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Matrix-Free Inexact Preconditioning Techniques for Isogeometric Tensor-Product Discretizations.

Michał Ł Mika1, René R Hiemstra2, Dominik Schillinger1

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Summary
This summary is machine-generated.

We developed a new matrix-free preconditioning method for solving elliptic partial differential equations using isogeometric analysis. This approach significantly speeds up computations for problems like the Poisson equation.

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Area of Science:

  • Numerical Analysis
  • Computational Science
  • Scientific Computing

Background:

  • Elliptic partial differential equations (PDEs) are fundamental in modeling various physical phenomena.
  • Isogeometric Galerkin (IGG) methods using tensor-product spline spaces offer advantages in geometric representation and analysis.
  • Efficiently solving the large, sparse linear systems arising from IGG discretizations is computationally challenging.

Purpose of the Study:

  • To introduce a novel matrix-free inexact preconditioning strategy for IGG methods applied to elliptic PDEs.
  • To reduce the computational cost and improve the efficiency of solving these discretized systems.
  • To demonstrate the effectiveness and performance gains of the proposed method compared to existing techniques.

Main Methods:

  • A matrix-free inexact preconditioning strategy is proposed, approximating the discrete linear operator with Kronecker products.
  • An inner preconditioned conjugate gradient (PCG) solve approximates the action of the inverse operator.
  • The forward problem is solved using an inexact PCG method, leveraging the efficient Kronecker products for inner iterations.

Main Results:

  • The proposed method demonstrates robustness and effectiveness on test problems, including the Poisson equation and linear elasticity.
  • Significant performance gains are observed due to reduced iteration counts compared to fast diagonalization preconditioning.
  • The computational complexity of Kronecker matrix-vector products in the inner iteration is lower than in the forward problem.

Conclusions:

  • The developed matrix-free inexact preconditioning strategy offers a computationally efficient solution for elliptic PDEs discretized with IGG methods.
  • The approach provides substantial performance improvements, making it a valuable tool for complex simulations.
  • The method is implemented in an open-source Julia framework, promoting accessibility and further research.