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 or similar conditions div u = g , curl u = 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 or similar conditions div u = g , curl u = 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 on 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 div v = 0 (in general Λ v = 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 div v = 0 (in general Λ v = 0). Simple example: stationary Stokes problem − ν ∆ u = f − ∇ p in Ω , u = u 0 on ∂ Ω , div u 0 = 0 . � � ∀ w ∈ S 1 , 2 ν ∇ u : ∇ w dx = f · w dx 0 . Ω Ω S 1 , 2 = closure of smooth div free functions with compact supports 0 with respect to H 1 -norm.

Simplest version of the functional a posteriori error estimate for Stokes For any v ∈ S 1 , 2 + u 0 , q ∈ � L 2 (Ω) , τ ∈ H (Ω , Div ) , 0 � ν ∇ ( u − v ) � ≤ � τ − ν ∇ v + q I � + C F Ω � f + Div τ � =: 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 0 0 and the exact solution="Deviation estimate".

Simplest version of the functional a posteriori error estimate for Stokes For any v ∈ S 1 , 2 + u 0 , q ∈ � L 2 (Ω) , τ ∈ H (Ω , Div ) , 0 � ν ∇ ( u − v ) � ≤ � τ − ν ∇ v + q I � + C F Ω � f + Div τ � =: 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 0 0 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). α u − Div σ + Div ( a ⊗ u ) = f in Ω , (1) σ = ν ∇ u − p I in Ω , (2) div u = 0 in Ω , (3) u = u 0 on ∂ Ω . (4)

For any v ∈ S 1 , 2 + u 0 , q ∈ � L 2 (Ω) , and τ ∈ H (Ω , Div ) , the 0 following estimate holds: � � � � � � � � � µ 1 / 2 r ( v , τ ) � ν − 1 / 2 ( τ − ν ∇ v + q I ) | � �� � ≤ C F Ω | | u − v | | | � + � , (5) where 1 µ ( x ) = F Ω α ( x ) , (6) ν ( x ) + C 2 r ( v , τ ) = Div τ − a · ∇ v − α v + f , � | 2 := ( ν |∇ w | 2 + α w 2 ) dx and | | | w | | Ω

Our goal is to extend this approach to a wider set V 0 + u 0 where V 0 = W 1 , 2 0 Key point: we need an estimate d ( v , S 1 , 2 0 ) := inf �∇ ( v − w 0 ) � ≤ Π S 1 , 2 0 ( v ) , w 0 ∈ S 1 , 2 (Ω , R d ) 0 where Π S 1 , 2 0 ( v ) is a computable measure/functional.

Our goal is to extend this approach to a wider set V 0 + u 0 where V 0 = W 1 , 2 0 Key point: we need an estimate d ( v , S 1 , 2 0 ) := inf �∇ ( v − w 0 ) � ≤ Π S 1 , 2 0 ( v ) , w 0 ∈ S 1 , 2 (Ω , R d ) 0 where Π S 1 , 2 0 ( v ) is a computable measure/functional. Then simple manipulations yield a guaranteed bound for v ∈ V 0 + u 0 for the Stokes problem: ν �∇ ( u − v ) �≤ ν �∇ ( u − w 0 ) � + ν �∇ ( w 0 − v ) � ≤ � ν ∇ w 0 − τ − q I � + C F Ω � div τ + f � + ν �∇ ( w 0 − v ) � ≤ � ν ∇ v − τ − q I � + C F Ω � div τ + f � + 2 ν �∇ ( w 0 − v ) � ≤ � ν ∇ v − τ − q I � + C F Ω � div τ + f � + 2 ν Π S 1 , 2 0 ( v ))

Our goal is to extend this approach to a wider set V 0 + u 0 where V 0 = W 1 , 2 0 Key point: we need an estimate d ( v , S 1 , 2 0 ) := inf �∇ ( v − w 0 ) � ≤ Π S 1 , 2 0 ( v ) , w 0 ∈ S 1 , 2 (Ω , R d ) 0 where Π S 1 , 2 0 ( v ) is a computable measure/functional. Then simple manipulations yield a guaranteed bound for v ∈ V 0 + u 0 for the Stokes problem: ν �∇ ( u − v ) �≤ ν �∇ ( u − w 0 ) � + ν �∇ ( w 0 − v ) � ≤ � ν ∇ w 0 − τ − q I � + C F Ω � div τ + f � + ν �∇ ( w 0 − v ) � ≤ � ν ∇ v − τ − q I � + C F Ω � div τ + f � + 2 ν �∇ ( w 0 − v ) � ≤ � ν ∇ v − τ − q I � + C F Ω � div τ + f � + 2 ν Π S 1 , 2 0 ( v ))

Stability Theorem/Lemma Theorem (Aziz-Babuska, Ladyzhenskaya–Solonnikov) For any f ∈ L 2 (Ω) such that { f } Ω = 0 , there exists a function ∈ W 1 , 2 (Ω , R d ) such that w f 0 div w f = f and �∇ w f � ≤ κ Ω � f � , (7) where κ Ω 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 of 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 c Ω such that � Ω p div w dx inf sup ≥ c Ω . (8) � p � �∇ w � p ∈ � L 2 (Ω) w ∈ V 0 { p } Ω = 0 , p � = 0 w � = 0 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 , 129–151 (1974).

Also can be viewed as a weak form of the Poincar´ e inequality: � p � ≤ C �∇ p � − 1 , { p } Ω = 0 . eorie des ´ J. Ne ˇ c as. Les M´ Equations Elliptiques , ethodes Directes en Th´ Masson et Cie, ´ Editeurs, Paris; Academia, ´ Editeurs, Prague 1967.

Generalisations 1 < q < + ∞ Theorem can be extended to L q spaces. Theorem (Bogovskii, Piletskas (79’-80’)) Let f ∈ L q (Ω) . If { f } Ω = 0 , then there exists v f ∈ W 1 , q (Ω , R d ) 0 such that div v f = f �∇ v f � q , Ω ≤ κ Ω , q � div v f � q , Ω , (9) and where κ Ω , q ( κ Ω , 2 = κ Ω ) is a positive constant, which depends only on Ω .

Generalisations 1 < q < + ∞ Theorem can be extended to L q spaces. Theorem (Bogovskii, Piletskas (79’-80’)) Let f ∈ L q (Ω) . If { f } Ω = 0 , then there exists v f ∈ W 1 , q (Ω , R d ) 0 such that div v f = f �∇ v f � q , Ω ≤ κ Ω , q � div v f � q , Ω , (9) and 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 0 , Rend. Mat. Acc. Lincei, 2003

Inf–Sup ⇒ an upper bound of Π S 1 , 2 0 ( v ) � � 1 � 1 ◦ 2 �∇ ( w − v ) � 2 − φ div w v ) � 2 = inf inf 2 �∇ ( v − sup dx ◦ w ∈ V 0 v ∈ S 1 , 2 φ ∈ � L 2 (Ω) 0 Ω   � � 1 � 2 |∇ w | 2 − φ div ( w + v )  inf  . = sup dx w ∈ V 0 φ ∈ � L 2 (Ω) Ω Consider the term [...]. Let ¯ w � = 0 be an element of V 0 ; t ¯ w ∈ V 0 ∀ t ∈ R . Therefore, � � 1 � 2 |∇ w | 2 − φ div ( w + v ) inf dx ≤ w ∈ V 0 Ω � � ≤ 1 2 t 2 �∇ ¯ w � 2 − t φ div ¯ w dx − φ div v dx . (10) Ω Ω

�� � t ∗ := w � − 2 , minimizes the right–hand side and φ div ¯ �∇ ¯ w dx Ω   2 �   φ div ¯ wdx � � ( ... ) dx ≤ − 1 Ω − inf φ div v dx . (11) w � 2 2 �∇ ¯ w ∈ V 0 Ω Ω