Towards pressure-robust mixed methods for the incompressible Navier Stokes equations

Transcription

1 Mohrenstrasse Berlin Germany Tel , Pittsburgh Weierstrass Institute for Applied Analysis and Stochastics Towards pressure-robust mixed methods for the incompressible Navier Stokes equations Alexander Linke & Christian Merdon

2 Discretizing the incompressible Navier Stokes equations (I) Model v t ν v + (v )v + 2Ω v + p = f v = 0. Question How many sources for velocity errors (space discretization)? Pressure-robust mixed methods , Pittsburgh Page 2 (32)

3 Discretizing the incompressible Navier Stokes equations (II) Model v t ν v + (v )v + 2Ω v + p = f v = 0. Answer Sources for velocity errors (space discretization) are difficult to count About 8! Velocity errors = instabilities??? Pressure-robust mixed methods , Pittsburgh Page 3 (32)

4 Discretizing the incompressible Navier Stokes equations (III) Model v t ν v + (v )v + 2Ω v + p = f v = 0. Remark (Potential velocity errors) 3 velocity errors known from scalar PDEs: dominant advection dominant reaction in Coriolis term mass lumping necessary in v t = 5 vectorial velocity errors: discrete inf-sup stability L 2 -orthogonality of divergence-free vector fields and gradients fields Pressure-robust mixed methods , Pittsburgh Page 4 (32)

5 Discretizing the incompressible Navier Stokes equations (IV) Question (40 years ago) Rather naive approach: quest for optimal (high order) methods! Scalar-type sources for velocity errors were known But how to treat the vectorial character of the Navier Stokes equations? Answer (50 years ago) Discrete inf-sup stability convergence & optimal convergence order! Pressure-robust mixed methods , Pittsburgh Page 5 (32)

6 Inf-sup stability & discrete divergence Remark (Discrete divergence) discrete divergence div h : X h Q h conforming methods like Taylor Hood: div h w h = π Qh ( w h ) nonconforming methods like Crouzeix Raviart,... discrete inf-sup stability g h Q h wh X h : div h w h = g h w h 0 1 β h g h 0 V 0,h = {w h X h : div h w h = 0} discrete divergence replaces divergence! divergence no sense! problem: φ H 1 (D), w h X h : conforming methods: D φ w h dx = D φ w h dx D φ div hw h dx nonconforming methods: D φ w h dx = D φ div hw h dx +Ch l+1 φ l+1 w h 1,h vector L 2 scalar product can excite arbitrarily large velocity errors! Pressure-robust mixed methods , Pittsburgh Page 6 (32)

7 A Stokes model Model ν v + p = f, v = g, v = v D, x Ω x Ω x Ω. Theorem (Stokes error estimate) v v h L 2 2(1 +C F ) inf v w h L w h X h ν inf p q h L 2 q h Q h Remark Locking for ν 1!!! Velocity error small only, when v and ν 1 p well-resolved simultaneously! Pressure-robust mixed methods , Pittsburgh Page 7 (32)

8 A motivating example Model Steady incompressible Stokes equations: ν v + p = f, x Ω v = 0, x Ω v = 0 x Ω. ν = 10 3 ξ = x 2 (1 x) 2 y 2 (1 y) 2, v = ξ, p = x 3 + y f := ν v + p ν small f nearly a gradient! Pressure-robust mixed methods , Pittsburgh Page 8 (32)

9 Two Crouzeix Raviart mixed FEMs (I) Crouzeix-Raviart, h1 error norm CR CR+ 1/sqrt(n) Crouzeix-Raviart, l2 error norm CR CR+ 1/n h1 error l2 error e e+06 ndof 1e e+06 ndof Remark classical Crouzeix Raviart element: first-order convergent right hand side: f w h dx (modified) pressure-robust Crouzeix Raviart element (A. L.: CMAME 2014): first-order convergent right hand side: f Π F (w h )dx Pressure-robust mixed methods , Pittsburgh Page 9 (32)

10 Two Crouzeix Raviart mixed FEMs (II) Isolines of the vertical velocity component: Remark Example: Pressure-robust mixed method is 10 refinement levels more accurate. 2D: reduction of numerical effort: ! Pressure-robust mixed methods , Pittsburgh Page 10 (32)

11 Classical & pressure-robust mixed methods (I) Model ν v + p = f, v = g, v = v D, x Ω x Ω x Ω. Theorem (Stokes velocity error estimate) Classical: Pressure-robust: v v h L 2 2(1 +C F ) inf v w h L w h X h ν inf p q h L 2 q h Q h v v h L 2 2(1 +C F ) inf w h X h v w h L 2 +C # h k v k+1 Remark Pressure-robust mixed method velocity error is pressure-independent! velocity error small, when only v is well-resolved! Pressure-robust mixed methods , Pittsburgh Page 11 (32)

12 Classical & pressure-robust mixed methods (II) Remark poor mass conservation : grad-div stabilization! incompressible Navier Stokes equations: two momentum balances for divergence-free & irrotational forces orthogonal in L 2 scalar product (up to gradients of harmonic potentials) poor mass conservation : V 0,h L 2 { φ}! pressure-robust methods: V 0,h L 2 { φ}! D f w h dx = D ( φ + ξ ) w h dx }{{} Helmholtz decomposition = ( ξ ) w h dx D D φ w }{{} h dx + div h w h message: poor mass conservation can be repaired without stabilization! Pressure-robust mixed methods , Pittsburgh Page 12 (32)

13 Pressure-robust mixed methods Remark construction of pressure-robust mixed methods rather easy since 70ies: it was thought: pressure-robust = divergence-free first divergence-free / pressure-robust 3D Stokes discretization in 2005 (!) by S. Zhang: Math. Comp., three classes of inf-sup stable, pressure-robust methods: divergence-free H 1 -conforming, div h w h = w h, X h = Q h : S. Zhang, M. Neilan, J. Guzman, R. Falk, T. Hughes, J. Evans, A. Buffa,... divergence-free H(div)-conforming, div h w h = w h, X h = Q h : G. Kanschat, B. Cockburn, D. Schötzau, J. Wang, J. Schöberl, C. Lehrenfeld, classical MAC scheme (V. Girault) inf-sup stable pressure-robust sibling methods: use H(div)-conforming velocity reconstructions Π F with div h w h = (Π F w h ): FVM: A. L. FEM: C. Merdon, J. Schöberl, G. Matthies, L. Tobiska, W. Wollner, A. L. HDG: A. Ern, D. di Pietro, F. Schieweck, A. L. Pressure-robust mixed methods , Pittsburgh Page 13 (32)

14 Pressure-robust sibling methods (I) Promise you give me an inf-sup stable Stokes discretization I give you back a pressure-robust Stokes discretization! does not add any artificial viscosity! Remark works for conforming/nonconforming FEM discretizations with discontinuous pressures on simplices (Crouzeix Raviart, Bernardi Raugel, P2b-P disc 1,... ) works for conforming/nonconforming FEM discretizations with discontinuous pressures on bricks (Q k+1 -P disc k,... ) (under construction for mapped elements on quadrilaterals/hexahedra) seems to work for conforming/nonconforming FEM discretizations with continuous pressures on simplices (Mini, Taylor Hood,... ) probably useful in practice, at least good for comparison with well-established methods,... Pressure-robust mixed methods , Pittsburgh Page 14 (32)

15 Pressure-robust sibling methods (II) References A. L.: On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. CMAME, A. L., G. Matthies, L. Tobiska: Robust arbitrary order mixed finite element methods for the incompressible Stokes equations. M2AN, Idea Crouzeix Raviart, Bernardi Raugel, P2b-P 1, all simplicial H 1 -conforming, inf-sup stable FEMs with discontinuous pressures,... major difficulty in Navier Stokes equations: double orthogonality! H 1 -orthogonality: appropriate by definition L 2 -orthogonality: f, (v h )v h,2ω v h,... must be improved! mixed methods &relaxation of divergence condition: good idea in trial functions bad idea in some test functions! Pressure-robust mixed methods , Pittsburgh Page 15 (32)

16 Pressure-robust sibling methods (III) Scheme (Pressure-robust sibling method) For all w h X h holds ν v h : w h dx p h div h w h dx = f Π F (w h )dx, div h v h = 0. Pressure-robust mixed methods , Pittsburgh Page 16 (32)

17 Pressure-robust sibling methods (III) Scheme (Pressure-robust sibling method) For all w h X h holds ν v h : w h dx p h div h w h dx = f Π F (w h )dx, div h v h = 0. Properties of Π F : Π F : X h H(div,D) q h Q h w h X h : ( w h,q h ) = ( Π F w h,q h ) ( v,π F w h ) + ( v, w h ) Ch k v k+1 w h 1 new pressure-robust sibling methods deliver in the case f = 0 exactly the same discrete solutions like classical inf-sup stable mixed methods, since Π F 0 = 0! Remark v v h L 2 2(1 +C F ) inf w h X h v w h L 2 +C # h k v k+1 Pressure-robust mixed methods , Pittsburgh Page 16 (32)

18 Reconstruction for nonlinear & Coriolis terms Remark pressure-robust convection form (not skew-symmetric) (v h )v h Π F w h dx D pressure-robust rotational form (skew-symmetric) v h ) Π F v h ) Π F w h dx D(( 1 v 2 h 2 div hw h dx D pressure-robust methods: rotational form can be safely used! note for conforming inf-sup stable methods we have (v h )v h Π F w h dx = v h ) v h ) Π F w h dx D D(( 1 v 2 h 2 div hw h dx D pressure-robust, skew-symmetric Coriolis term 2 (Ω Π F v h ) Π F w h dx. D Pressure-robust mixed methods , Pittsburgh Page 17 (32)

19 Classical mixed methods and real world flows (I) Idea (Benchmark construction) construct physically stable (!) benchmarks, where pressure-robust discretizations are more efficient than classical discretizations! better pressure approximation, smaller poor mass conservation first order method and v H 2, p H 1 : maximize F 1 := 1 ν p πh L2 p L 2 v πh H1 v 1 ν second order method and v H 3, p H 2 : maximize F 2 := 1 ν p πh L2 p L 2 v πh H1 v 1 ν third order method and v H 4, p H 3 : maximize F 3 := 1 ν p πh L2 p L 2 v πh H1 v 1 ν p 1 v 2. p 2 v 3. p 3 v 4. Pressure-robust mixed methods , Pittsburgh Page 18 (32)

20 Classical mixed methods and real world flows (II) Conjecture Classical inf-sup stable mixed methods perform well for momentum balance: error term ν v + (v )v+ p = 0. }{{} 0 F 1 = 1 ν p = O(ν) and easier than v is relatively small! C 2 ν p 1 1 νv 2 = 1! v 2 ν v 2 inf q h Q h p q h L 2 new pressure-robust sibling methods deliver in this case exactly the same discrete solutions like classical mixed methods, since Π F 0 = 0! Stokes flow & low Reynolds number flows not to speak about: dominant advection, nonlinear convection, turbulence,... Pressure-robust mixed methods , Pittsburgh Page 19 (32)

21 Classical mixed methods and real world flows (III) Conjectures Classical mixed methods perform poorly, when pressure large & complicated: (quasi-)hydrostatics with complicated f: ν v+(v )v+ p = f. }{{}}{{} 0 0 Reference: K. Galvin, A. L., L. Rebholz, N. Wilson: CMAME, (quasi-)geostrophic flows (meteorology): ν v+(v )v+ f yv }{{} + p = 0. }{{}}{{} 0 0 latitude-dependent Coriolis force high Reynolds number flows with momentum balance: ν v+(v )v + p = 0. }{{} 0 Pressure-robust mixed methods , Pittsburgh Page 20 (32)

22 Some simple benchmarks (I) Example (Quasi-)hydrostatics for: ν v + p = f: v = (0,0) T, f = (3x 2,0) T f (P k ) d p P k+1. Taylor Hood element: Reference: A. L., C. Merdon: JCP, F 1 = F 2 = F 3 =, F 4 = 0 Pressure-robust mixed methods , Pittsburgh Page 21 (32)

23 Some simple benchmarks (II) Example West wind for: ν v + yv + p = 0: ( ) 1 v =, p = y 2 0 v (P k ) d p P k+2 Taylor Hood element: Reference: A. L., C. Merdon: JCP, F 1 = F 2 =, F 3 = F 4 = 0 Pressure-robust mixed methods , Pittsburgh Page 22 (32)

24 Some simple benchmarks (III) Example Rigid body rotation for ν v + (v )v+ p = 0: ( ) y v =, p = 1 x 2 (x2 + y 2 ) v (P k ) d p P 2k Taylor Hood element (p more complicated than v): Reference: A. L., C. Merdon: JCP, F 1 = F 2 =, F 3 = F 4 = 0 Pressure-robust mixed methods , Pittsburgh Page 23 (32)

25 Some simple benchmarks (IVa) - exact Navier-Stokes solutions Remark many exact Navier Stokes solutions fulfill (v )v = φ, potential flows: (v )v = 2 1 (v2 ) generic case: v (P k ) d p P 2k a classical (huge) class of flow problems: potential flows h: harmonic function with h = 0 v := h v = h = 0 ν v = 0 (v )v = ( v) v (v2 ) = 1 2 (v2 ) v t = (h t ) p := h t 1 2 v2 v t ν v + (v )v + p = 0. Pressure-robust mixed methods , Pittsburgh Page 24 (32)

26 Some simple benchmarks (IVb) - exact Navier-Stokes solutions Remark Compare Bernardi Raugel element with pressure-robust sibling (with RT 0 ) (v h )v h w h dx vs. D 2D: h = y 5 5x 4 y 10x 2 y 3 (v h )v h Π F (w h )dx D ν ndof e e e e e e e e Table: Reduction of L 2 gradient error Speedup: 6 refinement levels factor 4 6 = 4096! Pressure-robust mixed methods , Pittsburgh Page 25 (32)

27 Some simple benchmarks (IVc) - exact Navier-Stokes solutions Remark Compare P2b-P disc 1 with pressure-robust sibling (with RT 1 ) (v h )v h w h dx vs. D 2D: h = y 5 5x 4 y 10x 2 y 3 (v h )v h Π F (w h )dx D ν ndof e e e e e e e e Table: Reduction of L 2 gradient error Speedup: 3 refinement levels factor 4 3 = 64! Pressure-robust mixed methods , Pittsburgh Page 26 (32)

28 Some simple benchmarks (IVd) - exact Navier-Stokes solutions Remark Compare Bernardi Raugel element with pressure-robust sibling (with RT 0 ) 3D: (v h )v h w h dx vs. D h = xyz (v h )v h Π F (w h )dx D ν ndof e e e e e e e Table: Reduction of L 2 gradient error Speedup: 8 refinement levels factor millions! Pressure-robust mixed methods , Pittsburgh Page 27 (32)

29 Grad-div stabilization Remark classical idea against poor mass conservation : grad-div stabilization, γ = O(1): ν v γ v + p = f pressure-robust sibling methods show: artificial diffusion not necessary! trade-off, conceptually strange idea: divergence has no sense in inf-sup stable methods discrete divergence is zero (stable!) non-locking-free error estimate: v v h L 2 C 1 ν inf v w h L 2 + C 2 w h X h ν inf q h Q h p q h L 2 can safely work, when additional inf-sup condition holds! M. Case, V. Ervin, A. L., L. Rebholz : SINUM 2011, E. Jenkins, V. John, A. L., L. Rebholz: Adv. Comput. Math., however: div h -div h -term can be good for linear solvers (does not change the discrete velocity solution)! Pressure-robust mixed methods , Pittsburgh Page 28 (32)

30 Conclusions Conjecture mixed methods for Navier Stokes: lot of confusion concerning divergence!!! discrete inf-sup stability: part of the solution Navier Stokes: two momentum balances, no poor mass conservation relaxing divergence constraint: very dangerous, but only in test functions! L 2 -orthogonality of divergence-free and gradients fields important, can be repaired via changing some test functions! no artificial viscosity necessary! dozens/hundreds of new pressure-robust sibling methods! pressure-robust sibling mixed methods: universal tool (FEM, FVM, DG,... ) pressure-robust Bernardi Raugel (3D)! classification of flows according to momentum balance better benchmarks! large to arbitrary speedups in some benchmarks in reach high Reynolds numbers: nonlinear convection excites two different oscillations! pressure-robust mixed methods: new research field in CFD! Pressure-robust mixed methods , Pittsburgh Page 29 (32)

31 Open questions Questions pressure-robust siblings on general quadrilaterals & hexahedra (work in progress: G. Matthies, L. Tobiska, N. Ahmed, A. L.) pressure-robust siblings for methods with continuous pressures like Taylor Hood (work in progress: J. Schöberl, P. Lederer, C. Merdon, A. L.) numerical analysis for time-dependent Navier Stokes (work in progress: G. Matthies, C. Merdon, N. Ahmed, A. L.) more and better benchmarks in 2D and 3D quadratures & coupled problems compressible Euler/Navier Stokes: low Mach numbers, well-balanced schemes,... Pressure-robust mixed methods , Pittsburgh Page 30 (32)

32 References (I) References (Pressure-robust mixed siblings) A. L.: On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. CMAME, A. L., G. Matthies, L. Tobiska: Robust arbitrary order mixed finite element methods for the incompressible Stokes equations. M2AN, A. L.: A divergence-free velocity reconstruction for incompressible flows. CRAS, C. Brennecke, A. L., C. Merdon, J. Schöberl: Optimal and pressure-independent L2 velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions. JCM, A. L., C. Merdon: Guaranteed energy error estimators for a modified robust Crouzeix-Raviart Stokes element. J. Sci. Comp., D. di Pietro, A. Ern, A. L., F. Schieweck: A discontinuous skeletal method for the viscosity-dependent Stokes problem. CMAME, Pressure-robust mixed methods , Pittsburgh Page 31 (32)

33 References (II) References (Poor mass conservation & benchmarks) V. John, A. L., C. Merdon, M. Neilan, L. Rebholz: On the divergence constraint in mixed finite element methods for incompressible flows. WIAS Preprint 2177, A. L., C. Merdon: On velocity errors due to irrotational forces in the Navier-Stokes momentum balance. JCP, A. L., C. Merdon: On pressure robustness and discrete Helmholtz projectors in mixed methods for the incompressible Navier Stokes equations. WIAS Preprint 2250, K. Galvin, A. L., L. Rebholz, N. Wilson: Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection. CMAME, A. L.: Collision in a cross-shaped domain - A steady 2d Navier-Stokes example demonstrating the importance of mass conservation in CFD. CMAME, Pressure-robust mixed methods , Pittsburgh Page 32 (32)