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

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, an enhanced successor to the standard discrete Green preconditioner that preserves the quasilinear complexity, $\mathcal{O}(N \log N)$, of conventional FFT-based solvers. Through numerical experiments, we demonstrate the effectiveness of the Jacobi-accelerated FFT (J-FFT) solver within a linear elastic framework. For problems characterized by smooth data and high material contrast, J-FFT significantly reduces the iteration count of the conjugate gradient method compared to the standard Green preconditioner. These findings are particularly relevant for phase-field fracture simulations, density-based topology optimization, and solvers that use adaption of the grid, which all introduce smooth variations in the material properties that challenge conventional FFT-based solvers.