Resolvent estimates for high-contrast elliptic problems with periodic coefficients

Transcription

1 Resolvent estimates for high-contrast elliptic problems with periodic coefficients Joint work with Shane Cooper (University of Bath) 25 August 2015, Centro de Ciencias de Benasque Pedro Pascual Partial differential equations, optimal design and numerics

2 Key ingredients:

3 Key ingredients: Homogenisation of periodic elliptic DO

4 Key ingredients: Homogenisation of periodic elliptic DO High-contrast periodic media

5 Key ingredients: Homogenisation of periodic elliptic DO High-contrast periodic media Gelfand transform

6 Key ingredients: Homogenisation of periodic elliptic DO High-contrast periodic media Gelfand transform Multiscale analysis, matched asymptotics

7 Homogenisation setting

8 Homogenisation setting Classical homogenisation ( ( x ) ) div A u + u = f, f L 2 (R d ), ε A νi > 0

9 Homogenisation setting Classical homogenisation ( ( x ) ) div A u + u = f, f L 2 (R d ), ε A νi > 0 Convergence (two-scale expansions, compensated compactness, two-scale convergence, periodic unfolding, Bloch decomposition): u = u ε u 0 in H 1 (R d ), diva hom u 0 + u 0 = f.

10 Problem under study: high contrast We study the problem ( div Here: A ε = A 1 + ε 2 A 0 A ε( x ε ) ) u + u = f, f L 2 (R d ). A 0, A 1 [L (Q)] d d, Q periodic, symmetric A 1 νi, ν > 0 on Q 1 Q := [0, 1) d, ( stiff component) A 0 νi, A 1 = 0 on Q 0 = Q \ Q 1 ( soft component) n Z d ( Q1 + n ) is connected in R d

11 Notation b hom ( ) ε,θ (c, u), (d, v) := A hom θ θcd+ A 0 ( +iεθ)u ( + iεθ)v, (c, u), (d, v) H 0, Q A hom ξ ξ = min A 1 (ξ + u) (ξ + u), ξ R d, u H # 1 (Q) Q

12 Notation b hom ( ) ε,θ (c, u), (d, v) := A hom θ θcd+ A 0 ( +iεθ)u ( + iεθ)v, (c, u), (d, v) H 0, Q A hom ξ ξ = min A 1 (ξ + u) (ξ + u), ξ R d, u H # 1 (Q) Q L := {c + ũ : c C, ũ L 2 (Q), ũ Q1 = 0} L 2 (Q). Identification map I : C L 2 (Q 0 ) (c, u) c + ũ L, ũ = u on Q 0, ũ = 0 on Q 1.

13 Notation b hom ( ) ε,θ (c, u), (d, v) := A hom θ θcd+ A 0 ( +iεθ)u ( + iεθ)v, (c, u), (d, v) H 0, Q A hom ξ ξ = min A 1 (ξ + u) (ξ + u), ξ R d, u H # 1 (Q) Q L := {c + ũ : c C, ũ L 2 (Q), ũ Q1 = 0} L 2 (Q). Identification map I : C L 2 (Q 0 ) (c, u) c + ũ L, ũ = u on Q 0, ũ = 0 on Q 1. Consider C L 2 (Q 0 ) with inner product ( (c, u), (d, v) ) 0 = ( I(c, u), I(d, v) ) L 2 (Q). Operators Bε,θ hom in C L 2 (Q 0 ) : ( B hom ε,θ (c, u), (d, v)) 0 = ( ) bhom ε,θ (c, u), (d, v), (d, v) H0,

14 Notation b hom ( ) ε,θ (c, u), (d, v) := A hom θ θcd+ A 0 ( +iεθ)u ( + iεθ)v, (c, u), (d, v) H 0, Q A hom ξ ξ = min A 1 (ξ + u) (ξ + u), ξ R d, u H # 1 (Q) Q L := {c + ũ : c C, ũ L 2 (Q), ũ Q1 = 0} L 2 (Q). Identification map I : C L 2 (Q 0 ) (c, u) c + ũ L, ũ = u on Q 0, ũ = 0 on Q 1. Consider C L 2 (Q 0 ) with inner product ( (c, u), (d, v) ) 0 = ( I(c, u), I(d, v) ) L 2 (Q). Operators Bε,θ hom in C L 2 (Q 0 ) : ( B hom ε,θ (c, u), (d, v)) 0 = ( ) bhom ε,θ (c, u), (d, v), (d, v) H0, P orthogonal projection of L 2 (ε 1 Q Q) onto { c + g : c L 2 (ε 1 Q ), g L 2 (ε 1 Q Q), g(θ, y) = 0 a.e. (θ, y) ε 1 Q Q 1 }.

15 Gelfand transform Scaled version: (G εf)(θ, z) := ( ) ε d/2 2π f(z + εn)e iθ(z+εn), θ ε 1 [0, 2π) d. n Z d

16 Gelfand transform Scaled version: (G εf)(θ, z) := ( ) ε d/2 2π f(z + εn)e iθ(z+εn), θ ε 1 [0, 2π) d. n Z d Consider scaling of space variable: (T εh)(θ, y) := ε d/2 h(θ, εy) and unitary ( ε 2 ) d/2 ( ) (U εf)(θ, y) := ε(y + n) e iεθ(y+n). 2π n Z d f

17 Gelfand transform Scaled version: (G εf)(θ, z) := ( ) ε d/2 2π f(z + εn)e iθ(z+εn), θ ε 1 [0, 2π) d. n Z d Consider scaling of space variable: (T εh)(θ, y) := ε d/2 h(θ, εy) and unitary ( ε 2 ) d/2 ( ) (U εf)(θ, y) := ε(y + n) e iεθ(y+n). 2π n Z d f Lemma Define ( Bε,θ u, v ) = ( ε 2 A 1 + A 0 )( + iεθ ) u ( + iεθ ) v, u, v H 1 # (Q) Q Note: Then u ε θ = (B ε,θ + 1) 1 F, F L 2 (Q), ε 2 ( + iεθ) A ε (y) ( + iεθ) u ε θ + uε θ = F, U ε(a ε + 1) 1 Uε 1 = (B ε,θ + 1) 1 dθ. ε 1 Q

18 The main convergence result Theorem The resolvents of the operator family A ε are asymptotically close as ε 0 to the family R ε := Uε 1 I ( B hom ε 1 Q ε,θ + I) 1 I 1 dθ PU ε, where the corresponding approximation error is of order O(ε). More precisely, there exists a constant C > 0, independent of ε, such that (A ε + 1) 1 R ε L 2 (R d ) L 2 (R d ) Cε

19 An asymptotic expansion Then: u ε θ = n=0 ε n u (n) θ, u (n) θ H# 1 (Q), n = 0, 1, 2,...

20 An asymptotic expansion u ε θ = n=0 ε n u (n) θ, u (n) θ H# 1 (Q), n = 0, 1, 2,... Then: A 1 u (0) θ = 0, hence A 1 u (0) θ = 0.

21 An asymptotic expansion u ε θ = n=0 ε n u (n) θ, u (n) θ H# 1 (Q), n = 0, 1, 2,... Then: A 1 u (0) θ = 0, hence A 1 u (0) θ = 0. d A 1 u (1) θ = i A 1 θu (0) θ, hence u(1) θ = i N j θ j u (0) θ for some N 1, N 2,..., N d. j=1

22 An asymptotic expansion u ε θ = n=0 ε n u (n) θ, u (n) θ H# 1 (Q), n = 0, 1, 2,... Then: A 1 u (0) θ = 0, hence A 1 u (0) θ = 0. d A 1 u (1) θ = i A 1 θu (0) θ, hence u(1) θ = i N j θ j u (0) θ for some N 1, N 2,..., N d. j=1 A 1 u (2) θ = i A 1 θu (1) θ + A 0 u (0) θ θ A 1 θu (0) θ + u (0) θ.

23 An asymptotic expansion u ε θ = n=0 ε n u (n) θ, u (n) θ H# 1 (Q), n = 0, 1, 2,... Then: A 1 u (0) θ = 0, hence A 1 u (0) θ = 0. d A 1 u (1) θ = i A 1 θu (0) θ, hence u(1) θ = i N j θ j u (0) θ for some N 1, N 2,..., N d. j=1 A 1 u (2) θ = i A 1 θu (1) θ + A 0 u (0) θ θ A 1 θu (0) θ + u (0) θ. Solvability = u (0) θ = c (0) θ + v (0) θ, where ( c (0) θ, ) v(0) θ H0 satisfies ( (c b hom (0) 0,ε θ, ) ) ( v(0) (0) ) θ, (d, ϕ) + c θ +v(0) θ (d + ϕ) = P f F (d + ϕ), (d, ϕ) H 0, Q Q ( c (0) θ, ) ( v(0) θ = B hom 0,θ + I) 1 I 1 P f F.

24 Justification of asymptotics: error estimates Theorem Solution u ε θ, denote Uθ ε := u(0) θ + εũ (1) θ, Then there exists C > 0 : u ε θ Uθ ε H 1 (Q) Cε F L 2 (Q) θ 1. Corollary u ε θ u(0) θ L 2 (Q) Cε F L 2 (Q) θ 1.

25 Main ingredients of the proof Proposition Define V := { v H 1 # (Q) A1 v = 0 }. There exists C > 0 : w H 1 (Q) C A 1 w L 2 (Q) w V.

26 Main ingredients of the proof Proposition Define V := { v H 1 # (Q) A1 v = 0 }. There exists C > 0 : w H 1 (Q) C A 1 w L 2 (Q) w V. Lemma u (0) θ = c (0) θ + v (0) θ : ( c (0) θ, v(0) θ ) = ( B hom 0,θ + I) 1 I 1 P f F, corrector u (1) θ Then C > 0 : u (0) θ H 1 (Q) C( 1 + θ 2) 1 F L 2 (Q), u (1) θ H 1 (Q) C θ ( 1 + θ 2) 1 F L 2 (Q). = i d j=1 N j θ j u (0) θ.

27 Extension to all θ Q Theorem Consider u (0) ε,θ := c(0) ε,θ + v(0) ε,θ where ( c (0) ε,θ, ε,θ) v(0) H0 satisfies ( (c b hom (0) ε,θ ε,θ, ) ) ( v(0) (0) ε,θ, (d, ϕ) + c ε,θ + ) v(0) ε,θ (d + ϕ) = P f F (d + ϕ), (d, ϕ) H 0. Q Q Then C > 0 : u ε θ u(0) ε,θ L 2 (Q) Cε F L 2 (Q) θ.

28 Extension to all θ Q Theorem Consider u (0) ε,θ := c(0) ε,θ + v(0) ε,θ where ( c (0) ε,θ, ε,θ) v(0) H0 satisfies ( (c b hom (0) ε,θ ε,θ, ) ) ( v(0) (0) ε,θ, (d, ϕ) + c ε,θ + ) v(0) ε,θ (d + ϕ) = P f F (d + ϕ), (d, ϕ) H 0. Q Q Then C > 0 : u ε θ u(0) ε,θ L 2 (Q) Cε F L 2 (Q) θ. Additional ingredients in the proof: Proposition Define V (κ) := { v H 1 κ (Q) A1 v = 0 }. There exists C > 0 : w H 1 (Q) Cd(κ) A 1 w L 2 (Q) w V (κ), d(κ) = { 1, κ = 0, κ 1, κ 0.

29 Extension to all θ Q Theorem Consider u (0) ε,θ := c(0) ε,θ + v(0) ε,θ where ( c (0) ε,θ, ε,θ) v(0) H0 satisfies ( (c b hom (0) ε,θ ε,θ, ) ) ( v(0) (0) ε,θ, (d, ϕ) + c ε,θ + ) v(0) ε,θ (d + ϕ) = P f F (d + ϕ), (d, ϕ) H 0. Q Q Then C > 0 : u ε θ u(0) ε,θ L 2 (Q) Cε F L 2 (Q) θ. Additional ingredients in the proof: Proposition Define V (κ) := { v H 1 κ (Q) A1 v = 0 }. There exists C > 0 : w H 1 (Q) Cd(κ) A 1 w L 2 (Q) w V (κ), d(κ) = { 1, κ = 0, κ 1, κ 0. Lemma H ε,θ H 1 # (Q), H ε,θ, ϕ = 0 ϕ H 1 0 (Q0) = R ε,θ H 1 # (Q) : ( + iεθ ) ( ) A 1 + iεθ Rε,θ = H ε,θ, [ R H 1 ε,θ 1 (Q) C H ε,θ H ε,θ, 1 εθ H H # (Q) εθ 2 ε,θ, 1 ].

30 The outer asymptotics for θ > ε 1/2 Theorem Let w ε,κ H 1 κ(q) be solution to (ε 2 A 1 + A 0 ) wε,κ + w ε,κ = F, F L 2 (Q). Then N C N > 0 : wε,κ U ε,κ (N) H 1 (Q) C N Truncation of formal series: In particular, U (N) ε,κ = ( ) ε 2(N+1) F κ L 2(Q). N n=0 ε 2n w (n) κ. ( ) wε.κ w κ (0) ε 2 H 1 (Q) C 1 F κ L 2(Q), Q 0 ( A0 w (0) κ ϕ + w (0) κ ϕ ) = F, ϕ ϕ H 1 0 (Q 0 ).

31 Matched asymptotic expansions in the Bloch space Inner : u ε θ N ε n u (n) θ C N εθ N+1, θ ε α ; n=0

32 Matched asymptotic expansions in the Bloch space Inner : Outer : u ε θ N w ε,κ N ε n u (n) θ C N εθ N+1, θ ε α ; n=0 ε 2n w κ (n) ε D N κ 2(N+1), κ > ε 1 α. n=0

33 Matched asymptotic expansions in the Bloch space Inner : Outer : u ε θ N w ε,κ N ε n u (n) θ C N εθ N+1, θ ε α ; n=0 ε 2n w κ (n) ε D N κ 2(N+1), κ > ε 1 α. n=0 Error: C N ε (1 α)(n+1) + D N ε 2α(N+1). Optimal for α = 1/3.

34 Matched asymptotic expansions in the Bloch space Inner : Outer : u ε θ N w ε,κ N ε n u (n) θ C N εθ N+1, θ ε α ; n=0 ε 2n w κ (n) ε D N κ 2(N+1), κ > ε 1 α. n=0 Error: C N ε (1 α)(n+1) + D N ε 2α(N+1). Optimal for α = 1/3. Matching region : ε γ θ ε β 1 (0 < γ < γ + β < 1) (Equivalently, ε 1 γ κ ε β.)

35 Matched asymptotic expansions in the Bloch space Inner : Outer : u ε θ N w ε,κ N ε n u (n) θ C N εθ N+1, θ ε α ; n=0 ε 2n w κ (n) ε D N κ 2(N+1), κ > ε 1 α. n=0 Error: C N ε (1 α)(n+1) + D N ε 2α(N+1). Optimal for α = 1/3. Matching region : ε γ θ ε β 1 (0 < γ < γ + β < 1) (Equivalently, ε 1 γ κ ε β.) N F N common part of ε n u (n) θ and N n=1 n=1 Both replaced by N n=1 ε 2n w (n) κ. ε n u (n) θ + N ε 2n w κ (n) F N. n=1

36 Matched asymptotic expansions in the Bloch space Inner : Outer : u ε θ N w ε,κ N ε n u (n) θ C N εθ N+1, θ ε α ; n=0 ε 2n w κ (n) ε D N κ 2(N+1), κ > ε 1 α. n=0 Error: C N ε (1 α)(n+1) + D N ε 2α(N+1). Optimal for α = 1/3. Matching region : ε γ θ ε β 1 (0 < γ < γ + β < 1) (Equivalently, ε 1 γ κ ε β.) N F N common part of ε n u (n) θ and N n=1 n=1 Both replaced by N n=1 ε 2n w (n) κ. ε n u (n) θ + N ε 2n w κ (n) F N. n=1 This yields error E N ε σ(n+1) (σ to be determined) Our result: matching with σ = 1 for N = 0.

37 Spectra of the operators B hom ε,θ (c, u) H 0 is an eigenvector of Bε,θ hom with eigenvalue λ A hom θ θcd+ A 0 ( +iεθ)u ( + iεθ)v = λ (c+u)(d + v) (d, v) H 0. Q Q

38 Spectra of the operators B hom ε,θ (c, u) H 0 is an eigenvector of Bε,θ hom with eigenvalue λ A hom θ θcd+ A 0 ( +iεθ)u ( + iεθ)v = λ (c+u)(d + v) (d, v) H 0. Q Q Either λ S 0 eigenvalues of A 0 with Dirichlet b.c. s on Q 0,

39 Spectra of the operators B hom ε,θ (c, u) H 0 is an eigenvector of Bε,θ hom with eigenvalue λ A hom θ θcd+ A 0 ( +iεθ)u ( + iεθ)v = λ (c+u)(d + v) (d, v) H 0. Q Q Either λ S 0 eigenvalues of A 0 with Dirichlet b.c. s on Q 0, or ( A hom θ θ = λ 1+λ (λ j λ) 1 j=0 Function β appeared in Zhikov (2000). Q 0 ϕ j 2) =: β(λ), ϕ j (y) := ϕ j(y) exp(iεθy).

40 Spectra of the operators B hom ε,θ (c, u) H 0 is an eigenvector of Bε,θ hom with eigenvalue λ A hom θ θcd+ A 0 ( +iεθ)u ( + iεθ)v = λ (c+u)(d + v) (d, v) H 0. Q Q Either λ S 0 eigenvalues of A 0 with Dirichlet b.c. s on Q 0, or ( A hom θ θ = λ 1+λ (λ j λ) 1 j=0 Function β appeared in Zhikov (2000). Q 0 ϕ j 2) =: β(λ), ϕ j (y) := ϕ j(y) exp(iεθy). Theorem The spectra of A ε converge in the Hausdorff sense to the union of S 0 and { lim λ : β(λ) = A hom θ θ } = { λ : β(λ) 0 }. ε 0 θ ε 1 Q

41 The Zhikov function β(λ)

42 Examples of the family A ε Classical homogenisation: Q 0 =

43 Examples of the family A ε Classical homogenisation: Q 0 = V = {constants on Q}

44 Examples of the family A ε Classical homogenisation: Q 0 = V = {constants on Q} Poincaré inequality for functions with zero mean over Q

45 Examples of the family A ε Classical homogenisation: Q 0 = V = {constants on Q} Poincaré inequality for functions with zero mean over Q Family R ε consists of one element, the resolvent of A hom. = We recover earlier results of Birman and Suslina (2004), Griso (2006).

46 Examples of the family A ε Double porosity : Q 0, A 0 Q1 = 0

47 Examples of the family A ε Double porosity : Q 0, A 0 Q1 = 0 V = {constants + functions in H 1 0 (Q 0)}

48 Examples of the family A ε Double porosity : Q 0, A 0 Q1 = 0 V = {constants + functions in H 1 0 (Q 0)} Analogue of Poincaré inequality: Cooper (2012)

49 Examples of the family A ε Double porosity : Q 0, A 0 Q1 = 0 V = {constants + functions in H 1 0 (Q 0)} Analogue of Poincaré inequality: Cooper (2012) Zhikov (2000): (A ε + I) 1 f S ε (A dp + I) 1 f < C(f)ε, L 2 (R d )

50 Examples of the family A ε Double porosity : Q 0, A 0 Q1 = 0 V = {constants + functions in H 1 0 (Q 0)} Analogue of Poincaré inequality: Cooper (2012) Zhikov (2000): (A ε + I) 1 f S ε (A dp + I) 1 f < C(f)ε, L 2 (R d ) A hom v 1 ϕ 1 + A 0 yv 0 yϕ 0 R d R d Q + (v 1 + v 0 )(ϕ 1 + ϕ 0 ) = f(ϕ 1 + ϕ 0 ), R d Q R d Q (v 1, v 0 ), (φ 1, φ 0 ) H 1 (R d ) L 2( R d, H 1 0 (Q 0 ) ).

51 Examples of the family A ε Double porosity : Q 0, A 0 Q1 = 0 V = {constants + functions in H 1 0 (Q 0)} Analogue of Poincaré inequality: Cooper (2012) Zhikov (2000): (A ε + I) 1 f S ε (A dp + I) 1 f < C(f)ε, L 2 (R d ) A hom v 1 ϕ 1 + A 0 yv 0 yϕ 0 R d R d Q + (v 1 + v 0 )(ϕ 1 + ϕ 0 ) = f(ϕ 1 + ϕ 0 ), R d Q R d Q (v 1, v 0 ), (φ 1, φ 0 ) H 1 (R d ) L 2( R d, H 1 0 (Q 0 ) ). However: f ε bounded in L 2 (R d ) such that C(f ε ) as ε 0. NB: Difference of spectral projections of S ε (A dp + I) 1 and R ε does not go to zero near (1 + λ ) 1 such that β(λ) as λ λ.