A novel upscaling method with arbitrarily high convergence rate for the modeling error in numerical homogenization.

Guest speaker:

Edoardo Paganoni, PhD Student
Swiss Federal Institute of Technology Lausanne

Abstract:

Many multiscale problems in physics and engineering cannot be solved by direct numerical simulations, due to the huge amount of degrees of freedom that are necessary to capture all the scales in space and time. On the other hand, for most of the applications only the macroscopic behavior of the solution is of practical interest. Therefore, multiscale numerical methods (such as numerical homogenization) have been designed to capture the macroscopic behavior of multiscale differential problems without resolving the full model.
Numerical homogenization methods aim at approximating the so-called homogenized model derived by solving the correctors equations over local representative volume elements (RVEs) of size δ~ε, where ε is the finest oscillation length. A modeling error, proportional to ε/δ, occurs when the period of the heterogeneities is not perfectly captured, or if the medium is not periodic (e.g., random). In many applications, the modelling error dominates other discretization errors, but reducing it by enlarging the RVE size δ can be computationally very expensive. Hence, designing novel micro-macro coupling techniques that can reach higher orders of convergence for the modeling error is a practical need, especially when stochastic multiscale materials are investigated.

We propose a novel modified corrector equation to compute the homogenized coefficients of second order elliptic partial differential equations with oscillating coefficients. Our approach differs from the standard one due to the presence of an additional forcing term that depends on a time-dependent diffusion equation. Such a forcing term can be efficiently precomputed by calculating the exponential function of a low-rank approximation of the stiffness matrix, obtained by the Arnoldi iteration. An exponential convergence of the resonance error w.r.t. δ/ε can be proved by means of Green's function estimates, under certain modeling assumptions. The convergence result is proved rigorously in the smooth, periodic setting, but numerical experiments show that it still holds true in the case of discontinuous and even stochastic coefficients.

Bio: 

I am a PhD student in the Chair of “Numerical Analysis and Computational Mathematics” at the École Polytechnique Fédérale de Lausanne (EPFL), which I joined in 2016. Previously, I received a BSc in Mathematics for engineering from the Polytechnic University of Turin (PoliTo) in 2013, and a joint MSc in Mathematical Engineering from PoliTo and the Technical University of Eindhoven (TU/e) in 2015. In Eindhoven, I did an internship in the Department of Population Health at Philips Research, where I worked on my master thesis “Blood coagulation: modeling, analysis and simulation of clot formation”.

Reference: Assyr Abdulle, Doghonay Arjmand, Edoardo Paganoni, “Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems”, Comptes Rendus Mathematique, Volume 357, Issue 6, 2019, Pages 545-551, https://doi.org/10.1016/j.crma.2019.05.011