard fixed-point scheme. The arising sequence of symmetric positive definite matrices is solved by a preconditioned conjugate gradient method, preconditioned by an algebraic multigrid technique. Because of the relatively high setup time of algebraic multigrid, the preconditioner is kept constant as long as possible in order to minimize the overall CPU time. A practical control mechanism keeps the condition number of the overall preconditioned system as small as possible and reduces the total computational costs in terms of CPU time. Numerical studies involving the TEAM 20 and the TEAM 27 problem demonstrate the efficiency of the proposed technique. For comparison, the standard incomplete Cholesky preconditioner is used.