Estimates of the distance to the set of divergence free fields and applications to a posteriori error estimation

Transcription

1 Estimates of the distance to the set of divergence free fields and applications to a posteriori error estimation S. Repin V.A. Steklov Institute of Mathematics, St. Petersburg and University of Jyväskylä, Finland Special Semester on Computational Methods in Science and Engineering, RICAM, Linz, Austria, 216

2 Motivation The condition div u = or similar conditions divu = g, curlu =,... arise in various mathematical models (e.g., flow of incompressible fluids, Maxwell type equations). Such type conditions may generate serious difficulties (especially in 3D problems).

3 Motivation The condition div u = or similar conditions divu = g, curlu =,... arise in various mathematical models (e.g., flow of incompressible fluids, Maxwell type equations). Such type conditions may generate serious difficulties (especially in 3D problems). Usually, numerical solutions satisfy it only approximately. Typical approaches in the theory of incompressible fluids are based on minimax settings and mixed velocity pressure or velocity-stress-pressure formulations and discretizations subject to discrete inf sup conditions

4 Guaranteed error bounds for approximations Two different ways depending on what is taken as the basic space: A. Operate only with div free functions. B. Use estimates of the distance to sets of functions defined by the condition divv = (in general Λv = ).

5 Guaranteed error bounds for approximations Two different ways depending on what is taken as the basic space: A. Operate only with div free functions. B. Use estimates of the distance to sets of functions defined by the condition divv = (in general Λv = ). Simple example: stationary Stokes problem Ω ν u = f p in Ω, u = u on Ω, divu =. ν u : w dx = Ω f w dx w S 1,2. S 1,2 = closure of smooth div free functions with compact supports with respect to H 1 -norm.

6 Simplest version of the functional a posteriori error estimate for Stokes For any v S 1,2 + u, q L 2 (Ω), τ H(Ω, Div), ν (u v) τ ν v + qi + C F Ω f + Divτ =: M(v, q, τ) v, q, and τ can be viewed as approximations of the velocity, pressure, and stress. C F is the Friedrichs constant. M(v, q, τ) is a computable measure of the distance between ANY v S 1,2 + u and the exact solution="deviation estimate".

7 Simplest version of the functional a posteriori error estimate for Stokes For any v S 1,2 + u, q L 2 (Ω), τ H(Ω, Div), ν (u v) τ ν v + qi + C F Ω f + Divτ =: M(v, q, τ) v, q, and τ can be viewed as approximations of the velocity, pressure, and stress. C F is the Friedrichs constant. M(v, q, τ) is a computable measure of the distance between ANY v S 1,2 + u and the exact solution="deviation estimate". S. R. A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp., 2. S. R. book De Gruyter, 28 (RICAM series) evolutionary models U. Langer, S. Matculevich, M. Wolfmauer and S. R. exterior domains D. Pauly and S. R. modeling errors S. Sauter and S. R....

8 Other estimates (for generalized Stokes, Oseen, Navier Stokes) have similar structures. Example: the generalised Oseen problem (typically arises in time discretisation schemes for NS equations). αu Divσ + Div(a u) = f in Ω, (1) σ = ν u pi in Ω, (2) divu = in Ω, (3) u = u on Ω. (4)

9 For any v S 1,2 + u, q L 2 (Ω), and τ H(Ω, Div), the following estimate holds: u v }{{} C F Ω µ 1/2 r(v, τ) + ν 1/2 (τ ν v + qi), (5) where 1 µ(x) = ν(x) + CF 2 (6) Ωα(x), r(v, τ) = Divτ a v αv + f, and w 2 := Ω (ν w 2 + αw 2 ) dx

10 Our goal is to extend this approach to a wider set V + u where V = W 1,2 d(v, S 1,2 Key point: we need an estimate ) := inf w S 1,2 (Ω,Rd ) (v w ) Π S 1,2(v), where Π S 1,2(v) is a computable measure/functional.

11 Our goal is to extend this approach to a wider set V + u where V = W 1,2 d(v, S 1,2 Key point: we need an estimate ) := inf w S 1,2 (Ω,Rd ) (v w ) Π S 1,2(v), where Π S 1,2(v) is a computable measure/functional. Then simple manipulations yield a guaranteed bound for v V + u for the Stokes problem: ν ( u v) ν ( u w ) + ν ( w v) ν w τ qi +C F Ω divτ +f + ν ( w v) ν v τ qi +C F Ω divτ +f + 2ν ( w v) ν v τ qi +C F Ω divτ +f + 2νΠ S 1,2(v))

12 Our goal is to extend this approach to a wider set V + u where V = W 1,2 d(v, S 1,2 Key point: we need an estimate ) := inf w S 1,2 (Ω,Rd ) (v w ) Π S 1,2(v), where Π S 1,2(v) is a computable measure/functional. Then simple manipulations yield a guaranteed bound for v V + u for the Stokes problem: ν ( u v) ν ( u w ) + ν ( w v) ν w τ qi +C F Ω divτ +f + ν ( w v) ν v τ qi +C F Ω divτ +f + 2ν ( w v) ν v τ qi +C F Ω divτ +f + 2νΠ S 1,2(v))

13 Stability Theorem/Lemma Theorem (Aziz-Babuska, Ladyzhenskaya Solonnikov) For any f L 2 (Ω) such that {f } Ω =, there exists a function w f W 1,2 (Ω, R d ) such that divw f = f and w f κ Ω f, (7) where κ Ω is a positive constant depending on Ω. I. Babuška and A. K. Aziz. Surway lectures on the mathematical foundations of the finite element method. The mathematical formulations of the FEM with applications to partial differential equations, Academic Press, New York, 1972, O. A. Ladyzenskaja and V. A. Solonnikov. Some problems of vector analysis, and generalized formulations of boundary value problems for the Navier-Stokes equation, Zap. Nauchn, Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 59(1976),

14 Inf-Sup condition There exists a positive constant c Ω such that Ω p divw dx sup p w inf p L 2 (Ω) {p} Ω =, p w V w c Ω. (8) In view of Stability Lemma, the condition c Ω = (κ Ω ) 1. F. Brezzi. On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers, R.A.I.R.O., Annal. Numer. R2, (1974).

15 Also can be viewed as a weak form of the Poincaré inequality: p C p 1, {p} Ω =. J. Nečas. Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague 1967.

16 Generalisations 1 < q < + Theorem can be extended to L q spaces. Theorem (Bogovskii, Piletskas (79-8 )) Let f L q (Ω). If {f } Ω =, then there exists v f W 1,q (Ω, R d ) such that divv f = f and v f q,ω κ Ω,q divv f q,ω, (9) where κ Ω,q ( κ Ω,2 = κ Ω ) is a positive constant, which depends only on Ω.

17 Generalisations 1 < q < + Theorem can be extended to L q spaces. Theorem (Bogovskii, Piletskas (79-8 )) Let f L q (Ω). If {f } Ω =, then there exists v f W 1,q (Ω, R d ) such that divv f = f and v f q,ω κ Ω,q divv f q,ω, (9) where κ Ω,q ( κ Ω,2 = κ Ω ) is a positive constant, which depends only on Ω. For q = 1 and q = + in general similar estimates do not hold, e.g., B. Dacorogna, N. Fusco, L. Tartar, On the solvability of the equation divu = f in L 1 and in C, Rend. Mat. Acc. Lincei, 23

18 Inf Sup an upper bound of Π S 1,2(v) inf v S 1,2 1 2 ( v v) 2 = inf = sup inf φ L 2 (Ω) sup w V φ L 2 (Ω) Ω w V Ω ( ) 1 2 (w v) 2 φdivw dx ( 1 2 w 2 φdiv(w + v) Consider the term [...]. Let w be an element of V ; t w V t R. Therefore, ( ) 1 inf 2 w 2 φ div(w + v) dx w V Ω 1 2 t2 w 2 t Ω Ω ) dx. φ div w dx φ divv dx. (1)

19 ( t := Ω inf w V Ω ) φ div w dx w 2, minimizes the right hand side and φ div wdx (... ) dx 1 Ω 2 w 2 2 φ divv dx. (11) Ω

20 Since w is an arbitrary function, we can take sup over w for the quotient. In view of Inf Sup condition, φ div w dx Hence inf w V Ω sup w V w Ω w c Ω φ φ L(Ω). ( 1 2 w 2 φ div(w+v)) dx c2 Ω 2 φ 2 Ω φ divv dx c2 Ω 2 φ 2 + φ divv. Take supremum over φ and conclude that inf v S 1,2 (v v) c 1 Ω divv =: Π S 1,2(v). (12)

21 Another way Lemma (Distance to S 1,q (Ω, R d )) For any v W 1,q (Ω, R d ), d(v, S 1,q (Ω, R d )) κ Ω,q divv q,ω. (13) Set f = divv, {f } Ω =. Then, a function v f W 1,q (Ω, R d ) exists such that divv f = divv, so that and obtain v := v v f S 1,q (Ω) (v v ) q,ω = v f q,ω κ Ω,q divv q,ω.

22 We need estimates c Ω..., κ Ω,q...? They are known for some domains (mainly for d = 2 and q = 2) C. Horgan and L. Payne, On inequalities of Korn, Friedrichs and Babuska Aziz., Arch. Ration. Mech. Anal., G. Stoyan. Towards Discrete Velte Decompositions and Narrow Bounds for Inf-Sup Constants, Computers and Mathematics with Applications, 1999 M. Dobrowolski. On the LBB constant on stretched domains, Math. Nachr., 23 M. Kessler. Die Ladyzhenskaya Konstante in der numerischen Behandlung von Stromungsproblemen. Doktorgrades der Bayerischen Univ. Wurzburg 2. M.A. Olshanskii and E.V. Chizhonkov. On the best constant in the inf sup condition for prolonged rectangular domains, Matematicheskie Zametki, 2 L. E. Payne. A bound for the optimal constant in an inequality of Ladyzhenskaya and Solonnikov, IMA Journal of Appl. Math., 27 M. Costabel and M. Dauge. On the inequalities of Babuska Aziz, Friedrichs and Horgan Payne, Arch. Ration. Mech. Anal., 215

23 Summary: Constants κ Ω,q are not known except some special cases. So far we do not know a method able to compute guaranteed and realistic bounds of these constants for arbitrary three dimensional Lipshitz domains or, at least, for polygonal 3D domains. Moreover, it is too optimistic to hope on getting some simple methods for complicated domains in 3D. In practice, we need estimates for spaces of functions vanishing only on a part of the boundary.

24 Estimates for functions vanishing on Γ D Γ Typical situation: Γ consists of Γ D and Γ N, meas d 1 Γ D, Γ N >. Approximation satisfies v W 1,q,Γ D (Ω, R d ) := {v W 1,q (Ω, R d ) v = on Γ D }. We need an upper bound of d(v, S 1,q,Γ D (Ω, R d )) := inf (v v ) q,ω, (14) v S 1,q,Γ (Ω,R d ) D where { } S 1,q,Γ D (Ω, R d ) = v W 1,q,Γ D (Ω, R d ) divv =,

25 Lemma Let v W 1,q,Γ D (Ω, R d ) and {divv} Ω =. (15) Then, d(v, S 1,q,Γ D (Ω, R d )) κ Ω,q divv q,ω. (16) Here v must be post processed in order to satisfy the integral (zero mean divergence) condition. For q = 2 it was used in S. R. and R. Stenberg. A posteriori error estimates for the generalized Stokes problem. J. Math. Sci., (27)

26 Estimates without "preconditions". First, the most interesting case: q = 2 Theorem For v W 1,2,Γ D (Ω, R d ) d(v, S 1,2,Γ D (Ω, R d )) κ Ω {divu 1 } Ω {divu 1 } Ω divv divu 1 {divv} Ω + divv dx Ω u 1, where u 1 is the solution of an auxiliary elliptic problem u 1 = in Ω, u 1 = on Γ D, u 1 n + n = on Γ N. S. R. Estimates of deviations from the exact solution of the generalized Oseen problem. J. Math. Sci., 195(213), 1, S. R. Estimates of the distance to the set of solenoidal vector fields and applications to a posteriori error control. Comput. Meth. Appl. Math., 215

27 Corollary. We have a somewhat different estimate: d(v, S 1,2,Γ D (Ω, R d )) κ Ω divv + C 1 divv dx, (17) Ω where C 1 = ( ) 1 divu 1 κ Ω u 1 u

28 L q case d(v, S 1,2,Γ D (Ω, R d )) = inf (v v ) q,ω v S 1,q,Γ (Ω,R d ) D κ Ω,q divv q,ω + C,q divv dx, where ( ) 1 divu 1 q,ω C,q = u 1 q 1 κ Ω,q + 1. u q,ω 1 q,ω Ω

29 Remark: Constants can be evaluated by finite dimensional auxiliary problems. Let u,h, solves the problem ( u,h : w h + divw h ) dx = w V,Γ h D (Ω) V,ΓD (Ω), Ω Then, repeating above arguments, we find that inf (ṽ v) (v u,h) = ṽ V,ΓD (Ω) 1 u,h Ω divv dx and the respective constant is C h 1 = 1 u,h ( ) divu,h κ Ω u,h + 1.

30 Estimates of the distance to S 1,2 based upon decomposition of Ω A modified L q stability Lemma Ω is divided into non-overlapping Lipschitz subdomains Ω i, i = 1, 2,...N. Assumption: subdomains Ω i are relatively simple, so that the respective constants (or suitable estimates of them) are known. Lemma Let f L q (Ω). If f satisfies {f } Ωi = for i = 1, 2,..., N, then there exists v f W 1,q (Ω, R d ) such that N divv f = f and v f q q,ω κ q Ω i,q f q q,ω i i=1 max{κ q i Ω i,q } f q q,ω, (18) where κ Ωi,q are positive constants associated with subdomains Ω i.

31 v Corollary: Let { } 1,q W N;,Γ D (Ω, R d )= W 1,q,Γ D (Ω, R d ) {divv} Ωi = i = 1, 2,..., N d(v, S 1,q,Γ D (Ω)) Then, ( N ) 1/q κ q Ω i,q divv q q,ω i. (19) i=1 Satisfaction of N integral conditions can be performed without essential difficulties unlike the methods based on constructing a sufficiently wide subspace of divergence free functions and computing the estimate directly (especially in the three dimensional case).

32 Comment: orthogonal projection to W 1,2 N;,Γ D (Ω, R d ) is not required! We can deduce fully computable estimates of the distance between 1,2 v W,Γ D (Ω, R d ) and S 1,2,Γ D by combining two steps. Define the vector µ: µ i = divv dx. Ω i "Post processing"of v construct a correction function w µ W 1,2,Γ D such that Ω i divw µ dx = µ i for i = 1, 2,..., N. Then d(v, S 1,2,Γ D (Ω)) d(v w µ, S 1,2,Γ D (Ω)) + w µ 2,Ω and (19) yields a simple estimate d(v, S 1,2,Γ D (Ω)) ( N ) 1/2 κ 2 Ω i div(v w µ ) 2 Ω i + w µ 2,Ω. i=1 (2)

33 Decomposition into overlapping domains Ω is decomposed into a collection of overlapping Lipschitz subdomains D k, k = 1, 2,..., K, C Dk,q are the respective constants. Ω = K D k = k=1 N Ω i, Ω i Ω j = for i j (21) i=1 D k D l is either empty or consists of one or several subdomains Ω i. For any Ω i there exists at least one D k such that Ω i D k.

34 Lemma (Stability Lemma for overlapped subdomains, q = 2) Let f L 2 (Ω) be such that {f } Ωi = i = 1, 2,..., N. (22) Then, there exists a function v f W 1,2 (Ω, R d ) such that divv f = f in Ω (23) and where v f Ω N C i f Ωi (24) i=1 C i = { inf ρ CDk,q if Ω k, ρ k = i D k, k=1,...,k + if Ω i D k, (25)

35 Proof. Define f i (x) = { f if x Ωi, if x Ω i, There exists at least one D k such that Ω i D k. If there are several D k containing Ω i, then we select k such that C Dk,q is minimal (see (25)).

36 Proof. Define f i (x) = { f if x Ωi, if x Ω i, There exists at least one D k such that Ω i D k. If there are several D k containing Ω i, then we select k such that C Dk,q is minimal (see (25)). Since {f i } Dk =, and D k is a Lipschitz domain, we can find v fi W 1,q (D k, R d ) such that divv fi = f i in D k (26) and v fi q,dk C i f i q,dk = C i f q,ωi. (27)

37 We extend v fi by zero to Ω \ D k and find that (26) holds in Ω. Moreover, v fi Ω,q C i f q,ωi. (28)

38 We extend v fi by zero to Ω \ D k and find that (26) holds in Ω. Moreover, v fi Ω,q C i f q,ωi. (28) Set v f = N i=1 v fi W 1,q (Ω, R d ). Then divv f = f. Since v f q,ω n v fi q,dk i=1 N C i f i q,dk = i=1 N C i f i q,ωi, i=1 we arrive at (24).

39 Similar result holds for L q. It implies the corollary Lemma (Distance to the set of divergence free fields) Assume v W 1,q (Ω, R d ) satisfies {divv} Ωi = i = 1, 2,..., N, and divv L δ (Ω), where δ q. Then, there exists v W 1,γ (Ω, R d ) such that divv =, v = v on Γ, and (v v ) Ω,q N C i Ω i 1 q 1 δ divv Ωi,q. (29) i=1

40 Examples a 1 D 2 b 2 b 1 Ω D1 1 Ω 4 Ω 5 Ω 2 Ω 3 a 2 a 3 D 3 b3 Ω = D 1 D 2 D 3, D i are rectangles, D 2 = Ω 2 Ω 4, and D 3 = Ω 3 Ω 5. For a rectangular domain a,b := (, a) (, b) a, b >, a > b C a,b 1 b 2d(a + d), (3) where d = a 2 + b 2 (length of the diagonal).

41 Let v V (Ω) be such that {divv} Ωi = i = 1, 2, 3, 4, 5. (31) There exists a divergence free field v vanishing on Γ such that (v v ) C D1 divv Ω1 + C D2 ( divv Ω2 + divv Ω4 ) + C D3 ( divv Ω3 + divv Ω5 ) where C Dk = 1 2d b k 2 + 2a kd k, k k = 1, 2, 3.

42 Another example a D D 2 D 1 3 r a Let Ω = D 1 D 2 D 3, where D 1 and D 2 are isosceles triangles and D 2 is a circle (Fig. 1 right). Let Ω 1 = D 1 D 2 (measω 1 > ), Ω 2 = D 1 \ Ω 1 Ω 3 = D 2 \ (D 1 D 3, Ω 4 = D 3 D 2 (measω 4 > ), Ω 5 = D 3 \ Ω 4 and v satisfies zero mean conditions. Then, there exists v such that divv =, v = v on Γ and (v v ) C D1 divv Ω1 + C D2 ( divv Ω2 + divv Ω3 + divv Ω4 ) +C D3 divv Ω5

43 Thank you for attention