Estimates of the distance to the set of divergence free fields and - - PowerPoint PPT Presentation

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 Jyv¨ askyl¨ a, Finland Special Semester on Computational Methods in Science and Engineering, RICAM, Linz, Austria, 2016

Motivation

The condition div u = 0

• r similar conditions divu = g, curlu = 0,...

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).

Motivation

The condition div u = 0

• r similar conditions divu = g, curlu = 0,...

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

• n minimax settings

and mixed velocity–pressure or velocity-stress-pressure formulations and discretizations subject to discrete inf–sup conditions

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 = 0 (in general Λv = 0).

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 = 0 (in general Λv = 0). Simple example: stationary Stokes problem − ν∆u = f − ∇p in Ω, u = u0

• n ∂Ω,

divu0 = 0.

• Ω

ν∇u : ∇w dx =

• Ω

f · w dx ∀w ∈ S1,2

0 .

S1,2 = closure of smooth div free functions with compact supports with respect to H1-norm.

Simplest version of the functional a posteriori error estimate for Stokes

For any v ∈ S1,2 + u0, q ∈ L2(Ω), τ ∈ H(Ω, Div), ν∇(u−v) ≤ τ −ν∇v + qI + CFΩf + Divτ =: M(v, q, τ) v, q, and τ can be viewed as approximations of the velocity, pressure, and stress. CF is the Friedrichs constant. M(v, q, τ) is a computable measure of the distance between ANY v ∈ S1,2 + u0 and the exact solution="Deviation estimate".

Simplest version of the functional a posteriori error estimate for Stokes

For any v ∈ S1,2 + u0, q ∈ L2(Ω), τ ∈ H(Ω, Div), ν∇(u−v) ≤ τ −ν∇v + qI + CFΩf + Divτ =: M(v, q, τ) v, q, and τ can be viewed as approximations of the velocity, pressure, and stress. CF is the Friedrichs constant. M(v, q, τ) is a computable measure of the distance between ANY v ∈ S1,2 + u0 and the exact solution="Deviation estimate".

• S. R. A posteriori error estimation for variational problems with uniformly convex functionals. Math.

Comp., 2000.

• S. R. book De Gruyter, 2008 (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. ...............

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 = 0 in Ω, (3) u = u0

• n ∂Ω.

(4)

For any v ∈ S1,2 + u0, q ∈ L2(Ω), and τ ∈ H(Ω, Div), the following estimate holds: | | |u − v| | | ≤ CFΩ

• µ1/2r(v, τ)
• +
• ν−1/2(τ − ν∇v + qI)
• ,

(5) where µ(x) = 1 ν(x) + C 2

FΩα(x),

(6) r(v, τ) = Divτ − a · ∇v − αv + f, and | | |w| | |2 :=

• Ω

(ν|∇w|2 + αw2) dx

Our goal is to extend this approach to a wider set V0 + u0 where V0 = W 1,2 Key point: we need an estimate d(v, S1,2

0 ) :=

inf

w0∈S1,2 (Ω,Rd)

∇(v − w0) ≤ ΠS1,2

0 (v),

where ΠS1,2

0 (v) is a computable measure/functional.

Our goal is to extend this approach to a wider set V0 + u0 where V0 = W 1,2 Key point: we need an estimate d(v, S1,2

0 ) :=

inf

w0∈S1,2 (Ω,Rd)

∇(v − w0) ≤ ΠS1,2

0 (v),

where ΠS1,2

0 (v) is a computable measure/functional.

Then simple manipulations yield a guaranteed bound for v ∈ V0 + u0 for the Stokes problem: ν ∇( u−v)≤ν ∇( u−w0) + ν ∇( w0−v) ≤ ν∇w0−τ −qI+CFΩdivτ +f + ν ∇( w0−v) ≤ ν∇v −τ −qI+CFΩdivτ +f + 2ν ∇( w0−v) ≤ ν∇v −τ −qI+CFΩdivτ +f + 2νΠS1,2

0 (v))

Our goal is to extend this approach to a wider set V0 + u0 where V0 = W 1,2 Key point: we need an estimate d(v, S1,2

0 ) :=

inf

w0∈S1,2 (Ω,Rd)

∇(v − w0) ≤ ΠS1,2

0 (v),

where ΠS1,2

0 (v) is a computable measure/functional.

Then simple manipulations yield a guaranteed bound for v ∈ V0 + u0 for the Stokes problem: ν ∇( u−v)≤ν ∇( u−w0) + ν ∇( w0−v) ≤ ν∇w0−τ −qI+CFΩdivτ +f + ν ∇( w0−v) ≤ ν∇v −τ −qI+CFΩdivτ +f + 2ν ∇( w0−v) ≤ ν∇v −τ −qI+CFΩdivτ +f + 2νΠS1,2

0 (v))

Stability Theorem/Lemma

Theorem (Aziz-Babuska, Ladyzhenskaya–Solonnikov) For any f ∈ L2(Ω) such that {f }Ω = 0, there exists a function wf ∈ W 1,2 (Ω, Rd) such that divwf = f and ∇wf ≤ κΩf , (7) where κΩ is a positive constant depending on Ω.

• I. Babuˇ

ska and A. K. Aziz. Surway lectures on the mathematical foundations of the finite element method. The mathematical formulations

• f the FEM with applications to partial differential equations, Academic

Press, New York, 1972, 5–359.

• 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), 81–116.

Inf-Sup condition

There exists a positive constant cΩ such that inf

p∈ L2(Ω) {p}Ω=0, p=0

sup

w∈V0 w=0

• Ω p divw dx

p ∇w ≥ cΩ. (8) In view of Stability Lemma, the condition cΩ = (κΩ)−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, 129–151 (1974).

Also can be viewed as a weak form of the Poincar´ e inequality: p ≤ C∇p−1, {p}Ω = 0.

• J. Neˇ
• cas. Les M´

ethodes Directes en Th´ eorie des ´ Equations Elliptiques,

Masson et Cie, ´ Editeurs, Paris; Academia, ´ Editeurs, Prague 1967.

Generalisations 1 < q < +∞

Theorem can be extended to Lq spaces. Theorem (Bogovskii, Piletskas (79’-80’)) Let f ∈ Lq(Ω). If {f }Ω = 0, then there exists vf ∈ W 1,q (Ω, Rd) such that divvf = f and ∇vf q,Ω ≤ κΩ,qdivvf q,Ω, (9) where κΩ,q ( κΩ,2 = κΩ) is a positive constant, which depends only

• n Ω.

Generalisations 1 < q < +∞

Theorem can be extended to Lq spaces. Theorem (Bogovskii, Piletskas (79’-80’)) Let f ∈ Lq(Ω). If {f }Ω = 0, then there exists vf ∈ W 1,q (Ω, Rd) such that divvf = f and ∇vf q,Ω ≤ κΩ,qdivvf q,Ω, (9) where κΩ,q ( κΩ,2 = κΩ) is a positive constant, which depends only

• n 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 L1 and in C0, Rend. Mat. Acc. Lincei, 2003

Inf–Sup ⇒ an upper bound of ΠS1,2

0 (v)

inf

• v∈S1,2

1 2∇(

• v −

v)2 = inf

w∈V0

sup

φ∈ L2(Ω)

• Ω

1 2∇(w −v)2 − φdivw

• dx

= sup

φ∈ L2(Ω)

  inf

w∈V0

• Ω

1 2|∇w|2 − φdiv(w + v)

• dx

  . Consider the term [...]. Let ¯ w = 0 be an element of V0; t ¯ w ∈ V0 ∀ t ∈ R. Therefore, inf

w∈V0

• Ω

1 2|∇w|2 − φ div(w + v)

• dx ≤

≤ 1 2 t2 ∇ ¯ w2 − t

• Ω

φ div ¯ w dx −

• Ω

φ divv dx. (10)

t∗ :=

• Ω

φ div ¯ w dx

• ∇ ¯

w−2 , minimizes the right–hand side and inf

w∈V0

• Ω

( ... ) dx ≤ −1 2  

• Ω

φ div ¯ wdx  

2

∇ ¯ w2 −

• Ω

φ divv dx. (11)

Since ¯ w is an arbitrary function, we can take sup over ¯ w for the

• quotient. In view of Inf–Sup condition,

− sup

¯ w∈V0 ¯ w=0

• Ω

φ div ¯ w dx ∇ ¯ w ≤ −cΩ φ ∀ φ ∈ L(Ω). Hence inf

w∈V0

• Ω

(1 2|∇w|2− φ div(w+v)) dx ≤ −c2

Ω

2 φ2−

• Ω

φ divv dx ≤ ≤ −c2

Ω

2 φ2 + φ divv . Take supremum over φ and conclude that inf

• v∈S1,2

∇(v−

• v) ≤ c−1

Ω

divv =: ΠS1,2

0 (v).

(12)

Another way

Lemma (Distance to S1,q

0 (Ω, Rd))

For any v ∈ W 1,q (Ω, Rd), d(v, S1,q

0 (Ω, Rd)) ≤ κΩ,qdivvq,Ω.

(13) Set f = divv, {f }Ω = 0. Then, a function vf ∈ W 1,q (Ω, Rd) exists such that divvf = divv, so that v0 := v − vf ∈ S1,q

0 (Ω)

and obtain ∇(v − v0)q,Ω = ∇vf q,Ω ≤ κΩ,qdivvq,Ω.

We need estimates cΩ ≥ ..., κΩ,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., 1983.

• 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.,

2003

• M. Kessler. Die Ladyzhenskaya–Konstante in der numerischen Behandlung von
• Stromungsproblemen. Doktorgrades der Bayerischen Univ. Wurzburg 2000.

M.A. Olshanskii and E.V. Chizhonkov. On the best constant in the inf sup condition for prolonged rectangular domains, Matematicheskie Zametki, 2000

• L. E. Payne. A bound for the optimal constant in an inequality of

Ladyzhenskaya and Solonnikov, IMA Journal of Appl. Math., 2007

• M. Costabel and M. Dauge. On the inequalities of Babuska–Aziz, Friedrichs

and Horgan–Payne, Arch. Ration. Mech. Anal., 2015

Summary: Constants κΩ,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

• nly on a part of the boundary.

Estimates for functions vanishing on ΓD ⊂ Γ

Typical situation: Γ consists of ΓD and ΓN, measd−1ΓD, ΓN > 0. Approximation satisfies v ∈ W 1,q

0,ΓD(Ω, Rd) := {v ∈ W 1,q(Ω, Rd) | v = 0 on ΓD}.

We need an upper bound of d(v, S1,q

0,ΓD(Ω, Rd)) :=

inf

v0∈S1,q

0,ΓD (Ω,Rd)

∇(v − v0)q,Ω, (14) where S1,q

0,ΓD(Ω, Rd) =

• v ∈ W 1,q

0,ΓD(Ω, Rd) | divv = 0

• ,

Lemma Let v ∈ W 1,q

0,ΓD(Ω, Rd) and

{divv}Ω = 0. (15) Then, d(v, S1,q

0,ΓD(Ω, Rd)) ≤ κΩ,qdivvq,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., (2007)

Estimates without "preconditions". First, the most interesting case: q = 2 Theorem For v ∈ W 1,2

0,ΓD(Ω, Rd)

d(v, S1,2

0,ΓD(Ω, Rd)) ≤

κΩ |{divu1 }Ω| {divu1 }Ω divv − divu1 {divv}Ω +

• Ω

divv dx

• ∇u1

, where u1 is the solution of an auxiliary elliptic problem ∆u1 = 0 in Ω, u1 = 0

• n ΓD,

∇u1 n + n = 0

• n ΓN.
• S. R. Estimates of deviations from the exact solution of the generalized Oseen problem. J. Math. Sci. ,

195(2013), 1, 64–75.

• S. R. Estimates of the distance to the set of solenoidal vector fields and applications to a posteriori error
• control. Comput. Meth. Appl. Math. , 2015

• Corollary. We have a somewhat different estimate:

d(v, S1,2

0,ΓD(Ω, Rd)) ≤ κΩdivv + C1

• Ω

divv dx

• ,

(17) where C1 = 1 ∇u1

• κΩ

divu1 ∇u1 + 1

• .

Lq case

d(v, S1,2

0,ΓD(Ω, Rd)) =

inf

v0∈S1,q

0,ΓD (Ω,Rd)

∇(v − v0)q,Ω ≤ κΩ,qdivvq,Ω + C∗,q

• Ω

divv dx

• ,

where C∗,q = 1 ∇u1 q−1

q,Ω

• κΩ,q

divu1 q,Ω ∇u1 q,Ω + 1

• .

Remark: Constants can be evaluated by finite dimensional auxiliary

• problems. Let u∗,h, solves the problem
• Ω

(∇u∗,h : ∇wh + divwh) dx = 0 ∀w ∈ V h

0,ΓD(Ω) ⊂ V0,ΓD(Ω),

Then, repeating above arguments, we find that inf

• v∈V0,ΓD (Ω) ∇(

v − v) ≤ ∇(v − u∗,h) = 1 ∇u∗,h

• Ω

divv dx

• and the respective constant is

C h

1 =

1 ∇u∗,h

• κΩ

divu∗,h ∇u∗,h + 1

• .

Estimates of the distance to S1,2 based upon decomposition of Ω A modified Lq stability Lemma Ω is divided into non-overlapping Lipschitz subdomains Ωi, i = 1, 2, ...N. Assumption: subdomains Ωi are relatively simple, so that the respective constants (or suitable estimates of them) are known. Lemma Let f ∈ Lq(Ω). If f satisfies {f }Ωi = 0 for i = 1, 2, ..., N, then there exists vf ∈ W 1,q (Ω, Rd) such that divvf = f and ∇vf q

q,Ω ≤ N

• i=1

κq

Ωi,qf q q,Ωi ≤ max i

{κq

Ωi,q}f q q,Ω, (18)

where κΩi,q are positive constants associated with subdomains Ωi.

Corollary: Let v ∈ W 1,q

N;0,ΓD(Ω, Rd)=

• W 1,q

0,ΓD(Ω, Rd) | {divv}Ωi = 0 i = 1, 2, ..., N

• Then,

d(v, S1,q

0,ΓD(Ω)) ≤

N

• i=1

κq

Ωi,qdivvq q,Ωi

1/q . (19) 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).

Comment: orthogonal projection to W 1,2

N;0,ΓD(Ω, Rd) is not required!

We can deduce fully computable estimates of the distance between v ∈ W 1,2

0,ΓD(Ω, Rd) and S1,2 0,ΓD by combining two steps.

Define the vector µ: µi =

• Ωi

divv dx. "Post processing"of v ⇒ construct a correction function wµ ∈ W 1,2

0,ΓD such that

• Ωi

divwµ dx = µi for i = 1, 2, ..., N. Then d(v, S1,2

0,ΓD(Ω)) ≤ d(v − wµ, S1,2 0,ΓD(Ω)) + ∇wµ2,Ω

and (19) yields a simple estimate d(v, S1,2

0,ΓD(Ω)) ≤

N

• i=1

κ2

Ωidiv(v − wµ)2 Ωi

1/2 + ∇wµ2,Ω. (20)

Decomposition into overlapping domains

Ω is decomposed into a collection of overlapping Lipschitz subdomains Dk, k = 1, 2, ..., K, CDk,q are the respective constants. Ω =

K

• k=1

Dk =

N

• i=1

Ωi, Ωi ∩ Ωj = ∅ for i = j (21) Dk ∩ Dl is either empty or consists of one or several subdomains Ωi. For any Ωi there exists at least one Dk such that Ωi ⊂ Dk.

Lemma (Stability Lemma for overlapped subdomains, q = 2) Let f ∈ L2(Ω) be such that {f }Ωi = 0 ∀i = 1, 2, ..., N. (22) Then, there exists a function vf ∈ W 1,2 (Ω, Rd) such that divvf = f in Ω (23) and ∇vf Ω ≤

N

• i=1

Cif Ωi (24) where Ci = inf

k=1,...,K ρk,

ρk = CDk,q if Ωi ⊂ Dk, +∞ if Ωi ⊂ Dk, (25)

Proof.

Define fi(x) = f if x ∈ Ωi, if x ∈ Ωi, There exists at least one Dk such that Ωi ⊂ Dk. If there are several Dk containing Ωi, then we select k such that CDk,q is minimal (see (25)).

Proof.

Define fi(x) = f if x ∈ Ωi, if x ∈ Ωi, There exists at least one Dk such that Ωi ⊂ Dk. If there are several Dk containing Ωi, then we select k such that CDk,q is minimal (see (25)). Since {fi}Dk = 0, and Dk is a Lipschitz domain, we can find vfi ∈ W 1,q (Dk, Rd) such that divvfi = fi in Dk (26) and ∇vfiq,Dk ≤ Cifiq,Dk = Cif q,Ωi. (27)

We extend vfi by zero to Ω \ Dk and find that (26) holds in Ω. Moreover, ∇vfiΩ,q ≤ Cif q,Ωi. (28)

We extend vfi by zero to Ω \ Dk and find that (26) holds in Ω. Moreover, ∇vfiΩ,q ≤ Cif q,Ωi. (28) Set vf =

N

• i=1

vfi ∈ W 1,q (Ω, Rd). Then divvf = f . Since ∇vf q,Ω ≤

n

• i=1

∇vfiq,Dk ≤

N

• i=1

Cifiq,Dk =

N

• i=1

Cifiq,Ωi, we arrive at (24).

Similar result holds for Lq. It implies the corollary Lemma (Distance to the set of divergence free fields) Assume v ∈ W 1,q(Ω, Rd) satisfies {divv}Ωi = 0 i = 1, 2, ..., N, and divv ∈ Lδ(Ω), where δ ≥ q. Then, there exists v0 ∈ W 1,γ(Ω, Rd) such that divv0 = 0, v0 = v on Γ, and ∇(v − v0)Ω,q ≤

N

• i=1

Ci|Ωi|

1 q − 1 δ divvΩi,q.

(29)

Examples

D2 D3

b1

D1

Ω

a1 a2 a3 b3 b2

Ω2

4

Ω Ω 3 Ω5

1

Ω = D1 ∪ D2 ∪ D3, Di are rectangles, D2 = Ω2 ∪ Ω4, and D3 = Ω3 ∪ Ω5. For a rectangular domain ✷a,b := (0, a) × (0, b) a, b > 0, a > b C✷a,b ≤ 1 b

• 2d(a + d),

(30) where d = √ a2 + b2 (length of the diagonal).

Let v ∈ V0(Ω) be such that {divv}Ωi = 0 i = 1, 2, 3, 4, 5. (31) There exists a divergence free field v0 vanishing on Γ such that ∇(v − v0) ≤ CD1divvΩ1 + CD2(divvΩ2 + divvΩ4) + CD3(divvΩ3 + divvΩ5) where CDk = 1 bk

• 2d2

k + 2akdk,

k = 1, 2, 3.

Another example

D D D

1 3 2

a r a

Let Ω = D1 ∪ D2 ∪ D3, where D1 and D2 are isosceles triangles and D2 is a circle (Fig. 1 right). Let Ω1 = D1 ∩ D2 (measΩ1 > 0), Ω2 = D1 \ Ω1 Ω3 = D2 \ (D1 ∪ D3, Ω4 = D3 ∩ D2 (measΩ4 > 0), Ω5 = D3 \ Ω4 and v satisfies zero mean conditions. Then, there exists v0 such that divv0 = 0, v = v0 on Γ and ∇(v − v0) ≤ CD1divvΩ1 + CD2(divvΩ2 + divvΩ3 + divvΩ4) +CD3divvΩ5

Thank you for attention