Local pointwise a posteriori gradient error bounds for the Stokes equations. Stig Larsson. Heraklion, September 19, 2011 Joint work with A.

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1 Local pointwise a posteriori gradient error bounds for the Stokes equations Stig Larsson Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg Heraklion, September 19, 2011 Joint work with A. Demlow 1 / 29

2 The stationary Stokes equations u + p = f, u = g, u = 0, in Ω in Ω on Ω Ω R n, n = 2, 3, polygonal domain with Lipschitz boundary f (L (Ω)) n g Wq 1 (Ω) for some q > n g dx = 0 Ω p dx = 0 Ω 2 / 29

3 Weak formulation Find (u, p) V X such that L((u, p), (v, λ)) = (f, v) + (g, λ) (v, λ) V X V = (H 1 0 (Ω)) n, Ω i,j=1 X = L 2 (Ω) L((u, p), (v, λ)) = a(u, v) + b(v, p) b(u, λ) n u i v i a(u, v) = dx, b(v, p) = ( v)p dx x j x j (f, v) = n f i v i dx, (g, λ) = Ω i,j=1 Ω gλ dx Ω 3 / 29

4 Taylor-Hood FEM Find (u h, p h ) V h X h such that L((u h, p h ), (v h, λ h )) = (f, v h ) + (g, λ h ) (v h, λ h ) V h X h {T h } regular family of triangulations of Ω V h V continuous piecewise poly of degree k 2 X h X continuous piecewise poly of degree k 1 p h dx = 0 Ω 4 / 29

5 Global a posteriori error estimates W 1 -type residual error indicator: η 1, (T ) = h T f + u h p h L (T ) h T = diam(t ), + u h L ( T ) + g u h L (T ), T T h h = min T T h h T Theorem (S. L. and E. D. Svensson (2006)) ( u u h L (Ω) C ln 1 ) αn max h T η 1, (T ) h T T h ( (u u h ) L (Ω) C ln 1 ) αn max η 1, (T ) h T T h Here α 2 = 2 and α 3 = 4/3. (for Ω convex) u is unbounded near reentrant corners and edges Proof based on: L p (Ω), L p (Ω) duality argument regularity estimates in L p (Ω) and p 1, p 5 / 29

6 Local a posteriori error estimate D Ω target domain d = dist(d, reentrant corner of edge of Ω) D d = {x Ω : dist(d, x) < d}, D ρ = {x Ω : dist(d, x) < ρ} T Dd = {T T h : T D d } mesh d-neighborhood of D h T = diam(t ), h = min T T h h T Theorem (S. L. and A. Demlow) Let ρ c 0 min{d, h} for a sufficiently small constant c 0. Assume (u, p) C 1,β (D ρ ) C 0,β (D ρ ) for some β (0, 1). Then (u u h ) L (D) + p p h L (D) C ln d ρ max T T Dd h T h T + dist(t, D) η 1, (T ) + C 1 ( ) u u h L (Ω) + max h T η 1, (T ) d T T h + Cρ β( ) u C 1,β (D + p ρ) C 0,β (D ρ). 6 / 29

7 The domains d D c 1 d x 0 B D d Ω 7 / 29

8 Local residual term C ln d ρ max T T Dd Here: max T T Dd h T h T + dist(t, D) η 1, (T ) h T h T + dist(t, D) η 1, (T ) = η 1, (T ), T in D h T d η 1, (T ), T near D d \ Ω ρ regularization parameter W -type 1 residual error indicator: η 1, (T ) = h T f + u h p h L (T ) + u h L ( T ) + g u h L (T ), T T h 8 / 29

9 Global pollution term C 1 d ( ) ( u u h L (Ω) + max h T η 1, (T ) C ln 1 ) αn max h T η 1, (T ) T T h h T T h by S. L. and E. D. Svensson (2006). Here α 2 = 2 and α 3 = 4/3. Behaves like an L -estimator. 9 / 29

10 Regularization penalty Cρ β( ) u C 1,β (D + p ρ) C 0,β (D ρ) Caused by the use of a regularized Green s function. Can be bounded a priori for some β (0, 1). May take ρ arbitralily small. This leads to a completely a posteriori estimate. 10 / 29

11 Corollary: Completely a posteriori error estimate If n = 3 assume that the incompressibility g = 0 near Ω D. Define F (f, g, d) = f Lq(Ω) + g W 1 q (Ω) + d 2( f W 1 q (Ω) + g L q(ω) E = max T T Dd h T h T + dist(t, D) η 1, (T ), β = 1 n q, ρ = min(d, h, (E/F (f, g, d))1/β ). Then there exists q > n such that: If E = 0, then (u u h ) L (D) + p p h L (D) C If E 0, then ), ( ln 1 h ) αn 1 d max T T h h T η 1, (T ). (u u h ) L (D)+ p p h L (D) ( C 1 + ln d ρ ) ( E + C ln 1 ) αn 1 h d max h T η 1, (T ). T T h 11 / 29

12 Idea of proof regularized Green s function duality argument on convex polygonal subdomain bounds for Green s function on convex polygonal domains guided by A. Demlow, Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems, Math. Comp. 76 (2007), / 29

13 Regularized delta function e u = u h u, e p = p h p. Select x 0 D and i, j so that ( ) e u L (D) + e p L (D) 2 sup max D k e ul (x) + e p (x) x D k,l ( ) = 2 D i e uj (x 0 ) + e p (x 0 ) ( ) = 2 D i e u (x 0 ) k j + e p (x 0 ). Approximate delta function at x 0 : D i e u (x 0 ) k j (D i e u k j, δ) = (D i e u, δk j ), e p (x 0 ) (e p, δ) 13 / 29

14 Regularized delta function shape-regular simplex T 0 T h diam(t 0 ) = ρ x 0 T 0 T x0 T h δ C0 (T 0 ) P(x 0 ) = δ(x)p(x) dx P Π k 1 T 0 ( ) e u L (D) + e p L (D) = 2 D i e u (x 0 ) k j + e p (x 0 ) = 2 D i e u (x 0 ) k j ± e p (x 0 ) e u L (D) + e p L (D) 2 (D i e u, δk j ) ± (e p, δ) +Cρ β( u C 1,β (D + p ) ρ) C 0,β (D ρ) where the sign of ±(e p, δ) is chosen to match the sign of (D i e u, δk j ) 14 / 29

15 Localization D c 1 d c 1 d/2 x 0 B The cut-off function ω is supported in B c1d(x 0 ) and ω 1 in B c1d/2(x 0 ). B is a convex polyhedron with D d B. 15 / 29

16 Localization (D i e u, δk j ) ± (e p, δ) = (e u, D i δk j ) ± (e p, δ) [assume minus ] = (e u, D i δk j ) (e p, δ) Choose smooth δ B with (δ δ B ) dx = 0 B Localized adjoint problem: find (v, q) V B X B = (H0 1(B))n L 2 (B) with q dx = 0 such that B L B ((w, λ), (v, q)) = (w, D i δk j ) + (λ, δ δ B ) (w, λ) V B X B The strong form: v q = D i δk j, in B, v = δ δ B, in B, v = 0, on B. We extend v by zero outside of B. Note that ωe u V B. 16 / 29

17 Duality argument Localized adjoint problem: find (v, q) V B X B with q dx = 0 such that B L B ((w, λ), (v, q)) = (w, D i δk j ) + (λ, δ δ B ) (w, λ) V B X B Choose (w, λ) = (ωe u, ωe p ) V B X B : (D i e u, δk j ) + (e p, δ) = (ωe u, D i δk j ) + (ωe p, δ δ B ) + (ωe p, δ B ) = L B ((ωe u, ωe p ), (v, q)) + (e p, ωδ B ) = a(ωe u, v) + b(v, ωe p ) b(ωe u, q) + (e p, ωδ B ) 17 / 29

18 Error representation e u L (D) + e p L (D) h T C max T T Dd h T + dist(t, D) η 1, (T ) ( v W 1 1 (B) + q L1(B) + B\B 2ρ(x0 ) x x 0 ( q + D 2 v ) ) dx ( ) + Cd 1 e u L (B) d 1 v L1(B\B d/2 ) + v L1(B\B d/2 ) + q L1(B\B d/2 ) + (e p, ωδ B + v ω) + Cρ β( ) u C 1,β (D + p ρ) C 0,β (D ρ) The pressure term: (e p, ωδ B + v ω) C 1 d max T T h h T η 1, (T ) 18 / 29

19 Regularity estimates Let (v, q) V B X B be the adjoint solution. Then v W 1 1 (B) + q L1(B) C ln d ρ, d 1 v L1(B\B d/2 ) + v L1(B\B d/2 ) + q L1(B\B d/2 ) C, x x 0 ( q(x) + D 2 v(x) ) dx C ln d ρ. B\B 2ρ(x 0) 19 / 29

20 Proof of regularity estimates Transformation to a reference domain: B B, translation and scaling by a factor cd Finitely many B suffice. Transformed localized adjoint problem: find (ṽ, q) V B X B such that L B((w, λ), (ṽ, q)) = (w, D xi δkj ) + (λ, δ δ B ) (w, λ) V B X B 20 / 29

21 Green s matrix Maz ya and Rossmann (n = 3 published, n = 2 private communication): Assume that B R n, n = 2, 3, is a convex polyhedral domain and let (ṽ, q) be the adjoint solution in B. There exists {G lj ( x, ξ)} 1 l,j n+1, ( x, ξ) B B, such that for x B and 1 l n, n ṽ l ( x) = G lj ( x, ξ) f j ( ξ) d ξ + G l,n+1 ( x, ξ) g( ξ) d ξ, B j=1 B n q( x) = G n+1,j ( x, ξ) f j ( ξ) d ξ + G n+1,n+1 ( x, ξ) g( ξ) d ξ. B B j=1 Moreover, there is a constant C such that, for δ l,n+1 + α 1 and δ n+1,j + β 1, { D α x D β ξ G lj( x, ξ) C x ξ κ, if κ > 0, C ln x ξ, if κ = 0, where δ l,j is Kronecker s delta and κ = n + δ l,n+1 + α + δ n+1,j + β / 29

22 Regularization penalty Cρ β( ) u C 1,β (D + p ρ) C 0,β (D ρ) Caused by the use of a regularized Green s function. Can be bounded a priori for some β (0, 1). May take ρ arbitralily small. This leads to a completely a posteriori estimate. 22 / 29

23 Local Hölder regularity Let q > n, β = 1 n q. n = 2: W 2 q ( B) W 1 q ( B) estimates and Sobolev s inequality yield ũ C 1,β ( B) + p C 0,β ( B)/R C( ũ W 2 q ( B) + p W 1 q ( B) ) C( f Lq( B) + g W 1 q (Ω) ) n = 3: Schauder estimates (need g = 0 on B, unknown in 2-D) We can now prove: ũ C 1,β ( B) + p C 0,β ( B) C( f C 1,β ( B) + g C 0,β ( B) ) C( f Lq( B) + g W 1 q (Ω) ) Let f and g be as in the completely a posteriori corollary. Then for q > n and β = 1 n q, the solution of the Stokes equation satisfies u C 1,β (D ρ) + p C 0,β (D ρ) C( f Lq(B) + g W 1 q (B) + d 2 f W 1 q (Ω) + d 2 g Lq(Ω)). 23 / 29

24 Corollary: Completely a posteriori error estimate If n = 3 assume that the incompressibility g = 0 near Ω D. Define F (f, g, d) = f Lq(Ω) + g W 1 q (Ω) + d 2( f W 1 q (Ω) + g L q(ω) E = max T T Dd h T h T + dist(t, D) η 1, (T ), β = 1 n q, ρ = min(d, h, (E/F (f, g, d))1/β ). Then there exists q > n such that: If E = 0, then (u u h ) L (D) + p p h L (D) C If E 0, then ), ( ln 1 h ) αn 1 d max T T h h T η 1, (T ). (u u h ) L (D)+ p p h L (D) ( C 1 + ln d ρ ) ( E + C ln 1 ) αn 1 h d max h T η 1, (T ). T T h 24 / 29

25 Adaptive algorithm solve estimate mark refine. Given D, d, and D d as above, we let h T η(t ) = h T + dist(t, D) η 1, (T ), T D d, h T d η 1, (T ), T D d =. We mark an element T T h for refinement if η(t ) 0.5 max T T h η(t ). We use polynomial degree k = 2. Software: ALBERTA. 25 / 29

26 Computational example Let (x 1)(x 0.5) 5 64(x 0.5) 6, x 0.5, w(x) = 1, 0 x < 0.5, w( x), x < 0. Letting (r, φ) be polar coordinates, we also define Finally, we let γ(r, θ) = r 1.5( 3 sin(θ/2) sin(3θ/2) ). u 1 (x, y) = y u 2 (x, y) = ( ) w(x)w(y)γ(r(x, y), θ(x, y)), ( ) w(x)w(y)γ(r(x, y), θ(x, y)), x u(x, y) = (u 1 (x, y), u 2 (x, y)). Then u = 0 on Ω and u = 0 in Ω. Finally, we let p(r, θ) = 6r 0.5 cos(θ/2). Regularity: u H 3/2 ɛ (Ω) for any ɛ > / 29

27 A crack domain D d D Figure: Initial mesh with D = B 1/4 ( 1, 1) and D d = B 2 ( 1, 1) outlined (left). Computational mesh with degrees of freedom (right). 27 / 29

28 Convergence. Error vs number of degrees of freedom Slope=-1 grad(u-u_h) _D 0.01 Estimator u-u_h _Omega / 29

29 Related works Y. He, J. Xu, A. Zhou, and J. Li, Local and parallel finite element algorithms for the Stokes problem, Numer. Math. 109 (2008), J. Guzmán and D. Leykekhman, Pointwise error estimates of finite element approximations to the Stokes problem on convex polyhedra, Tech. report, 2010, submitted. V. Girault, R. H. Nochetto, and R. L. Scott, Stability of the finite element Stokes projection in W 1,, C. R. Math. Acad. Sci. Paris 338 (2004), V. Girault, R. H. Nochetto, and R. L Scott, Maximum-norm stability of the finite element Stokes projection, J. Math. Pures Appl. (9) 84 (2005), / 29