Some remarks on grad-div stabilization of incompressible flow - PowerPoint PPT Presentation

Some remarks on grad-div stabilization of incompressible flow simulations Gert Lube Institute for Numerical and Applied Mathematics Georg-August-University G¨ ottingen M. Stynes Workshop Numerical Analysis for Singularly Perturbed Problems Dresden University of Technology, November 16-18, 2011 Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 1 / 27

Outline Incompressible Navier-Stokes model 1 Numerical analysis of grad-div stabilized Oseen problem 2 Some recent result on limit case γ → ∞ 3 A (potentially) new approach to parameter design 4 Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 2 / 27

Incompressible Navier-Stokes model Navier-Stokes problem Incompressible Navier-Stokes model: Find velocity u , pressure p ∂ t u − ∇ · ( 2 ν D u ) + ( u · ∇ ) u + ∇ p = f in ( 0 , T ] × Ω ∇ · u = 0 in [ 0 , T ] × Ω u | t = 0 = u 0 in Ω ⊆ R d no-slip boundary conditions u = 0 (for simplicity) deformation tensor D u = 1 2 ( ∇ u + ( ∇ u ) T ) viscosity ν (Reynolds number Re = U L ν ). Claude Louis George Gabriel Marie Henri Stokes Navier Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 4 / 27

Incompressible Navier-Stokes model Finite element approximation V = [ H 1 0 (Ω)] 3 , Q = L 2 0 (Ω) := { q ∈ L 2 (Ω) : � Ω q dx = 0 } T h – admissible (possibly anisotropic) mesh Ω = ∪ K ∈ T h K V h ⊂ V , Q h ⊂ Q Conforming finite element spaces: Basic Galerkin FE method: find ( u h , p h ): [ 0 , T ] − → V h × Q h s.t. ∀ ( v h , q h ) ∈ V h × Q h ( ∂ t u h , v h ) + ( 2 ν D u h , D v h ) + b S ( u h , u h , v h ) − ( p h , ∇ · v h ) = ( f , v h ) ( q h , ∇ · u h ) = 0 with skew-symmetric convective term b S ( u , v , w ) := 1 2 [(( u · ∇ ) v , w ) − (( u · ∇ ) w , v )] Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 5 / 27

Incompressible Navier-Stokes model Examples of inf-sup stable approximations Inf-sup stable velocity-pressure FE spaces V h × Q h ⊂ V × Q ( q h , ∇ · v h ) ∃ β � = β ( h ) s . t . q h ∈ Q h sup inf � q h � 0 �∇ v h � 0 ≥ β > 0 v h ∈ V h � No additional pressure stabilization required (at least for laminar flows) Taylor-Hood elements: for k ∈ N 0 (Ω)] d × [ P k ( T h ) ∩ C (Ω)] V TH × Q TH [ P k + 1 ( T h ) ∩ H 1 = or h h 0 (Ω)] d × [ Q k ( T h ) ∩ C (Ω)] , V TH × Q TH [ Q k + 1 ( T h ) ∩ H 1 = k ∈ N h h � problems with mass conservation with increasing order k Scott-Vogelius elements: on barycenter refined tetrahedral meshes T h 0 (Ω)] d × [ P disc V SV × Q SV h = [ P k + 1 ( T h ) ∩ H 1 ( T h ) ∩ L 2 0 (Ω)] , for k ≥ d − 1 h k with property ∇ · [ P k + 1 ( T h )] d ⊂ P disc ( T h ) � strong (pointwise) conservation of mass k Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 6 / 27

Incompressible Navier-Stokes model Grad-div stabilized Galerkin methods Galerkin FE method with grad-div stabilization: find ( u h , p h ): [ 0 , T ] − → V h × Q h s.t. ∀ ( v h , q h ) ∈ V h × Q h ( ∂ t u h , v h ) + ( 2 ν D u h , D v h ) + b S ( u h , u h , v h ) − ( p h , ∇ · v h ) + ( q h , ∇ · u h ) +( γ ∇ · u h , ∇ · v h ) = ( f , v h ) classical augmented Lagrangian approach, see e.g. F ORTIN /G LOWINSKI [1983] also applied to Maxwell problem introduced by H UGHES /F RANCA [1986] for equal-order interpolation with γ ∼ 0 ( h ) Numerical analysis for inf-sup stable interpolation by G ELHARD ET AL . [2005] and O LSHANSKII ET AL . [2009]: Choice γ ∼ 0 ( h ) is not correct in general case ! Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 7 / 27

Numerical analysis of grad-div stabilized Oseen problem Grad-div stabilized Oseen model Oseen model: For given b with ∇ · b = 0, find velocity u , pressure p C.W. Oseen − ν ∆ u + ( b · ∇ ) u + ∇ p + σ u = f in Ω ⊂ R d ∇ · u = 0 Grad-div stabilized Oseen problem: Find ( u h , p h ) ∈ V h × Q h ⊂ V × Q s.t. ∀ ( v , q ) ∈ V h × Q h : a γ ( u h , p h ; v h , q h ) := ( ν ∇ u h , ∇ v h ) + b S ( b , u h , v h ) + ( σ ∇ u h , v h ) � − ( p , ∇ · v ) + ( q , ∇ · u ) + γ K ( ∇ · u h , ∇ · v h ) K = ( f , v h ) K Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 9 / 27

Numerical analysis of grad-div stabilized Oseen problem Numerical analysis of grad-div stabilized Oseen problem: M. Olshanskii, G. Lube, T. Heister, J. L¨ owe: Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations , CMAME 198 (2009), pp. 3975-3988 Well-posedness: c p � q h � 2 0 + �√ γ ∇ · v h � 2 a γ ( v h , q h ; v h , q h ) ≥ 1 b ≡ 1 � � 2 | [ v h , q h ] | 2 ν �∇ v h � 2 0 0 + ν + γ max + ν − 1 � b � 2 2 L ∞ (Ω) A-priori estimate: �� h 2 K � b � 2 � � 1 � L ∞ ( K ) | [ u − u h , p − p h ] | 2 h 2 k | u | 2 ν + γ K | p | 2 b ≤ ν + γ K + H k + 1 ( K ) + H k ( K ) K ν K ∈T h Equilibration of error terms: leads to ”dynamic” parameter version 0 ; | p | H k ( K ) � � γ K ∼ max − ν | u | H k + 1 ( K ) Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 10 / 27

Numerical analysis of grad-div stabilized Oseen problem Example 1: Vortex pairs Oseen problem on Ω = ( 0 , 1 ) 2 with ν = 10 − 6 , σ = 0 and b = u pairs of vortices strong variation of mesh Reynolds number � u � ∞ , K h K ∈ [ 0 , h Re K := ν ] ν H 1 - and L 2 -errors vs. scaling parameter γ 0 of grad-div stabilization for Example 1 with σ = 0, h ≈ 1 64 Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 11 / 27

Numerical analysis of grad-div stabilized Oseen problem Example 1 ”Dynamic” vs. constant parameter design H 1 - and L 2 -errors vs. scaling parameter ˜ γ 0 of ”dynamic” grad-div stabilization for Example 1 H 1 - and L 2 -errors vs. scaling constant parameter γ 0 of grad-div stabilization for Example 1 with ν = 10 − 6 , σ = 0 Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 12 / 27

Numerical analysis of grad-div stabilized Oseen problem Example 2: Vortex in boundary layer B ERRONE [2001] Oseen problem on Ω = ( 0 , 1 ) 2 with b = u counter-clockwise vortex in boundary layer ν -dependent solution with �∇ u � 0 ∼ ν − 0 . 35 and � p � 0 ∼ ν − 0 . 12 . Errors in H 1 -seminorm and L 2 -norm vs. scaling parameter γ 0 of grad-div stabilization for Example 2 with ν = 10 − 4 Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 13 / 27

Numerical analysis of grad-div stabilized Oseen problem Example 3: Beltrami flow E ITHIER ET AL . [1994] Time-dependent Navier-Stokes flow in Ω = ( − 1 , 1 ) 3 with ν = 10 − 6 Series of counter-rotating vortices intersecting one another at oblique angles Diagonally implicit Runge-Kutta method of 1 order 2 with time step ∆ t = 64 L 2 -error vs. t ∈ [ 0 , 1 ] without stabilization for different h (left) and with grad-div stabilization for fixed h for ν = 10 − 6 Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 14 / 27

Numerical analysis of grad-div stabilized Oseen problem Example 3 (continued) L 2 (Ω) -error (as function of t ) for different values of Re = 1 ν for the Galerkin scheme, i.e. without grad-div stabilization (left) and with grad-div stabilization (right) Obvious improvement even with (time-independent) grad-div stabilization γ K = γ 0 Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 15 / 27

Numerical analysis of grad-div stabilized Oseen problem Some conclusions Constant value of grad-div parameter γ gives very often improvements of mass conservation and of other relevant norms � � | p | Hk ( K ) ”Dynamic” design of γ K ∼ max 0 ; | u | Hk + 1 ( K ) − ν is not feasible | p | H k ( K ) ≥ ν | u | H k + 1 ( K ) , Grad-div stabilization is not necessary in case of e.g. in shear flows ! Poiseuille-type flow: No improvement with grad-div stabilization ! Gert Lube (University of G¨ ottingen) Some remarks on grad-div stabilization Dresden, November 16-18, 2011 16 / 27