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Enhancing multigrid solvers for isogeometric analysis of nonlinear problems using polynomial extrapolation techniques

Abdellatif Mouhssine, Ahmed Ratnani, Hassane Sadok

TL;DR

This paper provides quadratic convergence results for polynomial extrapolation methods and 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.

Abstract

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-Ampère 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.

Enhancing multigrid solvers for isogeometric analysis of nonlinear problems using polynomial extrapolation techniques

TL;DR

This paper provides quadratic convergence results for polynomial extrapolation methods and 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.

Abstract

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-Ampère 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.
Paper Structure (18 sections, 5 theorems, 100 equations, 10 figures, 3 tables, 6 algorithms)

This paper contains 18 sections, 5 theorems, 100 equations, 10 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Multigrid methods
  3. Description of multigrid algorithms
  4. B-splines $\&$ IGA
  5. Picard iterative method for nonlinear problems
  6. The Bratu problem
  7. The Monge-Ampère equation
  8. Fixed-point iterations
  9. Vector extrapolation methods
  10. Description of polynomial extrapolation methods
  11. Algorithms of polynomial extrapolation methods
  12. Quadratic convergence results
  13. The Anderson acceleration method
  14. Numerical experiments
  15. Numerical study of the nonlinear one-dimensional Bratu's equation
  16. ...and 3 more sections

Key Result

Proposition 1

Figures (10)

  • Figure 1: B-spline basis functions of order $p=1,2,3$
  • Figure 2: Convergence results of Picard-MG method for different values of spline degree $p$ and $\lambda$
  • Figure 3: Convergence results of the Picard method with and without extrapolation acceleration techniques for $\lambda=3.5$
  • Figure 4: Behavior of the Picard method with and without extrapolation acceleration for $\lambda=7$ on different grid sizes
  • Figure 5: $L_2$ error norm of the Picard iterative method with and without extrapolation acceleration for $\lambda=7$ on different grid sizes
  • ...and 5 more figures

Theorems & Definitions (14)

  • Definition 1: B-Splines using Cox-DeBoor Formula
  • Proposition 1: B-Splines properties
  • proof
  • Remark 1
  • Example 1
  • Theorem 2
  • proof
  • Remark 2
  • Theorem 3: R38
  • proof
  • ...and 4 more