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Vector Extrapolation Methods Applied To Geometric Multigrid Solvers For Isogeometric Analysis

Abdellatif Mouhssine, Ahmed Ratnani, Hassane Sadok

TL;DR

The paper addresses solving large SPD systems from elliptic PDEs discretized by isogeometric analysis, where standard multigrid convergence deteriorates with increasing spline degree $p$. It introduces a hybrid approach that couples geometric multigrid with polynomial vector extrapolation methods (RRE and MPE) to accelerate fixed-point iterations, yielding RRE-V-cycle and MPE-V-cycle solvers. Numerical experiments on 1D and 2D Poisson problems and a full elliptic PDE demonstrate that these extrapolated schemes achieve faster convergence and maintain robustness as $p$ grows, often reducing iteration counts and CPU time significantly. The results suggest the proposed extrapolation-enhanced MG methods can serve as efficient solvers and preconditioners for Galerkin-discretized elliptic problems in IGA, with potential extensions to nonlinear problems through multicorrector strategies.

Abstract

In the present work, we study how to develop an efficient solver for the fast resolution of large and sparse linear systems that occur while discretizing elliptic partial differential equations using isogeometric analysis. Our new approach combines vector extrapolation methods with geometric multigrid schemes. Using polynomial-type extrapolation methods to speed up the multigrid iterations is our main focus. Several numerical tests are given to demonstrate the efficiency of these polynomial extrapolation methods in improving multigrid solvers in the context of isogeometric analysis.

Vector Extrapolation Methods Applied To Geometric Multigrid Solvers For Isogeometric Analysis

TL;DR

The paper addresses solving large SPD systems from elliptic PDEs discretized by isogeometric analysis, where standard multigrid convergence deteriorates with increasing spline degree . It introduces a hybrid approach that couples geometric multigrid with polynomial vector extrapolation methods (RRE and MPE) to accelerate fixed-point iterations, yielding RRE-V-cycle and MPE-V-cycle solvers. Numerical experiments on 1D and 2D Poisson problems and a full elliptic PDE demonstrate that these extrapolated schemes achieve faster convergence and maintain robustness as grows, often reducing iteration counts and CPU time significantly. The results suggest the proposed extrapolation-enhanced MG methods can serve as efficient solvers and preconditioners for Galerkin-discretized elliptic problems in IGA, with potential extensions to nonlinear problems through multicorrector strategies.

Abstract

In the present work, we study how to develop an efficient solver for the fast resolution of large and sparse linear systems that occur while discretizing elliptic partial differential equations using isogeometric analysis. Our new approach combines vector extrapolation methods with geometric multigrid schemes. Using polynomial-type extrapolation methods to speed up the multigrid iterations is our main focus. Several numerical tests are given to demonstrate the efficiency of these polynomial extrapolation methods in improving multigrid solvers in the context of isogeometric analysis.
Paper Structure (15 sections, 3 theorems, 40 equations, 11 figures, 8 tables, 5 algorithms)

This paper contains 15 sections, 3 theorems, 40 equations, 11 figures, 8 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Multigrid methods for IGA
  3. Multigrid algorithms
  4. Multigrid: A fixed-point analysis
  5. B-splines $\&$ IGA
  6. Investigating the limitations of standard multigrid techniques
  7. Multigrid iterations
  8. Vector extrapolation methods
  9. Polynomial extrapolation algorithms
  10. Restarted extrapolation methods
  11. Numerical experiments
  12. The $1D$ Poisson problem
  13. The $2D$ Poisson equation model
  14. The Full elliptic partial differential equation
  15. Conclusions

Key Result

Proposition 1

We denote by $S_{sm}$ the iteration matrix of the smoother. Hence, the iteration matrix of the Two-Grid method is given by: and the Two-Grid sequence is as follows: where $c = S_{\text{sm}}^{\nu_2} P_{2h}^{h} (A^{2h})^{-1} R_{h}^{2h} b^h.$

Figures (11)

  • Figure 1: B-spline basis functions of order $p=1,2,3$
  • Figure 2: Performance of the V-cycle scheme in $1D$ with $N=64$
  • Figure 3: Performance of the W-cycle scheme in $1D$ with $N=64$
  • Figure 4: The norms of the residual of the $1D$ Poisson problem as a function of the iteration number
  • Figure 5: Error evolution of the $1D$ Poisson problem as a function of the iteration number
  • ...and 6 more figures

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Proposition 2: C25
  • Definition 1: B-Splines using Cox-DeBoor Formula
  • Proposition 3: B-Splines properties
  • proof