GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Solving large nonlinear systems of equations by an adaptive condensation process
Numerische Mathematik
Hybrid Krylov methods for nonlinear systems of equations
SIAM Journal on Scientific and Statistical Computing
Stabilization of unstable procedures: the recursive projection method
SIAM Journal on Numerical Analysis
The superlinear convergence behaviour of GMRES
Journal of Computational and Applied Mathematics
SIAM Journal on Scientific Computing
Restarted GMRES preconditioned by deflation
Journal of Computational and Applied Mathematics
Parallel Computing - Special issue on applications: parallel computing methods in applied fluid mechanics
NITSOL: A Newton Iterative Solver for Nonlinear Systems
SIAM Journal on Scientific Computing
Accelerated Inexact Newton Schemes for Large Systems of Nonlinear Equations
SIAM Journal on Scientific Computing
An Adaptive Newton--Picard Algorithm with Subspace Iteration for Computing Periodic Solutions
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Nonlinearly Preconditioned Inexact Newton Algorithms
SIAM Journal on Scientific Computing
Tensor-GMRES Method for Large Systems of Nonlinear Equations
SIAM Journal on Optimization
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
BiCGStab, VPAStab and an adaptation to mildly nonlinear systems
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
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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.