Computational Multiscale Methods

Abstract

Numerical methods for multiscale parabolic and hyperbolic problems Assyr Abdulle In this report we summarize some recent developments of numerical homogenization methods for nonlinear parabolic equations and wave equations in heterogeneous medium. Monotone parabolic problems. Let Ω ⊂ R, d ≤ 3 be a convex polygonal domain. Consider for T > 0, f ∈ L(Ω) the following nonlinear parabolic problem (1) ∂tu e −∇ · (A(x,∇u)) = f in Ω× (0, T ), u|∂Ω×(0,T ) = 0, u|t=0 = g in Ω. Assumptions: • there exists C0 > 0 such that |Ae(x, 0)| ≤ C0 for a.e. x ∈ Ω; • Lipschitz continuity: |A(x, ξ)−A(x, η)| ≤ L|ξ − η| ∀ξ, η ∈ R, a.e. x ∈ Ω; • Strong monotonicity: [A(x, ξ)−A(x, η)] · (ξ − η) ≥ λ|ξ − η| ∀ξ, η ∈ R, a.e. x ∈ Ω. Under theses assumptions (1) has a unique solution u ∈ E, where E = {v ∈ L(0, T ;H 0(Ω)) | ∂tv ∈ L(0, T ;H(Ω))}, and {ue} is a bounded sequence in E which weakly converges (up to extracting a subsequence) to a function u ∈ E that is solution of a homogenized problem that takes a form similar to (1) with Ae(x,∇ue) replaced by A0(x,∇u0(x, t)) (see e.g. [10]). As A0(x,∇u0(x, t)) is usually not explicitly known in closed form, it must be approximated numerically. Multiscale methods. Let the time interval (0, T ) be uniformly divided into N subintervals of length ∆t = T/N and define tn = n∆t for 0 ≤ n ≤ N and N ∈ N>0. Let u0 ∈ S 0(Ω, TH) = {vH ∈ H 0 (Ω) | v |K ∈ P1(K), ∀K ∈ TH} be a given approximation of the initial condition g(x). Fully nonlinear method. Consider the multiscale method given by the recursion: for 0 ≤ n ≤ N − 1, find un+1 ∈ S 0(Ω, TH) such that