Jacobi-accelerated FFT-based solver for smooth high-contrast data

Abstract

The computational efficiency and rapid convergence of fast Fourier transform (FFT)-based solvers render them a powerful numerical tool for periodic cell problems in multiscale modeling. On regular grids, they tend to outperform traditional numerical methods. However, we show that their convergence slows down significantly when applied to microstructures with smooth, highly-contrasted coefficients. To address this loss of performance, we introduce a Green-Jacobi preconditioner, a complementary extension to the standard discrete Green preconditioner that preserves the quasilinear complexity, O(N log N ), of conventional FFT-based solvers. Through numerical experiments in both linear and nonlinear elastic frameworks, we demonstrate the effectiveness of the Jacobi-accelerated FFT (J-FFT) solver. The main disadvantage of the J-FFT solver is the loss of the mesh-size independence typical of standard FFT-based solvers, as its iteration count grows with mesh refinement. Nevertheless, for problems characterized by smooth data and high material contrast, J-FFT has the potential to significantly reduce the iteration count of the conjugate gradient method compared to the standard Green preconditioner for problem sizes of practical importance. These findings are particularly relevant for densitybased topology optimization, solvers that use adaptation of the grid, or nonlinear material laws, which all introduce smooth variations in the effective material properties that challenge conventional FFT-based solvers.

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BibTeX RIS

BibTeX

@article{ladecky2026jacobiaccelerated,
  title = {{Jacobi-accelerated FFT-based solver for smooth high-contrast data}},
  author = {Martin Ladecký and Ivana Pultarová and François Bignonnet and Indre Jödicke and Jan Zeman and Lars Pastewka},
  journal = {Computer Methods in Applied Mechanics and Engineering},
  year = {2026},
  month = aug,
  doi = {10.1016/j.cma.2026.119269},
  eprint = {2508.02613},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05743015v1},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - Jacobi-accelerated FFT-based solver for smooth high-contrast data
AU  - Martin Ladecký
AU  - Ivana Pultarová
AU  - François Bignonnet
AU  - Indre Jödicke
AU  - Jan Zeman
AU  - Lars Pastewka
PY  - 2026
DA  - 2026/08/06
JO  - Computer Methods in Applied Mechanics and Engineering
DO  - 10.1016/j.cma.2026.119269
UR  - https://hal.science/hal-05743015v1
AB  - The computational efficiency and rapid convergence of fast Fourier transform (FFT)-based solvers render them a powerful numerical tool for periodic cell problems in multiscale modeling. On regular grids, they tend to outperform traditional numerical methods. However, we show that their convergence slows down significantly when applied to microstructures with smooth, highly-contrasted coefficients. To address this loss of performance, we introduce a Green-Jacobi preconditioner, a complementary extension to the standard discrete Green preconditioner that preserves the quasilinear complexity, O(N log N ), of conventional FFT-based solvers. Through numerical experiments in both linear and nonlinear elastic frameworks, we demonstrate the effectiveness of the Jacobi-accelerated FFT (J-FFT) solver. The main disadvantage of the J-FFT solver is the loss of the mesh-size independence typical of standard FFT-based solvers, as its iteration count grows with mesh refinement. Nevertheless, for problems characterized by smooth data and high material contrast, J-FFT has the potential to significantly reduce the iteration count of the conjugate gradient method compared to the standard Green preconditioner for problem sizes of practical importance. These findings are particularly relevant for densitybased topology optimization, solvers that use adaptation of the grid, or nonlinear material laws, which all introduce smooth variations in the effective material properties that challenge conventional FFT-based solvers.
SN  - 0045-7825
SN  - 1879-2138
ER  -

Abstract

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