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

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.