Accelerating an inexact Newton/GMRES scheme by subspace decomposition

A technique for accelerating inexact Newton schemes is presented for the solution of nonlinear systems of algebraic equations that is based on the so-called recursive projection method (RPM) and is built as a computational shell around a Newton/Krylov code. The method acts directly on the ‘outer’ Newton iteration and does not act as a preconditioner to accelerate the solution of the linear system, i.e. the ‘inner’ Krylov iteration. The advantage of this approach is that it reduces the number of Newton iterations that are needed for convergence while still performing inexpensive Krylov iterations for the solution of the linear system at each Newton step. The method can be applied in conjunction with a preconditioned or unpreconditioned Krylov iterative solver, serial or parallel, of any discretized physical model. In addition, it enables the extraction of the dominant eigenspace with the help of a low-dimensional Jacobian matrix that is formulated in the course of the iterations making the solution of a large-scale eigenproblem, unnecessary. The proposed approach is applied on the 2-d Bratu problem and the lid-driven cavity problem. The equations are discretized with the Galerkin/finite element method and the resulting nonlinear algebraic equation set is solved by Newton's method. At each Newton step the restarted generalized minimum residual or GMRES(m) procedure is implemented for the solution of the resulting linear equation set.