Ralph E. Showalter http://www.math.oregonstate.edu/people/view/show Department of Mathematics Oregon State University Multiscale Summer School, August, 2008 DOE 98089 Modeling, Analysis, and Simulation of Multiscale Preferential Flow with Małgorzata Peszyńska
Introduction Two-scale approximation The Homogenized problem Origins Homogenization is a collection of methods to approximate a heterogeneous problem by a homogeneous one. Example (Elliptic) A ɛ (u ɛ ) = f : (a ɛ ij (x)uɛ x i ) xj = f, where a ɛ ij (x) = a ij(x/ɛ), and a ij ( ) are 1-periodic inr n. Exact model at microscale replaced by homogenized model a ij (x/ɛ) with constant ã ij
Introduction Two-scale approximation The Homogenized problem First Issues Question Can we find a limiting problem, Au = f, whose solution u characterizes the limit, lim ɛ 0 u ɛ = u? Example (Elliptic) A(u) = f : (ã ij u xi ) xj = f, where ã ij are constant. This is the homogenized equation with effective coefficients. Question Is A of the same type as A ɛ? The answer is frequently No!.
Introduction Two-scale approximation The Homogenized problem An Elliptic Problem Let Y be the unit cube in R N. Assume a( ) is Y -periodic. We want to approximate the solution to the singular problem u ɛ H0 1 (Ω) : a(x/ɛ) u ɛ (x) ϕ(x) dx = Ω Ω F(x)ϕ(x) dx for all ϕ H 1 0 (Ω). (1)
Introduction Two-scale approximation The Homogenized problem The approximation We seek an approximation u ɛ (x) = u(x, x/ɛ) + ɛu(x, x/ɛ) + O(ɛ 2 ), x Ω, in which each u(x, y) and U(x, y) is Y -periodic. The gradient is given (formally) by u ɛ (x) = x u(x, x/ɛ) + 1 ɛ yu(x, x/ɛ) + y U(x, x/ɛ) + O(ɛ).
Introduction Two-scale approximation The Homogenized problem The solution u ɛ has a bounded gradient, so y u(x, y) = 0 and u ɛ (x) = u(x) + ɛu(x, x/ɛ) + O(ɛ 2 ) (2a) u ɛ (x) = u(x) + y U(x, x/ɛ) + O(ɛ) (2b) Substitute (2) into (1) with a test function of the same form ϕ(x) + ɛφ(x, x/ɛ) where Φ(x, y) is Y -periodic for each x Ω.
Introduction Two-scale approximation The Homogenized problem The variational form Two-scale limits ɛ 0 give the system u H0 1 (Ω), U L 2 (Ω, H#(Y 1 )) : a(y)( u(x) + y U(x, y)) ( ϕ(x) + y Φ(x, y)) dy dx Ω Y = F (x)ϕ(x) dx for all ϕ H0 1 (Ω), Φ L 2 (Ω, H#(Y 1 )). (3) Ω
Introduction Two-scale approximation The Homogenized problem The homogenized system Decouple the system: U L 2 (Ω, H#(Y 1 )) : a(y)( y U(x, y) + u(x)) y Φ(x, y) dy dx = 0 Ω Y for all Φ L 2 (Ω, H 1 #(Y )). (4a) u H0 1 (Ω) : a(y)( u(x) + y U(x, y)) dy ϕ(x) dx = F (x)ϕ(x) dx Ω Y Ω for all ϕ H0 1 (Ω). (4b)
Introduction Two-scale approximation The Homogenized problem The Cell problem The local problem (4a) is equivalent to requiring U(x, ) H#(Y 1 ) : a(y)( y U(x, y) + u(x)) y Φ(y) dy = 0 for all Φ H#(Y 1 ). Y Define ω i (y) for each 1 i N to be the solution of the cell problem ω i H#(Y 1 ) : a(y)( y ω i (y)+e i ) y Φ(y) dy = 0 for all Φ H#(Y 1 ). Y where e i is the indicated coordinate vector in R N. (5)
Introduction Two-scale approximation The Homogenized problem The Homogenized problem By linearity the solution is given by i=n U(x, y) = i u(x)ω i (y) i=1 This is substituted into (4b) to obtain u H0 1 (Ω) : ã ij i u(x) j ϕ(x) dx = Ω Ω F(x)ϕ(x) dx for all ϕ H 1 0 (Ω). (6) where the constant coefficients are given by ã ij = a(y) ( δ ij + j ω i (y) ) dy. (7) Y
Introduction Two-scale approximation The Homogenized problem The approximate solution u ɛ (x) = u(x) + ɛ u(x) (ω 1 (x/ɛ), ω 2 (x/ɛ),..., ω N (x/ɛ)) + O(ɛ 2 ), u ɛ (x) = u(x) + u(x) ( ω 1 (x/ɛ), ω 2 (x/ɛ),..., ω N (x/ɛ)) + O(ɛ) of the singular elliptic problem.
Introduction Two-scale approximation The Homogenized problem Homogenization Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou. Asymptotic analysis for periodic structures, volume 5 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1978. Enrique Sánchez-Palencia. Nonhomogeneous media and vibration theory, volume 127 of Lecture Notes in Physics. Springer-Verlag, Berlin, 1980. V. V. Jikov, S. M. Kozlov, and O. A. Oleinik. Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin, 1994.
... the model u(x) = K(x) p(x), u(x) = 0 c(x, t) φ + (u(x)c(x, t) D(u(x)) c(x, t)) t = 0 K fast, K slow u fast, u slow D fast, D slow
The Parabolic Equation The Exact Model... with fine-scale coefficients φ(x) c t D(x) c = 0, x Ω, with D(x) = D slow on Ω slow and = D fast on Ω fast.... or in transmission form φ α c α t D α c α = 0, x Ω α, α = fast, slow with the interface conditions on Ω slow Ω fast : c fast = c slow, D fast c fast ν = D slow c slow ν
Homogenization The coefficients are D = [D fast, D slow ] : φ c t D c = 0 The fine-scale geometry is eliminated...... replaced by the constant effective coefficient D. The upscaled limit is of the same type... a single equation. The fast and slow regions are coupled by gradients of the solution (flux). Accurate only in this low contrast case.
... the algorithm Exact model at microscale Homogenization replaced by homogenized model D(x) = D slow, D fast with constant D Compute homogenized coefficient D = D(D fast, D slow ) D jk = 1 D jk (y)(δ jk + k ω j (y))da Ω 0 Ω 0 { D(y) ωj (y) = (D(y)e j ), ω j (y) is Ω 0 periodic.
... the algorithm Exact model at microscale Homogenization replaced by homogenized model D(x) = D slow, D fast with constant D Compute homogenized coefficient D = D(D fast, D slow ) D jk = 1 D jk (y)(δ jk + k ω j (y))da Ω 0 Ω 0 { D(y) ωj (y) = (D(y)e j ), ω j (y) is Ω 0 periodic. It works well for problems with low contrast D fast /D slow
The Obstacle Problem... a special case The coefficients are D = [D fast, 0] : φ 0 c0 t D 0 c 0 = 0 The slow region Ω slow is impervious.
Classical homogenization is not adequate for time-dependent problems with large contrast D fast /D slow Low contrast: classical homogenization High contrast: double porosity models u ɛ 0 + u ɛ 0 C local averages u ɛ 0 +ɛ u ɛ C local averages versus special averages
The Double-porosity model The coefficients are D = [D fast, ɛ 2 0 D slow] : φ 0 c t + i χ i q i (t) D 0 c = 0, q i (t) = 1 D slow c i νds, ˆΩ i Γ i c i φ slow D slow c i = 0, c i Γi = 1 c(x)da. t ˆΩ i ˆΩ i The upscaled model is a system, highly parallel. The coupling from the cell to global equation is via gradients. The coupling from the global equation to the cell is via the values. The contribution of the cells is a secondary storage. Accurate only in this high contrast case. It will not couple any advective effects to the local cell.
Double-porosity Micro-structure Model Exact model replaced by two layers + Global equation, x Ω φ 0 c t + χ i q i (t) D 0 c = 0 i q i (t) = Π 0,i (D slow c i (, t) ν) Cell problem at each x Ω i c i φ slow t D slow c i = 0 c i Γi = Π 0,i ( c)(t)
Notes Homogenization The coefficients in the global equation are precisely those of the obstacle problem. The cell input to the global equation is q i (t) = 1 φ slow c i (x, t) dx ˆΩ i t Ω i... the rate of secondary storage. The spatially-constant input to the cell problem will not transmit any advective transport. We shall more tightly couple the cell by replacing the constant coupling with an affine coupling to the surrounding fast medium. This will provide a gradient coupling.
Affine approximations Π 1 AVERAGE: Π 0 f = 1 Ω 0 Ω 0 f (x)da; assume here Ω 0 = 1. Denote x C - center of mass of Ω 0. General affine approximation f (x) Π 1 f := m + n x, x Ω 0 Choice of m, n Taylor (f C 1 (Ω 0 )) about midpoint f (x) f (x C ) + f (x C )(x x C ) L 2 (Ω 0 )-projection onto affines that is: (f, v) Ω0 = (m + n x, v) Ω0, affine v H 1 (Ω 0 ) projection: f (x) Π 1 f := Π 0 f + Π 0 f (x x C )
Summary: the affine coupling Π i and its dual Π i Affine coupling Π i : H 1 (Ω) H 1 (ˆΩ i ) Π i (w)(x) Π 0 w + Π 0 ( w) (x x C ) Its dual Π i : H 1 (ˆΩ i ) H 1 (Ω) pointwise Π i (q)(x) = χ i(x)m 0 i (q) χ i (x)m 1 i (q)
Double-porosity Model with secondary flux φ 0 c t + i χ i q i (t) D 0 c = 0, φ slow c i t D slow c i = 0, c i Γi = 1 ˆΩ i cda + 1 c da (x x C ). ˆΩ i ˆΩ i ˆΩ i q i (t) = 1 D slow c i νds χ i(x) D slow c i ν (x x C )ds ˆΩ i Γ i ˆΩ i Γ i This is t cells. (secondary storage) (secondary flux) from the local
Computational experiments at microscale GOAL: reproduce qualitatively experimental results
Porous Media Homogenization Ulrich Hornung, editor. Homogenization and Porous Media, volume 6 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York, 1997. Peszynska, M.; Showalter, R. E. Multiscale Elliptic-Parabolic Systems for Flow and Transport, Electron. J. Differential Equations 2007, No. 147, 30 pp. (electronic). http://ejde.math.txstate.edu/volumes/2007/147/abstr.html Brendan Zinn, Lucy C. Meigs, Charles F. Harvey, Roy Haggerty, Williams J. Peplinski, and Claudius Freiherr von Schwerin. Experimental visualization of solute transport and mass transfer processes in two-dimensional conductivity fields with connected regions of high conductivity. Environ Sci. Technol., 38:3916 3926, 2004.