When used to accelerate the convergence of fixed-point iterative methods, such as the Picard method, which is a kind of nonlinear fixed-point iteration, polynomial extrapolation techniques can be very effective. The numerical solution of nonlinear problems is further investigated in this study. Particularly, using multigrid with isogeometric analysis as a linear solver of the Picard iterative method, which is accelerated by applying vector extrapolation techniques, is how we address the nonlinear eigenvalue Bratu problem and the Monge-Amp\`ere equation. This paper provides quadratic convergence results for polynomial extrapolation methods. Specifically, a new theoretical result on the correlation between the residual norm and the error norm, as well as a new estimation for the generalized residual norm of some extrapolation methods, are given. We perform an investigation between the Picard method, the Picard method accelerated by polynomial extrapolation techniques, and the Anderson accelerated Picard method. Several numerical experiments show that the Picard method accelerated by polynomial extrapolation techniques can solve these nonlinear problems efficiently.

翻译：当用于加速诸如Picard方法（一种非线性不动点迭代）等不动点迭代方法的收敛时，多项式外推技术可以非常有效。本研究进一步探讨了非线性问题的数值求解。具体而言，我们通过将多重网格与等几何分析相结合，作为应用向量外推技术加速的Picard迭代法的线性求解器，来处理非线性特征值Bratu问题和Monge-Ampère方程。本文为多项式外推方法提供了二次收敛结果。具体来说，给出了关于残差范数与误差范数之间相关性的新理论结果，以及对某些外推方法的广义残差范数的新估计。我们对Picard方法、经多项式外推技术加速的Picard方法以及Anderson加速的Picard方法进行了比较研究。多项数值实验表明，经多项式外推技术加速的Picard方法能够高效求解这些非线性问题。