The Isogeometric Fast Fourier-based Diagonalization method

TL;DR

This paper introduces a novel Fast Fourier-based diagonalization preconditioner for isogeometric analysis that maintains robustness against grid size and spline degree while achieving near-linear computational complexity.

Contribution

It develops a new variant of the Fast Diagonalization method that leverages FFT and stable spline space splitting for efficient, robust preconditioning.

Findings

Preconditioner is robust to grid size and spline degree.

Computational complexity is nearly linear in degrees of freedom.

Method outperforms traditional approaches in efficiency.

Abstract

The construction of robust solvers for linear systems obtained from the discretization of partial differential equations using Isogeometric Analysis is challenging since the condition number of the system matrix not only grows with the reciprocal square of the grid size (for second order problems), but also exponentially with the spline degree. The Fast Diagonalization method allows the construction of a preconditioner that is robust both in grid size and spline degree. Although this method is efficient in practice, its computational complexity is superlinear in the number of degrees of freedom. In this work, we construct a variant of the Fast Diagonalization method that can exploit the Fast Fourier Transformation. Note that, because of boundary effects, a Fourier Transformation cannot diagonalize the overall problem. We circumvent this issue using a stable splitting of the spline space. The resulting preconditioner is still robust with respect to the grid size and spline degree and can be realized with a computational complexity that grows almost linearly with the number of degrees of freedom.