Uniform estimate of an iterative method for elliptic problems with rapidly oscillating coefficients
- PDF / 733,940 Bytes
- 32 Pages / 439.37 x 666.142 pts Page_size
- 19 Downloads / 183 Views
Uniform estimate of an iterative method for elliptic problems with rapidly oscillating coefficients Chenlin Gu1,2 Received: 25 January 2019 / Revised: 22 November 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We study the iterative algorithm proposed by Armstrong et al. (An iterative method for elliptic problems with rapidly oscillating coefficients, 2018. arXiv preprint arXiv:1803.03551) to solve elliptic equations in divergence form with stochastic stationary coefficients. Such equations display rapidly oscillating coefficients and thus usually require very expensive numerical calculations, while this iterative method is comparatively easy to compute. In this article, we strengthen the estimate for the contraction factor achieved by one iteration of the algorithm. We obtain an estimate that holds uniformly over the initial function in the iteration, and which grows only logarithmically with the size of the domain. Keywords Stochastic homogenization · Elliptic equation · Numerical algorithm · Iterative method
1 Introduction 1.1 Main theorem The problem of homogenization is a subject widely studied in mathematics and other disciplines for its applications and interesting properties. Let (a(x), x ∈ Rd ) be a random coefficient field, which takes values in the set of Rd×d symmetric matrices, and which we assume to be Zd -stationary, with a unit range of dependence and uniformly elliptic that −1 |ξ |2 ξ · a(x)ξ |ξ |2 for any x, ξ ∈ Rd . We give ourselves a bounded domain U ⊂ Rd with boundary C 1,1 , a scale parameter 0 < ε < 1, and for given f ∈ H −1 (U ) and g ∈ H 1 (U ), we consider the elliptic Dirichlet problem
B
Chenlin Gu [email protected]
1
Ecole Polytechnique, Palaiseau, France
2
DMA, Ecole Normale Supérieure, PSL Research University, Paris, France
123
Stoch PDE: Anal Comp
−∇ · a ε· ∇u ε = f uε = g
in
U,
on
∂U .
(1.1)
For the scale 0 < ε 1, a naive numerical algorithm for this problem is generally very expensive, due to the rapid oscillations of the coefficients (comparatively to the size of the domain) and we have to refine the mesh of the numerical schema. Thus, different methods have been proposed to approximate the solution and one of them is to replace the conductance matrix a by a constant effective conductance matrix a¯ in Eq. (1.1) and use its solution u¯ as an approximation, which can be solved quickly thanks to the multi-grid algorithm. However, u¯ is close to u ε in the sense L 2 (U ) or H −1 (U ), but not in some stronger topology, for example H 1 (U ). Furthermore, the approximation only becomes accurate in the limit ε → 0, but for a small finite scale ε, one can not expect a precision much smaller than ε with u. ¯ Recently, [3] proposed an iterative algorithm to solve the problem Eq. (1.2) efficiently for a given ε-scale and with a better precision. We recap at first their algorithm here with the same formulation in large scale: Instead of considering Eq. (1.1) with a small scale ε, we treat the Dirichlet problem with a dil
Data Loading...
