An iterative method for elliptic problems with rapidly oscillating coefficients

Scott Armstrong (1) , Antti Hannukainen (2) , Tuomo Kuusi (3) , Jean-Christophe Mourrat (4)

1 School of Biosciences
2 Department of Mathematics and Systems Analysis [Aalto]
3 Department of Mathematics [Aalto]
4 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address different length scales. However, we use here the homogenized equation on all scales larger than a fixed multiple of the scale of oscillation of the coefficients. While the performance of standard multigrid methods degrades rapidly under the regime of large scale separation that we consider here, we show an explicit estimate on the contraction factor of our method which is independent of the size of the domain. We also present numerical experiments which confirm the effectiveness of the method, with openly available source code.

Mots clés: Multiscale method; multigrid method; homogenization

Domaines: Mathématiques [math]

Scott Armstrong, Antti Hannukainen, Tuomo Kuusi, Jean-Christophe Mourrat. An iterative method for elliptic problems with rapidly oscillating coefficients. ESAIM: Mathematical Modelling and Numerical Analysis, 2021, 55 (1), pp.37-55. ⟨10.1051/m2an/2020080⟩. ⟨hal-03146216⟩