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Multigrid methods for isogeometric discretization.

K P S Gahalaut1, J K Kraus1, S K Tomar1

  • 1Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040 Linz, Austria.

Computer Methods in Applied Mechanics and Engineering
|February 11, 2014
PubMed
Summary
This summary is machine-generated.

We developed geometric multigrid methods for isogeometric analysis of elliptic problems. These methods ensure uniform convergence, offering efficient solutions for complex engineering simulations.

Keywords:
B-splinesGalerkin formulationIsogeometric methodMultigrid methodNURBS

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Scientific Computing

Background:

  • Isogeometric analysis (IGA) offers advantages in representing complex geometries using spline-based functions.
  • Second-order elliptic problems are fundamental in various scientific and engineering domains, including fluid dynamics and structural mechanics.
  • Multigrid methods are highly efficient iterative solvers for large systems of linear equations arising from discretizations.

Purpose of the Study:

  • To introduce and analyze geometric multigrid (GMG) methods tailored for isogeometric discretization of scalar second-order elliptic problems.
  • To establish the theoretical foundation for the uniform convergence of these GMG methods.
  • To provide comprehensive numerical validation of the proposed methods.

Main Methods:

  • Analysis of the smoothing property of relaxation methods within the IGA framework.
  • Analysis of the approximation property of intergrid transfer operators for isogeometric spaces.
  • Application of classical multigrid theory to prove uniform convergence.
  • Implementation and testing of V-, W-, and F-cycles for numerical experiments.

Main Results:

  • Theoretical guarantees for uniform convergence of two-grid and multigrid methods are derived.
  • Numerical results confirm the smoothing and approximation properties.
  • Convergence factors and iteration counts are reported for various cycle types (V, W, F).
  • Linear dependence of V-cycle convergence on smoothing steps is demonstrated.

Conclusions:

  • The presented geometric multigrid methods are theoretically sound and numerically effective for isogeometric analysis of elliptic problems.
  • The methods demonstrate uniform convergence, crucial for efficient and reliable simulations.
  • The study provides a robust framework for solving complex engineering problems using IGA and multigrid techniques.