Double Grid Integration with Projection (DoGIP): Application to numerical homogenization by Fourier‐Galerkin method

Abstract

In this contribution, the DoGIP approach is introduced as a method for decomposition of a linear system into a (block) diagonal matrix based on a double grid integration and an interpolation/projection operator that is never assembled but optimised for fast matrix-vector multiplication. The method reduces memory requirements, especially when higher order basis functions are used for discretisations. The method is explained for Fourier-Galerkin method within a numerical homogenisation.

Double-grid quadrature with interpolation-projection (DoGIP) as a novel discretisation approach: An application to FEM on simplexes

“…Those matrices are not stored because the required matrix-vector multiplication can be substituted with an efficient numerical routine. Particularly, the interpolation-projection matrix in the Fourier-Galerkin method is evaluated very efficiently with the fast Fourier transform using O(N log N ) operations [2,3]. In case of FEM, only the interpolation on the reference element is needed.…”