Polynomial Extrapolation Techniques for Accelerating Picard Iterations With Multigrid Solvers for Nonlinear Problems in Isogeometric Analysis

ABSTRACT

Vector extrapolation techniques can be highly effective when used to accelerate the convergence of fixed‐point iterative methods, such as the Picard method. In this study, we investigate the numerical solution of nonlinear problems. Specifically, we address the nonlinear eigenvalue Bratu problem and the Monge–Ampère equation using a multigrid solver with isogeometric analysis within a Picard iterative framework, accelerated by vector extrapolation techniques. The novel theoretical contribution of this work is a second‐order bound on the generalized residual generated by the reduced rank extrapolation (RRE) method in terms of the current error. This bound is established without the additional nondegeneracy condition required by the recalled quadratic‐convergence result. However, it does not establish quadratic convergence of the extrapolated iterates. Finally, we conduct a comparative study of the standard Picard method, the Anderson‐accelerated Picard method, and the Picard method accelerated by polynomial extrapolation techniques. Several numerical experiments demonstrate that the proposed approach efficiently solves these nonlinear problems.