Papers › Guaranteed upper-lower bounds on homogenized properties by FFT-based Galerkin method
Guaranteed upper-lower bounds on homogenized properties by FFT-based Galerkin method
Jaroslav Vondřejc, Jan Zeman, Ivo Marek
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
Guaranteed upper-lower bounds on homogenized coefficients, arising from the periodic cell problem, are calculated in a scalar elliptic setting. Our approach builds on the recent variational reformulation of the Moulinec-Suquet (1994) Fast Fourier Transform (FFT) homogenization scheme by Vond\v{r}ejc et al. (2014), which is based on the conforming Galerkin approximation with trigonometric polynomials. Upper-lower bounds are obtained by adjusting the primal-dual finite element framework developed independently by Dvo\v{r}\'{a}k (1993) and Wieckowski (1995) to the FFT-based Galerkin setting. We show that the discretization procedure differs for odd and non-odd number of grid points. Thanks to the Helmholtz decomposition inherited from the continuous formulation, the duality structure is fully preserved for the odd discretizations. In the latter case, a more complex primal-dual structure is observed due to presence of the trigonometric polynomials associated with the Nyquist frequencies. These theoretical findings are confirmed with numerical examples. To conclude, the main advantage of the FFT-based approach over conventional finite-element schemes is that the primal and the dual problems are treated on the same basis, and this property can be extended beyond the scalar elliptic setting.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections