Energy-based comparison between the Fourier--Galerkin method and the finite element method

TL;DR

This paper compares the Fourier-Galerkin method and finite element method in numerical homogenisation, focusing on accuracy, computational efficiency, and the influence of material coefficient regularity, revealing strengths and limitations of each approach.

Contribution

It provides a systematic energetic norm-based comparison of FFTH and FEM, highlighting their dependence on solution regularity and material properties, and clarifying their respective advantages.

Findings

FEM outperforms FFTH for problems with discontinuous material coefficients.

FFTH has better conditioning of the linear system regardless of degrees of freedom.

More degrees of freedom are generally needed by FFTH to achieve similar accuracy.

Abstract

The Fourier-Galerkin method (in short FFTH) has gained popularity in numerical homogenisation because it can treat problems with a huge number of degrees of freedom. Because the method incorporates the fast Fourier transform (FFT) in the linear solver, it is believed to provide an improvement in computational and memory requirements compared to the conventional finite element method (FEM). Here, we systematically compare these two methods using the energetic norm of local fields, which has the clear physical interpretation as being the error in the homogenised properties. This enables the comparison of memory and computational requirements at the same level of approximation accuracy. We show that the methods' effectiveness relies on the smoothness (regularity) of the solution and thus on the material coefficients. Thanks to its approximation properties, FEM outperforms FFTH for problems with jumps in material coefficients, while ambivalent results are observed for the case that the material coefficients vary continuously in space. FFTH profits from a good conditioning of the linear system, independent of the number of degrees of freedom, but generally needs more degrees of freedom to reach the same approximation accuracy. More studies are needed for other FFT-based schemes, non-linear problems, and dual problems (which require special treatment in FEM but not in FFTH).