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

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 compare the two methods using the energetic norm of local fields, which has the clear physical interpretation as being the error in the homogenized properties. This enables the comparison of memory and computational requirements at the same level of approximation accuracy. We show that the methods' effectiveness rely 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 linear system, independent of the number of degrees of freedom, but generally needs more degrees of freedom. More studies are needed for other FFT-based schemes, non-linear problems, and dual problems (which require special treatment in FEM). keywords:energy-based comparison, finite element method, Fourier-Galerkin method, fast Fourier transform, numerical homogenisation, computational effectiveness arXiv:1709.08477v1 [math.NA]

Effective Electronic and Ionic Conductivities of Dense EV-Designed NMC-Based Positive Electrodes using Fourier Based Numerical Simulations on FIB/SEM Volumes

“…The convolution product is advantageously computed in Fourier space, whereas the local constitutive laws are evaluated in the direct space, in regularly-spaced voxel grids [52] allowing for the use of complex microstructure derived from microtomography images [53]. Recent works have focused on reinterpreting the Fourier-Lippmann-Shwinger method in the context of Galerkin discretization and weak-form formulations of the mechanical or conductivity-type problems [54], establishing links between Fourier and Finite Element methods [55,56,57], and introducing new numerical schemes inspired by those employed in minimization problems such as the conjugate gradient or Barzilai and Borwein methods [58], to name a few. Studies have highlighted the improvement achieved by Fourier methods in terms of computational and memory requirements over finite element techniques [59,57].…”

“…Recent works have focused on reinterpreting the Fourier-Lippmann-Shwinger method in the context of Galerkin discretization and weak-form formulations of the mechanical or conductivity-type problems [54], establishing links between Fourier and Finite Element methods [55,56,57], and introducing new numerical schemes inspired by those employed in minimization problems such as the conjugate gradient or Barzilai and Borwein methods [58], to name a few. Studies have highlighted the improvement achieved by Fourier methods in terms of computational and memory requirements over finite element techniques [59,57]. Fourier-based methods, it has been argued, benefit from a better conditioning of the linear system provided by the Lippmann-Schwinger equation.…”

“…Fourier-based methods, it has been argued, benefit from a better conditioning of the linear system provided by the Lippmann-Schwinger equation. In certain situations, Finite element methods, however, require a lower number of degrees of freedom to achieve comparable accuracy [57].…”

A review of nonlinear FFT-based computational homogenization methods

“…Last but not least, we refer to a number of works [58][59][60], where a comparison to finite element methods was recorded.…”

FFT-based homogenisation accelerated by low-rank tensor approximations

“…This is chosen because the error in the homogenised properties corresponds to the square of energetic semi-norm (norm on zero-mean fields) of the algebraic error between the full solution and the low-rank approximation u N − u N ,r 2 A = a ∇u N − ∇u N ,r , ∇u N − ∇u N ,r = A H,N ,r − A H,N ; for the derivation see [54,Appendix D]. We also note that the full solution u N has been computed using conjugate gradients with high accuracy (tolerance 10 −10 on the norm of the residuum) to obtain a solution that is close to the exact one.…”