Details

Autor(en) / Beteiligte

• Vondřejc, Jaroslav
• de Geus, Tom W.J.

Titel

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

Ist Teil von

• Journal of computational and applied mathematics, 2020-08, Vol.374, p.112585, Article 112585

Ort / Verlag

Elsevier B.V

Erscheinungsjahr

2020

Link zum Volltext

Quelle

Elsevier ScienceDirect Journals Complete

Beschreibungen/Notizen

• 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). •Fourier–Galerkin (FFTH) and finite element method (FEM) are newly compared with energy-based criterion.•FEM outperforms FFTH for rough data (jumps in material coefficients).•The FFTH benefits from good condition number allowing to avoid preconditioning.•FEM benefits from fast matrix–vector multiplication in iterative linear solvers.