Accurate Algebraic Error Estimation in FFT-based Computational Homogenization
Please login to view abstract download link
We combine an accurate error estimation technique for the conjugate gradient method [1] with an FFT-accelerated finite element solver for computational homogenization. Meurant et al. [1]’s approach provides inexpensive and reliable estimates of the energy norm of the error in each iteration of the conjugate gradient method. The only input parameters for Meurant et al. [1]’s error estimates are guaranteed bounds on the eigenvalues of the linear system. However, eigenvalue bounds are readily available for linear systems reconditioned with the discrete Green’s operator [2]. The energy norm of the error in the local fields is equal to the error in the homogenized properties [3]. This clear physical interpretation makes it an appropriate measure for a stopping criterion of the iterative scheme in homogenization. In this talk, we will present key theoretical aspects of this approach and demonstrate the benefits and limitations of our technique on diffusion problems. REFERENCES [1] Meurant, G. and Papež, J. and Tichý, P., Accurate error estimation in CG, Numer. aAlgorithms, Vol. 88, pp.1337-1359, 2021. [2] M. Ladecký and I. Pultarová and J. Zeman, Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method, Appl. Math., Vol. 66, pp.21-42,2021. [3] Vondřejc, J. and de Geus, T.W.J., Energy-based comparison between the Fourier–Galerkin method and the finite element method, J. Comput. Appl. Math., Vol. 374, pp.112585, 2020
