A Newton-Raphson algorithm with adaptive accuracy control based on a block-preconditioned conjugate gradient technique

A fast Newton-Raphson algorithm is developed for the finite-element solution of nonlinear boundary value problems. The linearized equation systems in the nonlinear iteration steps are solved by a block-preconditioned conjugate gradient (CG) technique, in which the stopping criterion of the CG iteration is adaptively controlled by the nonlinear residual error. The Jacobian matrix is partitioned into linear and nonlinear blocks, thereby allowing the relatively rapid generation of an efficient multiplicative preconditioner for the CG iteration. The results obtained for a magnetostatic and a coupled steady conduction-static thermal problem confirm the effectiveness of the algorithm.