Besov regularity for the solution of the Stokes system in polyhedral cones

Transcription

1 the Stokes system Department of Mathematics and Computer Science Philipps-University Marburg Summer School New Trends and Directions in Harmonic Analysis, Fractional Operator Theory, and Image Analysis Inzell, September 2012

2 Outline 1. Motivation 2. Adaptive wavelet schemes 3. Polyhedral cones and weighted Sobolev spaces 4. for

3 Outline 1. Motivation

4 Motivation Numerical treatment of operator equations L : H m 0 (Ω) H m (Ω), L(u) = f.

5 Motivation Numerical treatment of operator equations L : H m 0 (Ω) H m (Ω), L(u) = f. Example: Poisson equation on a Lipschitz domain Ω R d : : H 1 0 (Ω) H 1 (Ω). u = f in Ω u = 0 on Ω.

6 Stokes system Consider on a Lipschitz domain Ω R 3 : u + p = f in Ω, divu = 0 in Ω, u = 0 on Ω.

7 Stokes system Consider on a Lipschitz domain Ω R 3 : u + p = f in Ω, divu = 0 in Ω, u = 0 on Ω. Weak formulation: For given f H 1 (Ω) 3, determine u H0 1(Ω)3 and p L 2,0 (Ω):= {q L 2 (Ω) : Ωq(x)dx = 0} such that where a(u,v)+b(v,p) = f(v) for all v H 1 0 (Ω)3 b(u,q) = 0 for all q L 2,0 (Ω), a(u,v) := 3 Ω i,j=1 Ω u i x j v i x j dx, b(v,q) := q(x)div(v)(x)dx.

8 Numerical Approaches: Nonadaptive schemes Based on uniform space refinements Approximation spaces a priori fixed

9 Numerical Approaches: Nonadaptive schemes Based on uniform space refinements Approximation spaces a priori fixed Easy to implement/analyze But: convergence might be slow

10 Numerical Approaches: Nonadaptive schemes Based on uniform space refinements Approximation spaces a priori fixed Easy to implement/analyze But: convergence might be slow Adaptive schemes Nonuniform space refinements Updating strategy

11 Numerical Approaches: Nonadaptive schemes Based on uniform space refinements Approximation spaces a priori fixed Easy to implement/analyze But: convergence might be slow Adaptive schemes Nonuniform space refinements Updating strategy A posteriori error estimator

12 Numerical Approaches: Nonadaptive schemes Based on uniform space refinements Approximation spaces a priori fixed Easy to implement/analyze But: convergence might be slow Adaptive schemes Nonuniform space refinements Updating strategy A posteriori error estimator Difficult to implement Goal: achieve a better convergence rate

13 Numerical Approaches: Nonadaptive schemes Based on uniform space refinements Approximation spaces a priori fixed Easy to implement/analyze But: convergence might be slow Adaptive schemes Nonuniform space refinements Updating strategy A posteriori error estimator Difficult to implement Goal: achieve a better convergence rate Convergence rate depends on regularity of the solution (which regularity?)

14 Outline 2. Adaptive wavelet schemes

15 Wavelets Ω R d : bounded domain Consider a Multiresolution analysis {V j } j 0 of L 2 (Ω):

16 Wavelets Ω R d : bounded domain Consider a Multiresolution analysis {V j } j 0 of L 2 (Ω): Vj V j+1, j 0,

17 Wavelets Ω R d : bounded domain Consider a Multiresolution analysis {V j } j 0 of L 2 (Ω): Vj V j+1, j 0, V j = L 2 (Ω), j 0

18 Wavelets Ω R d : bounded domain Consider a Multiresolution analysis {V j } j 0 of L 2 (Ω): Vj V j+1, j 0, V j = L 2 (Ω), j 0 f Vj f(2 j ) V 0.

19 Wavelets Ω R d : bounded domain Consider a Multiresolution analysis {V j } j 0 of L 2 (Ω): Vj V j+1, j 0, V j = L 2 (Ω), j 0 f Vj f(2 j ) V 0. V j+1 = V j W j+1, V 0 = W 0 L 2 (Ω) = j 0 W j, W j = span{ψ j,k : k I j }

20 Wavelets Ω R d : bounded domain Consider a Multiresolution analysis {V j } j 0 of L 2 (Ω): Vj V j+1, j 0, V j = L 2 (Ω), j 0 f Vj f(2 j ) V 0. V j+1 = V j W j+1, V 0 = W 0 L 2 (Ω) = j 0 W j, W j = span{ψ j,k : k I j } Get a orthonormalbasis for L 2 (Ω)

21 Solving with adaptive schemes I Determine approximations u Λ0,u Λ1,..., u Λj = λ Λ j c λ ψ λ, such that for a given ε > 0 after a finite number of steps we get a solution u Λkε with u u Λkε L2 (Ω) < ε.

22 Solving with adaptive schemes I Determine approximations u Λ0,u Λ1,..., u Λj = λ Λ j c λ ψ λ, such that for a given ε > 0 after a finite number of steps we get a solution u Λkε with u u Λkε L2 (Ω) < ε. The sets Λ i, i 0 are determined adaptively by the algorithm.

23 Solving with adaptive schemes II ideal algorithm: best m-term approximation. { M m := f = } λ ψ λ : Λ = m. λ Λc σ m (u) L2 (Ω) := inf g M m u g L2 (Ω) u g m L2 (Ω), g m = λ Λ m c λ ψ λ, Λ m =m biggest wavelet coefficients

24 Solving with adaptive schemes II ideal algorithm: best m-term approximation. { M m := f = } λ ψ λ : Λ = m. λ Λc σ m (u) L2 (Ω) := inf g M m u g L2 (Ω) u g m L2 (Ω), g m = λ Λ m c λ ψ λ, Λ m =m biggest wavelet coefficients σ m (u) L2 (Ω) = O(m α/d ) = u B α τ (L τ(ω)), 1 τ = α d.

25 Solving with adaptive schemes II ideal algorithm: best m-term approximation. { M m := f = } λ ψ λ : Λ = m. λ Λc σ m (u) L2 (Ω) := inf g M m u g L2 (Ω) u g m L2 (Ω), g m = λ Λ m c λ ψ λ, Λ m =m biggest wavelet coefficients σ m (u) L2 (Ω) = O(m α/d ) = u B α τ (L τ(ω)), 1 τ = α d. Implemented schemes from Cohen, Dahmen and DeVore.

26 Comparison with linear approximation Convergence using adaptive schemes: σ n (u) L2 (Ω) = O(n α/d ) = u B α τ (L τ (Ω)), 1 τ = α d.

27 Comparison with linear approximation Convergence using adaptive schemes: σ n (u) L2 (Ω) = O(n α/d ) = u B α τ (L τ (Ω)), Linear approximation: 1 τ = α d. E j (u) := inf g V j u g L2 (Ω) 2 βj u H β (Ω) E j (u) = O(n β/d ) = u H β (Ω).

28 Comparison with linear approximation Convergence using adaptive schemes: σ n (u) L2 (Ω) = O(n α/d ) = u B α τ (L τ (Ω)), Linear approximation: 1 τ = α d. E j (u) := inf g V j u g L2 (Ω) 2 βj u H β (Ω) E j (u) = O(n β/d ) = u H β (Ω). Natural question: α > Sobolev regularity β?

29 DeVore-Triebel diagram s 1/p

30 DeVore-Triebel diagram s β H β (Ω) 1/2 1/p

31 DeVore-Triebel diagram α s B α τ (L τ(ω)) β H β (Ω) 1/2 1/τ 1/p

32 DeVore-Triebel diagram α s B α τ (L τ(ω)) β H β (Ω) 1 τ = α d /2 1/τ 1/p

33 Outline 3. Polyhedral cones and weighted Sobolev spaces

34 Polyhedral cones 0 K := {x R 3 : x = ρ ω,0 < ρ <,ω Ω}, K 0 := {x K : x 2 r 0 }.

35 Polyhedral cones M j Γ j 0 Boundary of K consists of vertex x = 0,edges M j,faces Γ j.

36 Polyhedral cones θ j M j Γ j 0 θ j : angel at the edge M j.

37 Polyhedral cones θ j ρ(x) := x r j (x) := dist(x,m j ) M j Γ j 0

38 Weighted Sobolev Spaces Singularities at the edges M j and in the vertex x = 0.

39 Weighted Sobolev Spaces Singularities at the edges M j and in the vertex x = 0. Two weights: δ := (δ1,...,δ d ) R d belongs to the edges. β R belongs to the vertex.

40 Weighted Sobolev Spaces Singularities at the edges M j and in the vertex x = 0. Two weights: δ := (δ1,...,δ d ) R d belongs to the edges. β R belongs to the vertex. Weighted Sobolev norm K α l u W l,2 := β, (K) δ D α u(x) 2 dx 1/2

41 Weighted Sobolev Spaces Singularities at the edges M j and in the vertex x = 0. Two weights: δ := (δ1,...,δ d ) R d belongs to the edges. β R belongs to the vertex. Weighted Sobolev norm K α l u W l,2 := β, (K) δ d k=1 ( rk (x) ρ(x) ) 2δk D α u(x) 2 dx 1/2

42 Weighted Sobolev Spaces Singularities at the edges M j and in the vertex x = 0. Two weights: δ := (δ1,...,δ d ) R d belongs to the edges. β R belongs to the vertex. Weighted Sobolev norm d ρ(x) 2(β l+ α ) K α l u W l,2 := β, (K) δ k=1 ( rk (x) ρ(x) ) 2δk D α u(x) 2 dx 1/2

43 Weighted Sobolev Spaces Singularities at the edges M j and in the vertex x = 0. Two weights: δ := (δ1,...,δ d ) R d belongs to the edges. β R belongs to the vertex. Weighted Sobolev norm W l,2 d ρ(x) 2(β l+ α ) K α l u W l,2 := β, (K) δ k=1 β,δ (K) := C 0 (K\{0}) W ( rk (x) ρ(x) l,2 β,δ ) 2δk D α u(x) 2 dx 1/2

44 Outline 4. for

45 The Stokes system We consider u + p = f in K, divu = g in K, u = 0 on Γ j, j = 1,...,d.

46 The Stokes system We consider u + p = f in K, divu = g in K, u = 0 on Γ j, j = 1,...,d. ( p p =, p, p ) ( 3 ) 2 u j, u = x 1 x 2 x 3 2 x i i=1 j=1,2,3. divu = 3 i=1 u i x i

47 Sobolev regularity of the solution Consider u + p = f in K 0, divu = g in K 0, u = 0 on Γ j, j = 1,...,d. Proposition (M. Dauge, 89) Assume (f,g) L 2 (K 0 ) 3 H α 0 (K 0 ) for α 0 < 1/2. Let g fulfill K 0 g(x)dx = 0. Then there exists a unique solution (u,p) H α 0+1 (K 0 ) 3 H α 0 (K 0 ).

48 Weighted Sobolev estimates of the solution Proposition (V. Maz ya, J. Rossmann, 01) Suppose (f,g) W l 2,2 β,δ (K) 3 W l 1,2 β,δ (K) where l 2 is an integer. Then there exists a countable set E C such that the following holds. If β R and the vector δ (R\Z) d are chosen such that Re λ l β 3 2 for all λ E and ) max (0,l 1 πθk < δ k < l 1, k = 1,...,d, then there exists a uniquely determined solution (u,p) W l,2 β,δ (K) Wl 1,2 β,δ (K)

49 of the solution Theorem (E., 12) Assume that the conditions in the above propositions are fulfilled. For β < l 1 we obtain ( u B s 1 τ 1 (L τ1 (K 0 )) 3, s 1 < min l, 3 ) 2 (α 0 +1),3 (l δ ), p B s 2 τ 2 (L τ2 (K 0 )),s 2 < min 1 τ 1 = s , ( l 1, 3 2 α 0,3 (l ( δ +1)) 1 τ 2 = s ),

50 Summary Motivation: Justification of the use of adaptive schemes.

51 Summary Motivation: Justification of the use of adaptive schemes. Investigate the of the solution of the Stokes system.

52 Summary Motivation: Justification of the use of adaptive schemes. Investigate the of the solution of the Stokes system. Proof is performed by using characterization of Besov spaces by wavelet expansions.

53 Summary Motivation: Justification of the use of adaptive schemes. Investigate the of the solution of the Stokes system. Proof is performed by using characterization of Besov spaces by wavelet expansions. For valid parameters the is 3/2 times higher than its Sobolev regularity.

54 References A. Cohen, W. Dahmen, and R.A. DeVore. Adaptive wavelet methods for elliptic operator equations: Convergence rates. Math. Comput., 70:2775, M. Dauge. Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I: Linearized equations. SIAM J. Math. Anal, 20:7497, V. Mazya and J. Rossmann, Elliptic equations in polyhedral domains. Mathematical Surveys and Monographs 162. Providence, RI: American Mathematical Society (AMS). VII, 2010.

55 Thanks a lot for your attention!

56 Sketch of the proof I Idea: Estimate the coefficients of the wavelet decomposition of the solution. Proposition Let s R( and 0 < p,q <. ) Suppose ) r > max s,nmax (0, 1 p 1 s. Then f B s q(l p (R n )) ( ) 2 n 1 q/p 2 j(s+n(1/2 1/p))q f,ψ i,j,k p k Z n i=1 j=0 1/q <

57 Sketch of the proof II Idea: Estimate the coefficients of the wavelet decomposition of the solution. Wavelets are assumed to be compact supported. Then there exists a cube Q centered at the origin, s.t. Q j,k := 2 j k +2 j Q, j N 0, k Z 3 contains the support of the wavelets ψ i,j,k.

58 Sketch of the proof II Idea: Estimate the coefficients of the wavelet decomposition of the solution. Wavelets are assumed to be compact supported. Then there exists a cube Q centered at the origin, s.t. Q j,k := 2 j k +2 j Q, j N 0, k Z 3 contains the support of the wavelets ψ i,j,k. Consider the set of indices Λ ι := {(i,j,k) : Q j,k K 0,2 3ι 2 j 2 3ι+2 }.

59 Sketch of the proof III Estimate the coefficients v,ψ i,j,k in three steps: Consider for κ N the set Λ ι,κ := {(i,j,k) Λ ι : κ2 ι dist(q j,k,0) < (κ+1)2 ι }.

60 Sketch of the proof III Estimate the coefficients v,ψ i,j,k in three steps: Consider for κ N the set Λ ι,κ := {(i,j,k) Λ ι : κ2 ι dist(q j,k,0) < (κ+1)2 ι }. Weighted Sobolev esimates

61 Sketch of the proof III Estimate the coefficients v,ψ i,j,k in three steps: Consider the set Λ ι,0 := {(i,j,k) Λ ι : 0 < dist(q j,k,0) < 2 ι }.

62 Sketch of the proof III Estimate the coefficients v,ψ i,j,k in three steps: Consider the set Λ ι,0 := {(i,j,k) Λ ι : 0 < dist(q j,k,0) < 2 ι }. Sobolev regularity

63 Sketch of the proof III Estimate the coefficients v,ψ i,j,k in three steps: Consider the set Λ # ι := {(i,j,k) : Q j,k K 0,2 3ι 2 j 2 3ι+2 }.

64 Sketch of the proof III Estimate the coefficients v,ψ i,j,k in three steps: Consider the set Λ # ι := {(i,j,k) : Q j,k K 0,2 3ι 2 j 2 3ι+2 }. Sobolev regularity

65 Norm estimate I We obtain u H α 0 +1 (K 0 ) 3 + p H α 0(K 0 ) f L2 (K 0 ) 3 + g H α 0(K 0 ).

66 Norm estimate I We obtain u H α 0 +1 (K 0 ) 3 + p H α 0(K 0 ) f L2 (K 0 ) 3 + g H α 0(K 0 ). Define H β := { } u W l,2 β,0 (K)3 : u = 0 on Γ j, j = 1,...,d. The functional F(v) := K (f + g) v dx defines a linear and continuous mapping on H l 1 β.

67 Norm estimate I We obtain u H α 0 +1 (K 0 ) 3 + p H α 0(K 0 ) f L2 (K 0 ) 3 + g H α 0(K 0 ). Define H β := { } u W l,2 β,0 (K)3 : u = 0 on Γ j, j = 1,...,d. The functional F(v) := K (f + g) v dx defines a linear and continuous mapping on H l 1 β. The solution (u,p) found in the Proposition fulfills u W l,2 β,δ (K)3 + p W l 1,2 β,δ (K) ( F H l 1 β + g V 0,2 β l+1 (K) + f W l 2,2 β,δ (K) + g W l 1,2 β,δ (K) ).

68 Norm estimate II We show for β < l 1 u B s 1 τ1 (L τ1 (K 0 )) 3 + p B s 2 τ2 (L τ2 (K 0 )) u W l,2 β,δ (K)3 + u H α 0 +1 (K 0 ) 3+ p W l 1,2 β,δ (K) + p H α 0(K 0 ).

69 Norm estimate II We show for β < l 1 u B s 1 τ1 (L τ1 (K 0 )) 3 + p B s 2 τ2 (L τ2 (K 0 )) u W l,2 β,δ (K)3 + u H α 0 +1 (K 0 ) 3+ p W l 1,2 β,δ (K) + p H α 0(K 0 ). All in all this leads to u B s 1 τ1 (L τ1 (K 0 )) 3 + p B s 2 τ2 (L τ2 (K 0 )) F H l 1 β + g V 0,2 β l+1 (K)+ g W l 1,2 β,δ (K) + g H α 0(K 0 )+ f W l 2,2 β,δ (K) 3 + f L2 (K 0 ) 3