The Role of the Pressure in Finite Element Methods for Incompressible Flow Problems

Transcription

1 Weierstrass Institute for Applied Analysis and Stochastics The Role of the Pressure in Finite Element Methods for Incompressible Flow Problems Volker John Mohrenstrasse Berlin Germany Tel Fluids under Pressure: Summer School 2016, Prague, August 29 - September 2

2 Curriculum vitae study of Mathematics at the Martin Luther Universität Halle Wittenberg Diploma thesis Numerische Behandlung des Elektrischen Impedanz Tomographie Problems advisors: H. Schwetlick, O. Knoth scientific assistant at the Institut für Analysis und Numerik of the Otto von Guericke Universität Magdeburg Ph.D. thesis Parallele Lösung der inkompressiblen Navier Stokes Gleichungen auf adaptiv verfeinerten Gittern advisor: L. Tobiska; referees: H. G. Roos, G. Wittum School 2016, Prague, August 29 - September 2 Page 2 (102)

3 Curriculum vitae scientific assistant at the Institut für Analysis und Numerik of the Otto von Guericke Universität Magdeburg habilitation thesis Large Eddy Simulation of Turbulent Incompressible Flows. Analytical and Numerical Results for a Class of LES Models referees: M. Griebel, M.D. Gunzburger, W.J. Layton, L. Tobiska scientific assistant at the Institut für Analysis und Numerik of the Otto von Guericke Universität Magdeburg Professor for Applied Mathematics, Saarland University, Saarbrücken since head of the group Numerical Mathematics and Scientific Computing at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin Professor for Numerical Mathematics, Free University Berlin School 2016, Prague, August 29 - September 2 Page 3 (102)

4 WIAS Berlin Weierstrass Institute of Applied Analysis and Stochastics goal: project-oriented research in Applied Mathematics founded 1992 as successor of the Mathematical Institute of the Academy of Science of the G.D.R. member of the Leibniz Association 100 researchers in seven research groups situated in the center of Berlin has been hosting permanent office of the International Mathematical Union (IMU) since 2011 School 2016, Prague, August 29 - September 2 Page 4 (102)

5 Outline of the Lectures 1 On the Choice of Finite Element Spaces for Velocity and Pressure 2 Finite Element Error Analysis for Inf-Sup Stable Pairs of Finite Element Spaces 3 Reducing the Impact of the Pressure on the Finite Element Velocity Error 4 Finite Element Methods that Circumvent the Discrete Inf-Sup Condition School 2016, Prague, August 29 - September 2 Page 5 (102)

6 ISBN Literature Volker John Finite Element Methods for Incompressible Flow Problems This book explores finite element methods for incompressible flow problems: Stokes equations, stationary Navier-Stokes equations, and time-dependent Navier-Stokes equations. It focuses on numerical analysis, but also discusses the practical use of these methods and includes numerical illustrations. It also provides a comprehensive overview of analytical results for turbulence models. The proofs are presented step by step, allowing readers to more easily understand the analytical techniques. Mathematics SSCM 51 John 1 Finite Element Methods for Incompressible Flow Problems Springer Series in Computational Mathematics 51 Volker John Finite Element Methods for Incompressible Flow Problems to appear Nov./Dec around 750 pages here: Chapters 3 and 4 School 2016, Prague, August 29 - September 2 Page 6 (102)

7 1. On the Choice of Finite Element Spaces for Velocity and Pressure School 2016, Prague, August 29 - September 2 Page 7 (102)

8 1 Dimensionless Incompressible Navier Stokes Equations dimensionless Navier Stokes equations conservation of linear momentum conservation of mass u t 2Re 1 D(u) + (uu T ) + p = f in (0, T ] Ω u = 0 in [0, T ] Ω u(0, x) = u 0 in Ω given: Ω R d, d {2, 3}: domain T : final time u 0 : initial velocity boundary conditions + boundary conditions parameter: Reynolds number Re = ν 1 to compute: velocity u, with D(u) = u + ut 2 velocity deformation tensor pressure p School 2016, Prague, August 29 - September 2 Page 8 (102),

9 1 Model Problem: the Stokes Equations difficulties for mathematical analysis and numerical simulations coupling of velocity and pressure nonlinearity of the convective term the convective term dominates the viscous term, i.e., ν is small School 2016, Prague, August 29 - September 2 Page 9 (102)

10 1 Model Problem: the Stokes Equations difficulties for mathematical analysis and numerical simulations coupling of velocity and pressure nonlinearity of the convective term the convective term dominates the viscous term, i.e., ν is small with divergence constraint consider (scaled) Stokes equations 2ν D(u) = ν u ν u + p = f in Ω, u = 0 in Ω Ω R d, d {2, 3}, bounded domain with (polyhedral) Lipschitz boundary boundary condition u = 0 on Γ = Ω notion for second equation: velocity is (pointwise) divergence-free School 2016, Prague, August 29 - September 2 Page 9 (102)

11 1 Model Problem: the Stokes Equations weak formulation: Given f H 1 (Ω), find (u, p) H 1 0 (Ω) L 2 0(Ω) = V Q such that ( u, v) ( v, p) = f, v H 1 (Ω),H0 1(Ω) v H0 1 (Ω), ( u, q) = 0 q L 2 0(Ω) velocity space H 1 0 (Ω) = {v H 1 (Ω) : trace(v) = 0} to satisfy boundary condition pressure space { L 2 0(Ω) = q L 2 (Ω) : Ω } q(x) dx = 0 to fix additive constant for pressure notation for second equation: velocity is weakly divergence-free School 2016, Prague, August 29 - September 2 Page 10 (102)

12 1 Model Problem: the Stokes Equations existence and uniqueness of a solution (well-posed problem) velocity: consider problem in space of weakly divergence-free functions find u V div such that V div = {v V : ( v, q) = 0 q Q} ( u, v) = f, v H 1 (Ω),H 1 0 (Ω) v V div existence and uniqueness of u by representation theorem of Riesz School 2016, Prague, August 29 - September 2 Page 11 (102)

13 1 Model Problem: the Stokes Equations existence and uniqueness of a solution (well-posed problem) pressure: existence and uniqueness if and only if [1,2,3] inf sup q Q\{0} v V \{0} ( v, q) v L2 (Ω) q L 2 (Ω) β is > 0 inf-sup condition, Ladyzhenskaya Babuška Brezzi condition, LBB condition based on analysis of abstract linear saddle point problem in Hilbert spaces β is depends on the domain [4] [1] Babuška; Numer. Math. 20, , 1973 [2] Brezzi; Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8, , 1974 [3] Girault, Raviart; Finite Element Methods for Navier-Stokes Equations, Springer, 1986 [4] Galdi; An introduction to the mathematical theory of the Navier-Stokes equations, Springer, 2011 School 2016, Prague, August 29 - September 2 Page 12 (102)

14 1 FE Spaces for Incompressible Flow Problems idea: replace infinite-dimensional spaces V and Q by finite-dimensional spaces School 2016, Prague, August 29 - September 2 Page 13 (102)

15 1 FE Spaces for Incompressible Flow Problems idea: replace infinite-dimensional spaces V and Q by finite-dimensional spaces finite element spaces V h finite element velocity space Q h finite element pressure space V h /Q h pair conforming finite element spaces: V h V and Q h Q School 2016, Prague, August 29 - September 2 Page 13 (102)

16 1 Finite Element Spaces for Incompressible Flow Problems bilinear form a h : V h V h R a h ( v h, w h) := T h triangulation of Ω K T h mesh cells bilinear form b h K T h : V h Q h R b h ( v h, q h) := K T h norm in velocity finite element space v h 2 = ( v h, v h) = V h V h ( v h, w h) K ( v h, q h) K K T h if conf. = ( v h, v h) K ( v h, w h) if conf. = ( v h, q h) if conf. = ( v h, v h) School 2016, Prague, August 29 - September 2 Page 14 (102)

17 1 Well-Posedness of the Discrete Problem well-posedness of the finite element problem equip V h Q h with bases = finite element problem can be written in matrix-vector form ( ) ( ) A B T u = B 0 p ( ) f, 0 ( ) A B T R (n V +n Q ) (n V +n Q ), B 0 unique solution matrix has full rank necessary condition: n Q n V last rows of the system matrix span space of at most dimension n V assume that A is non-singular, then the system matrix is non-singular if and only if B has full rank, i.e., rank(b) = n Q rank(b) = n Q if and only if inf sup q R n Q \0 v R n V \0 v T B T q v 2 q 2 β > 0 proof much simpler as for infinite-dimensional case, board School 2016, Prague, August 29 - September 2 Page 15 (102)

18 1 The Discrete Inf-Sup Condition discrete inf-sup condition needed (finite-dimensional spaces are Hilbert spaces) inf sup q h Q h \{0} v h V h \{0} b ( h v h, q h) v h V h q h βis h > 0 L 2 (Ω) not inherited from inf-sup condition fulfilled by V and Q discussion: board School 2016, Prague, August 29 - September 2 Page 16 (102)

19 1 Discretely Divergence-Free Functions space of discretely divergence-free functions V h div = { v h V h : b h ( v h, q h) = 0 q h Q h} generally very hard to construct generally V h div V div finite element velocities not weakly or pointwise divergence-free conservation of mass violated = Part 3 best approximation estimate for Vdiv h. Let v V div and let the discrete inf-sup condition hold. Then inf v h V h div ( v v h ) ( L 2 (Ω) β h is ) inf w h V h ( v w h ) L 2 (Ω) for certain pairs of finite element spaces estimate with local inf-sup constant [1] [1] Girault, Scott; Calcolo 40, 1 19, 2003 School 2016, Prague, August 29 - September 2 Page 17 (102)

20 1 Finite Elements piecewise constant finite elements P 0, (Q 0 ) one degree of freedom (d.o.f.) per mesh cell continuous piecewise linear finite elements P 1 d d.o.f. per mesh cell School 2016, Prague, August 29 - September 2 Page 18 (102)

21 1 Finite Elements continuous piecewise quadratic finite elements P 2 (d + 1)(d + 2)/2 d.o.f. per mesh cell continuous piecewise bilinear finite elements Q 1 2 d d.o.f. per mesh cell and so on for continuous finite elements of higher order School 2016, Prague, August 29 - September 2 Page 19 (102)

22 1 Finite Elements nonconforming linear finite elements P1 nc, Crouzeix, Raviart (1973) continuous only in barycenters of faces d + 1 d.o.f. per mesh cell School 2016, Prague, August 29 - September 2 Page 20 (102)

23 1 Finite Elements rotated bilinear finite element Q rot 1, Rannacher, Turek (1992) continuous only in barycenters of faces 2d d.o.f. per mesh cell discontinuous linear finite element P1 disc defined by integral nodal functionals e.g., ϕ h P1 disc if ϕ h is linear on a mesh cell K (2d) and ϕ h (x) dx = 0, xϕ h (x) dx = 1, yϕ h (x) dx = 0 K d + 1 d.o.f. per mesh cell K School 2016, Prague, August 29 - September 2 Page 21 (102) K

24 1 FE Spaces Violating the Discrete Inf-Sup Condition criterion for violation of discrete inf-sup condition: there is non-trivial q h Q h such that = b h ( v h, q h) = 0 v h V h b ( h v h, q h) sup v h V h \{0} v h = 0 V h School 2016, Prague, August 29 - September 2 Page 22 (102)

25 1 FE Spaces Violating the Discrete Inf-Sup Condition criterion for violation of discrete inf-sup condition: there is non-trivial q h Q h such that = b h ( v h, q h) = 0 v h V h b ( h v h, q h) sup v h V h \{0} v h = 0 V h P 1 /P 1 pair of finite element spaces violates discrete inf-sup condition counter example: checkerboard instability, board School 2016, Prague, August 29 - September 2 Page 22 (102)

26 1 FE Spaces Violating the Discrete Inf-Sup Condition other pairs which violated discrete inf-sup condition P 1 /P 0 Q 1 /Q 0 P k /P k, k 1 Q k /Q k, k 1 P k /Pk 1 disc, k 2, on a special macro cell summary: many easy to implement pairs violate discrete inf-sup condition different finite element spaces for velocity and pressure necessary School 2016, Prague, August 29 - September 2 Page 23 (102)

27 1 FE Spaces Satisfying the Discrete Inf-Sup Condition pairs which fulfill discrete inf-sup condition P k /P k 1, Q k /Q k 1 : Taylor Hood finite elements [1] proofs: 2D, k = 2 [2], general [3,4] Q k /Pk 1 disc P k /Pk 1 disc, k d, on very special meshes (Scott Vogelius element) P1 bubble /P 1, mini element Pk bubble /Pk 1 disc [5] P1 nc /P 0, Crouzeix Raviart element [6] Q rot 1 /Q 0, Rannacher Turek element [7]... [1] Taylor, Hood; Comput. Fluids 1, , 1973 [2] Verfürth; RAIRO Anal. Numér. 18, , 1984 [3] Boffi; Math. Models Methods Appl. Sci. 4, , 1994 [4] Boffi; SIAM J. Numer. Anal. 34, , 1997 [5] Bernardi, Raugel; Math. Comp. 44, 71 79, 1985 [6] Crouzeix, Raviart; RAIRO. Anal. Numér. 7, 33 76, 1973 [7] Rannacher, Turek; Numer. Meth. Part. Diff. Equ. 8, , 1992 School 2016, Prague, August 29 - September 2 Page 24 (102)

28 1 FE Spaces Satisfying the Discrete Inf-Sup Condition techniques for proving the discrete inf-sup condition construction of Fortin operator [1] using projection to piecewise constant pressure [2] macroelement techniques [3,4] criterion from [5] for continuous finite element pressure: to show b(v h, q h ) sup v h V h \{0} v h β 2 V K T h h 2 K q h 2 L 2 (K) 1/2 q h Q h survey in [6] [1] Fortin; RAIRO Anal. Numér. 11, , 1977 [2] Brezzi, Bathe; Comput. Methods Appl. Mech. Engrg. 82, 27 57, 1990 [3] Boland, Nicolaides; SIAM J. Numer. Anal. 20, , 1983 [4] Stenberg; Math. Comp. 32, 9 23, 1984 [5] Verfürth; RAIRO Anal. Numér. 18, , 1984 [6] Boffi, Brezzi, Fortin; Lecture Notes in Mathematics 1939, Springer, , 2008 School 2016, Prague, August 29 - September 2 Page 25 (102)

29 1 Fortin Operator connection between the continuous and the discrete inf-sup condition let V, Q satisfy the continuous inf-sup condition let V h V and Q h Q = V h and Q h satisfy the discrete inf-sup condition if and only if there exists a constant γ h > 0, independent of h, such that for all v V there is an element P h For v V h with b ( v, q h) = b ( P h Forv, q h) q h Q h and proof =: board P h For v V γ h v V School 2016, Prague, August 29 - September 2 Page 26 (102)

30 1 Fortin Operator general approach for constructing Fortin operator P h For L(V, V h ) v P h Clev + P h 2 ( v P h Cle v ) with PCle h Clément operator (modification which preserves homogeneous Dirichlet boundary conditions) P h 2 v H1 (h C 1 (K) K v L 2 (K) + v H 1 (K) ), K T h, v V used to prove, e.g., inf-sup condition for MINI element P 1 V h bub /P 1 [1], Fortin operator approach can be extended to Crouzeix Raviart element P nc 1 /P 0 and Rannacher Turek element Q rot 1 /Q 0 [1] Arnold, Brezzi, Fortin; Calcolo 21, , 1984 School 2016, Prague, August 29 - September 2 Page 27 (102)

31 1 Macroelement Techniques goal: reduce proof of global inf-sup condition to proof of local inf-sup conditions general theory for both approaches somewhat technical Boland, Nicolaides (1983) Q k /Pk 1 disc, k 2 (unmapped [1], mapped [2]) Stenberg (1984) Taylor Hood P k /P k 1, Q k /Q k 1, k 2 [3] unmapped Q 2 /P1 disc in 2d [4] Scott Vogelius P k /Pk 1 disc, k 3, in 3d on barycentric meshes [5] [1] Girault, Raviart; Springer-Verlag, 1986 [2] Matthies, Tobiska; Computing 69, , 2002 [3] Stenberg; Math. Comp. 54, , 1990 [4] Stenberg; Math. Comp. 32, 9 23, 1984 [5] Zhang; Math. Comp. 74, , 2005 School 2016, Prague, August 29 - September 2 Page 28 (102)

32 1 Computing the Discrete Inf-Sup Constant from Riesz representation theorem: there is v h b V h such that for fixed q h Q h ( v h b, v h) V h = b h ( v h, q h) v h V h it follows for all v h V h b h ( v h, q h) v h b V h v h b ( h v h, q h) V h = sup v h V h \{0} v h v h V b h V h since b h ( v h b, qh) v h b V h = v h V b h = v h b ( h v h, q h) b = arg sup v h V h \{0} v h V h it follows ( β h is ) 2 = inf sup q h Q h \{0} v h V h \{0} ( b h ( v h, q h)) 2 v h 2 V h qh 2 Q v h 2 b V = inf h q h Q h \{0} q h 2 Q School 2016, Prague, August 29 - September 2 Page 29 (102)

33 1 Computing the Discrete Inf-Sup Constant equip spaces with bases = Gramian matrices (( ) M V = φ h j, φ h Nu ( (ψ ) h i, M Q = j, ψ )V h Np h i i,j=1 )Q h i,j=1 and bilinear forms and norms ( v h b, v h) V h = b T M T V v, b h ( v h, q h) = v T B T q, one obtains v h b 2 = b T M T V h V b b T M T V v = v T B T q v T M V b = v T B T q v R Nu, from what follows that inserting in discrete inf-sup condition M V b = B T q = b = M 1 V BT q ( ) β h 2 b T MV T is = inf b q R Nq \{0} q T MQ T q = = inf q R Nq \{0} q T BM 1 V BT q q T MQ T q inf q R Nq \{0} q T BM T V M V T M 1 V BT q q T M T Q q School 2016, Prague, August 29 - September 2 Page 30 (102)

34 1 Computing the Discrete Inf-Sup Constant M V and M Q are symmetric and positive definite ( ) ( ) ( q T M 1/2 Q M 1/2 Q BM T/2 V V ( q T M 1/2 Q ) ( ) ( B T M T/2 Q ) Q q M 1/2 M T/2 M T/2 Q q ) = Rayleigh quotient infimum is smallest eigenvalue of ( ) ( ) ( ) ( ) M 1/2 Q BM T/2 V M 1/2 V B T M T/2 Q M T/2 Q q = λ M T/2 Q q or (multiply with M 1/2 Q ) BM 1 V BT q = λmqq T discrete inf-sup constant is square root of smallest eigenvalue of this generalized eigenvalue problem School 2016, Prague, August 29 - September 2 Page 31 (102)

35 2. Finite Element Error Analysis for Inf-Sup Stable Pairs of Finite Element Spaces School 2016, Prague, August 29 - September 2 Page 32 (102)

36 2 Stokes and Scaled Stokes Equations continuous equation u + p = f in Ω, u = 0 in Ω (1) for simplicity: homogeneous Dirichlet boundary conditions difficulty: coupling of velocity and pressure properties linear form ν u + p = f in Ω, u = 0 in Ω becomes (1) by rescaling with new pressure, right-hand side School 2016, Prague, August 29 - September 2 Page 33 (102)

37 2 Finite Element Discretization finite element problem: Find (u h, p h ) V h Q h such that with a h ( v h, w h) = a h ( u h, v h) + b h ( v h, p h) = ( f, v h) v h V h, b h ( u h, q h) = 0 q h Q h K T h ( v h, w h) K, bh ( v h, q h) = K T h ( v h, q h) K School 2016, Prague, August 29 - September 2 Page 34 (102)

38 2 Finite Element Discretization finite element problem: Find (u h, p h ) V h Q h such that with a h ( v h, w h) = a h ( u h, v h) + b h ( v h, p h) = ( f, v h) v h V h, b h ( u h, q h) = 0 q h Q h K T h ( v h, w h) K, bh ( v h, q h) = K T h ( v h, q h) K only conforming inf-sup stable finite element spaces V h V and Q h Q inf sup q h Q h \{0} v h V h \{0} b h ( v h, q h) v h V h q h L2 (Ω) β h is > 0 School 2016, Prague, August 29 - September 2 Page 34 (102)

39 2 Finite Element Analysis existence and uniqueness of a solution same proof as for continuous problem School 2016, Prague, August 29 - September 2 Page 35 (102)

40 2 Finite Element Analysis existence and uniqueness of a solution same proof as for continuous problem stability u h f L 2 (Ω) H 1 (Ω), p h 2 L 2 (Ω) βis h f H 1 (Ω) same proof as for continuous problem, board School 2016, Prague, August 29 - September 2 Page 35 (102)

41 2 Finite Element Analysis existence and uniqueness of a solution same proof as for continuous problem stability u h f L 2 (Ω) H 1 (Ω), p h 2 L 2 (Ω) βis h f H 1 (Ω) same proof as for continuous problem, board goal of finite element error analysis: estimate error by best approximation errors best approximation errors depend only on finite element spaces, not on problem estimates for best approximation errors are known (interpolation errors) School 2016, Prague, August 29 - September 2 Page 35 (102)

42 2 Finite Element Analysis existence and uniqueness of a solution same proof as for continuous problem stability u h f L 2 (Ω) H 1 (Ω), p h 2 L 2 (Ω) βis h f H 1 (Ω) same proof as for continuous problem, board goal of finite element error analysis: estimate error by best approximation errors best approximation errors depend only on finite element spaces, not on problem estimates for best approximation errors are known (interpolation errors) reduction to a problem on the space of discretely divergence-free functions a ( u h, v h) = ( u h, v h) = ( f, v h) v h V h div School 2016, Prague, August 29 - September 2 Page 35 (102)

43 2 Finite Element Error Analysis finite element error estimate for the L 2 (Ω) norm of the gradient of the velocity Ω R d, bounded, polyhedral, Lipschitz-continuous boundary general case: V h div V div (u u h ) L2 (Ω) 2 inf proof: board velocity error depends on pressure v h V h div (u v h ) L2 (Ω) + inf q h Q h p q h L 2 (Ω) School 2016, Prague, August 29 - September 2 Page 36 (102)

44 2 Finite Element Error Analysis finite element error estimate for the L 2 (Ω) norm of the gradient of the velocity Ω R d, bounded, polyhedral, Lipschitz-continuous boundary general case: V h div V div (u u h ) L2 (Ω) 2 inf proof: board velocity error depends on pressure polyhedral domain in three dimensions which is not Lipschitzcontinuous v h V h div (u v h ) L2 (Ω) + inf q h Q h p q h L 2 (Ω) School 2016, Prague, August 29 - September 2 Page 36 (102)

45 2 Finite Element Error Analysis finite element error estimate for the L 2 (Ω) norm of the pressure same assumptions as for previous estimate proof: board p p h L 2 (Ω) 2 inf βis h v h Vdiv h + ( β h is (u v h ) L 2 (Ω) ) inf q h Q h p q h L 2 (Ω) School 2016, Prague, August 29 - September 2 Page 37 (102)

46 2 Finite Element Error Analysis error of the velocity in the L 2 (Ω) norm by Poincaré inequality not optimal u u h L 2 (Ω) C (u u h ) L 2 (Ω) School 2016, Prague, August 29 - September 2 Page 38 (102)

47 2 Finite Element Error Analysis error of the velocity in the L 2 (Ω) norm by Poincaré inequality not optimal u u h L 2 (Ω) C (u u h ) L 2 (Ω) regular dual Stokes problem: For given ˆf L 2 (Ω), find (φˆf, ξˆf ) V Q such that φˆf + ξˆf = ˆf in Ω, φˆf = 0 in Ω regular if mapping ( ) φˆf, ξˆf φˆf + ξˆf is an isomorphism from ( H 2 (Ω) V ) ( H 1 (Ω) Q ) onto L 2 (Ω) Γ of class C 2 bounded, convex polygons in two dimensions School 2016, Prague, August 29 - September 2 Page 38 (102)

48 2 Finite Element Error Analysis finite element error estimate for the L 2 (Ω) norm of the velocity same assumptions as for previous estimates dual Stokes problem regular with solution (φˆf, ξˆf ) u u h L2 (Ω) ( ( u u h ) + inf ) p L2(Ω) q h q h Q h L 2 (Ω) [( 1 sup ) ˆf L 2 (Ω) ˆf βis h inf (φˆf φ h) L2 φ h V h (Ω) L2 (Ω) ] + inf ξˆf r h L r h Q h 2 (Ω) velocity error depends on pressure School 2016, Prague, August 29 - September 2 Page 39 (102)

49 2 Finite Element Error Analysis finite element error estimates for conforming pairs of finite element spaces same assumptions on domain as for previous estimates solution sufficiently regular h mesh width of triangulation spaces Pk bubble /P k, k = 1 (mini element), P k /P k 1, Q k /Q k 1, k 2 (Taylor Hood element), Pk bubble /Pk 1 disc, Q k/pk 1 disc, k 2 (u u h ) ( ) L Ch k u 2 (Ω) H k+1 (Ω) + p H k (Ω) p p h ( ) L Ch k u 2 (Ω) H k+1 (Ω) + p H k (Ω) velocity error depends on pressure School 2016, Prague, August 29 - September 2 Page 40 (102)

50 2 Finite Element Error Analysis finite element error estimates for conforming pairs of finite element spaces (cont.) in addition: dual Stokes problem regular u u h ( ) L 2 (Ω) Chk+1 u H k+1 (Ω) + p H k (Ω) velocity error depends on pressure School 2016, Prague, August 29 - September 2 Page 41 (102)

51 2 Numerical Example Supporting Order of Convergence analytical example which supports the error estimates prescribed solution ( ) ( ) ( ) u1 y ψ x 2 (1 x) 2 y(1 y)(1 2y) u = = = 200 u 2 x ψ x(1 x)(1 2x)y 2 (1 y) 2 ( ( p = 10 x 1 ) 3 ( y 2 + (1 x) 3 y 1 ) ) School 2016, Prague, August 29 - September 2 Page 42 (102)

52 2 Numerical Example Supporting Order of Convergence initial grids (level 0) red refinement School 2016, Prague, August 29 - September 2 Page 43 (102)

53 2 Numerical Example Supporting Order of Convergences convergence of the errors (u u h ) L for different discretizations with 2 (Ω) different orders k h P bubble 1 /P h 2 P 2 /P 1 P bubble /P disc 2 1 Q 2 /Q 1 (u u h ) L 2 (Ω) 10 0 (u u h ) L 2 (Ω) Q 2 /P disc levels levels 10 0 h h 4 (u u h ) L 2 (Ω) P 3 /P 2 P bubble /P disc 3 2 Q 3 /Q 2 Q 3 /P disc 2 (u u h ) L 2 (Ω) P 4 /P 3 Q 4 /Q 3 Q 4 /P disc levels School 2016, Prague, August 29 - September 2 Page 44 (102) levels

54 2 Numerical Example Supporting Order of Convergence convergence of the errors p p h L for different discretizations with 2 (Ω) different orders k h 3/ h 2 P bubble 1 /P 1 P 2 /P P bubble /P disc 2 1 Q 2 /Q 1 p p h L 2 (Ω) 10-1 p p h L 2 (Ω) Q 2 /P disc levels levels 10 0 h h 4 p p h L 2 (Ω) P 3 /P 2 P bubble /P disc 3 2 Q 3 /Q 2 Q 3 /P disc 2 p p h L 2 (Ω) P 4 /P 3 Q 4 /Q 3 Q 4 /P disc levels School 2016, Prague, August 29 - September 2 Page 45 (102) levels

55 2 Numerical Example Supporting Order of Convergence convergence of the errors u u h L for different discretizations with 2 (Ω) different orders k h h P bubble 1 /P P 2 /P 1 P bubble /P disc 2 1 u u h L 2 (Ω) u u h L 2 (Ω) Q 2 /Q 1 Q 2 /P disc levels levels 10-2 h h 5 u u h L 2 (Ω) P 3 /P 2 P bubble /P disc 3 2 Q 3 /Q 2 Q 3 /P disc 2 u u h L 2 (Ω) P 4 /P 3 Q 4 /Q 3 Q 4 /P disc levels School 2016, Prague, August 29 - September 2 Page 46 (102) levels

56 2 Scaled Stokes Equations: Finite Element Error Analysis scaled Stokes equations ( p ) ν u + p = f u + = f ν ν error estimates (u u h ) ( L2 Ch k u (Ω) H k+1 (Ω) + 1 ) ν p H k (Ω) p p h ( ) L Ch k ν u 2 (Ω) H k+1 (Ω) + p H k (Ω) ( u u h L Ch k+1 u 2 (Ω) H k+1 (Ω) + 1 ) ν p H k (Ω) velocity error depends on pressure and inverse of viscosity School 2016, Prague, August 29 - September 2 Page 47 (102)

57 2 Numerical Example (cont.) Taylor Hood pair of finite element spaces P 2 /P h 2 h h 2 h (u u h ) L 2 (Ω) p p h L 2 (Ω) levels levels u u h L 2 (Ω) 10 7 h h ν =10 6 ν =10 2 ν =1 ν =10 2 ν =10 6 ν =10 10 levels velocity errors depend on inverse of viscosity School 2016, Prague, August 29 - September 2 Page 48 (102)

58 2 Finite Element Error Analysis: Special Case finite element error estimate for the L 2 (Ω) norm of the gradient of the velocity Ω R d, bounded, polyhedral, Lipschitz-continuous boundary special case: V h div V div (u u h ) L2 (Ω) 2 inf v h V h div proof: board velocity error does not depend on pressure same property for u u h L 2 (Ω) (u v h ) L2 (Ω) School 2016, Prague, August 29 - September 2 Page 49 (102)

59 2 Finite Element Error Analysis: Special Case most important example: Scott Vogelius [1] finite element barycentric-refined grid: P k /P disc k 1, k d [1] Scott, Vogelius; in Large-scale computations in fluid mechanics, Part 2, , 1985 School 2016, Prague, August 29 - September 2 Page 50 (102)

60 2 Numerical Example: No Flow Problem prescribed solution u = 0, p = 10 ( (x 0.5) 3 y + (1 x) 2 (y 0.5) 2 1 ) 36 Taylor Hood P 2 /P (u u h ) L 2 (Ω) h 2 ν =10 6 ν =10 2 ν =1 ν =10 2 ν =10 6 ν =10 10 u u h L 2 (Ω) h 3 ν =10 6 ν =10 2 ν =1 ν =10 2 ν =10 6 ν = levels levels velocity errors scale with inverse of viscosity School 2016, Prague, August 29 - September 2 Page 51 (102)

61 2 Numerical Example: No Flow Problem (cont.) Scott Vogelius P 2 /P disc 1, barycentric-refined grid (u u h ) L 2 (Ω) h 2 ν =10 6 ν =10 2 ν =1 ν =10 2 ν =10 6 ν =10 10 u u h L 2 (Ω) h 3 ν =10 6 ν =10 2 ν =1 ν =10 2 ν =10 6 ν = levels levels round-off errors from high condition number of matrix of linear system of equations School 2016, Prague, August 29 - September 2 Page 52 (102)

62 2 Summary large velocity errors can occur in the standard situation V h div V div in the presence of large pressure or small viscosity School 2016, Prague, August 29 - September 2 Page 53 (102)

63 3. Reducing the Impact of the Pressure on the Finite Element Velocity Error School 2016, Prague, August 29 - September 2 Page 54 (102)

64 3 Related Problem: Conservation of Mass continuous problem: ( u, q) = 0 q Q infinitely many conditions inf-sup condition equivalent to V = Q = take q = u 0 = u L 2 (Ω) weakly divergence-free School 2016, Prague, August 29 - September 2 Page 55 (102)

65 3 Related Problem: Conservation of Mass continuous problem: ( u, q) = 0 q Q infinitely many conditions inf-sup condition equivalent to V = Q = take q = u 0 = u L 2 (Ω) weakly divergence-free finite element problem ( u h, q h ) = 0 q h Q h discretely divergence-free finite number of conditions usually V h Q h = no mass conservation not tolerable in certain applications School 2016, Prague, August 29 - September 2 Page 55 (102)

66 3 Numerical Example 1 no-flow problem: prescribed solution for Stokes problem with ν = 1 u = 0, p = Ra (y 3 y2 2 + y 7 ) 12 finite elements P 2 /P 1 Taylor Hood P nc 1 /P 0 Crouzeix Raviart Crouzeix-Raviart Taylor-Hood P 2 /P 1 (u u h ) L 2 (Ω) Ra =1 Ra =100 Ra =10 4 Ra =10 6 (u u h ) L 2 (Ω) Ra =1 Ra =100 Ra =10 4 Ra = level level velocity error scales with the pressure School 2016, Prague, August 29 - September 2 Page 56 (102)

67 3 Numerical Example 2 stationary vortex: prescribed solution for Navier Stokes problem with ν = 1 ( ) ( y x 2 + y 2 u =, p = Re 1 ), Re > 0 x 2 3 balance of nonlinear term of the Navier Stokes equations and pressure term P bubble 1 /P 1 MINI element 10 3 Crouzeix-Raviart 10 2 mini element (u u h ) L 2 (Ω) Re =1 Re =10 Re =100 (u u h ) L 2 (Ω) Re =1 Re =10 Re = Re = level 10-5 Re = level velocity error scales with the pressure School 2016, Prague, August 29 - September 2 Page 57 (102)

68 3 Conclusions error scales also with Coriolis force if this term is present important in meteorology School 2016, Prague, August 29 - September 2 Page 58 (102)

69 3 Conclusions error scales also with Coriolis force if this term is present important in meteorology fundamental invariance property for continuous equations (with Dirichlet boundary conditions) physically correct behavior f f + ψ = (u, p) (u, p + ψ) obviously not satisfied for the considered discretizations no-flow problem: f = p but change of f changed u h unphysical behavior connected with Helmholtz decomposition of vector fields in L 2 (Ω) School 2016, Prague, August 29 - September 2 Page 58 (102)

70 3 Grad-Div Stabilization very popular technique add in continuous problem 0 = µ ( u) gives in finite element problem µ( u h, v h ) does not vanish if u h 0 stabilized finite element Stokes problem: Find ( u h, p h) V h Q h such that ν ( u h, v h) ( v h, p h) + ( u h, v h) = f, K vh V,V v h V h K T h µ K {µ K } with µ K 0 stabilization parameters ( u h, q h) = 0 q h Q h School 2016, Prague, August 29 - September 2 Page 59 (102)

71 3 Grad-Div Stabilization fixed triangulation: lim µ min u h L2 (Ω) = 0 Proof: consider µ min = µ = µ K for all K T h use u h as test function, dual pairing, Young s inequality consequently statement follows ν u h 2 + µ u h 2 L 2 (Ω) L 2 (Ω) = f, u h V,V f H 1 (Ω) u h L 2 (Ω) 1 4ν f 2 H 1 (Ω) + ν u h 2 L 2 (Ω) u h L 2 (Ω) 1 2ν 1/2 µ 1/2 f H 1 (Ω) Question: Large stabilization parameters good for other errors? School 2016, Prague, August 29 - September 2 Page 60 (102)

72 3 Grad-Div Stabilization answer by finite element error analysis different situations can be distinguished, see [1] standard situation leads to estimate ( u u h ) 2 L 2 (Ω) inf v h Vdiv h ( 4 ( u v h ) 2 L 2 (Ω) + 2µ ) v h 2 ν L 2 (Ω) + 2 µν inf q h Q h p q h 2 L 2 (Ω) present at board velocity error depends still on pressure dependency on viscosity is ν 1/2 instead of ν 1 without grad-div stabilization [1] Jenkins, J., Linke, Rebholz; Adv. Comput. Math. 40, , 2014 School 2016, Prague, August 29 - September 2 Page 61 (102)

73 3 Grad-Div Stabilization concrete estimate for Taylor Hood pair of finite element spaces V h /Q h = P k /P k 1, k 2 ( u u h) ( µ ) C 2 L 2 (Ω) V ν divh 2k u 2 h H k+1 (Ω) + 2C2 Q h µν h2k p 2 H k (Ω). optimal choice of stabilization parameter by minimizing error bound depends on unknown norms of the solution depends on unknown constants for the best approximation error estimates if one assumes that all unknown quantities are O (1) = µ = O (1) similar considerations for MINI element: µ = O (h) optimal stabilization parameter for velocity error is not large School 2016, Prague, August 29 - September 2 Page 62 (102)

74 3 Grad-Div Stabilization summary lot of experience in literature with grad-div stabilization and parameters that are appropriate for good error bounds improves mass conservation somewhat, but usually not essential velocity error still depends on pressure grad-div stabilization is not the solution of the problem School 2016, Prague, August 29 - September 2 Page 63 (102)

75 3 Divergence-Free Stable Finite Elements stability and mass conservation in finite element methods are conflicting requirements let V h be fixed stability: discrete inf-sup condition inf sup q h Q h \{0} v h V h \{0} given if Q h is sufficiently small ( v h, p h ) v h β L2 (Ω) qh is h > 0 L2 (Ω) School 2016, Prague, August 29 - September 2 Page 64 (102)

76 3 Divergence-Free Stable Finite Elements stability and mass conservation in finite element methods are conflicting requirements let V h be fixed stability: discrete inf-sup condition inf sup q h Q h \{0} v h V h \{0} ( v h, p h ) v h β L2 (Ω) qh is h > 0 L2 (Ω) given if Q h is sufficiently small mass conservation: V h Q h : take q h = u h given if Q h is sufficiently large 0 = ( u h, q h) = u h 2 L 2 (Ω) School 2016, Prague, August 29 - September 2 Page 64 (102)

77 3 Divergence-Free Stable Finite Elements stability and mass conservation in finite element methods are conflicting requirements let V h be fixed stability: discrete inf-sup condition inf sup q h Q h \{0} v h V h \{0} ( v h, p h ) v h β L2 (Ω) qh is h > 0 L2 (Ω) given if Q h is sufficiently small mass conservation: V h Q h : take q h = u h given if Q h is sufficiently large 0 = ( u h, q h) = u h 2 L 2 (Ω) V h L 2 (Ω) normal components of finite element functions are continuous note: not satisfied for Crouzeix Raviart finite element School 2016, Prague, August 29 - September 2 Page 64 (102)

78 3 Divergence-Free Stable Finite Elements analytical tool: smooth de Rham complex or Stokes complex in two dimensions R H 2 (Ω) curl H 1 (Ω) div L 2 (Ω) 0 with curl v(x) = ( ) y v (x) x v sequence of spaces and maps de Rham complex is called exact if the range of each operator is the kernel of the succeeding operator if w H 2 (Ω) is curl-free, then w is constant function if v H 1 (Ω) is divergence-free, then v = curl w for some w H 2 (Ω) the map div : H 1 (Ω) L 2 (Ω) is surjective, since the kernel of the last operator is L 2 (Ω) School 2016, Prague, August 29 - September 2 Page 65 (102)

79 3 Divergence-Free Stable Finite Elements finite element sub-complex R W h curl V h div Q h 0 if finite element sub-complex is exact V h /Q h satisfies the discrete inf-sup condition weakly divergence-free velocity fields are computed, since div V h = Q h goal: construction of exact finite element sub-complex [1] Ciarlet; The finite element method for elliptic problems, 1978, Chapter 6.1 School 2016, Prague, August 29 - September 2 Page 66 (102)

80 3 Divergence-Free Stable Finite Elements finite element sub-complex R W h curl V h div Q h 0 if finite element sub-complex is exact V h /Q h satisfies the discrete inf-sup condition weakly divergence-free velocity fields are computed, since div V h = Q h goal: construction of exact finite element sub-complex example consider barycentric refinement of triangles Hsieh Clough Tocher finite element [1] composite element of third order polynomials on each fine mesh cell requirement: continuously differentiable = finite element space W h belongs to H 2 (Ω) [1] Ciarlet; The finite element method for elliptic problems, 1978, Chapter 6.1 School 2016, Prague, August 29 - September 2 Page 66 (102)

81 3 Divergence-Free Stable Finite Elements example (cont.) differentiation reduces the polynomial degree by one and also the regularity by one = V h = curl W h H 1 (Ω) and V h = P 2 (on barycentric-refined mesh) = Q h = div V h L 2 (Ω) and Q h = P disc 1 (on barycentric-refined mesh) [1] Scott, Vogelius; in Large-scale computations in fluid mechanics, Part 2, , 1985 School 2016, Prague, August 29 - September 2 Page 67 (102)

82 3 Divergence-Free Stable Finite Elements example (cont.) differentiation reduces the polynomial degree by one and also the regularity by one = V h = curl W h H 1 (Ω) and V h = P 2 (on barycentric-refined mesh) = Q h = div V h L 2 (Ω) and Q h = P1 disc (on barycentric-refined mesh) to show exactness: div : V h Q h is a surjection finite-dimensional spaces: by counting the number of degrees of freedom (somewhat longer) P 2 /P disc 1 Scott Vogelius pair of spaces [1] on barycentric refined grids is stable and computes weakly divergence-free solutions [1] Scott, Vogelius; in Large-scale computations in fluid mechanics, Part 2, , 1985 School 2016, Prague, August 29 - September 2 Page 67 (102)

83 3 Divergence-Free Stable Finite Elements similar construction can be started with other H 2 (Ω) conforming finite element spaces lead to high polynomial spaces, of little importance in practice situation in two dimensions more or less clear [1] Zhang; Math. Comp. 74, , 2005 School 2016, Prague, August 29 - September 2 Page 68 (102)

84 3 Divergence-Free Stable Finite Elements similar construction can be started with other H 2 (Ω) conforming finite element spaces lead to high polynomial spaces, of little importance in practice situation in two dimensions more or less clear 3d case much more challenging possible de Rham complex R H 2 (Ω) grad H 1 (curl; Ω) curl H 1 (Ω) div L 2 (Ω) 0 leads to velocity space with polynomials of degree 6 Scott Vogelius pair of spaces P k /Pk 1 disc is stable on barycentric refined meshes for k 3, [1] [1] Zhang; Math. Comp. 74, , 2005 School 2016, Prague, August 29 - September 2 Page 68 (102)

85 3 Divergence-Free Stable Finite Elements summary Scott Vogelius pair of spaces P k /Pk 1 disc, k d, so far only pair that is used sometimes little hope to construct any other lower order pair School 2016, Prague, August 29 - September 2 Page 69 (102)

86 3 H (div, Ω)-Conforming Finite Element Methods abandon conformity of finite element velocity space: V h V require V h L 2 (Ω) = study H (div, Ω) conforming finite elements Raviart Thomas elements, BDM elements normal component of functions is continuous across faces difficulty: consistency error in discretizing the viscous term (ν u, v) no convergence for using just proposals K T h K ν u h : v h dx modify bilinear form modify H (div, Ω) to impose tangential continuity in a weak sense School 2016, Prague, August 29 - September 2 Page 70 (102)

87 3 H (div, Ω)-Conforming Finite Element Methods modify bilinear form: possible proposal u h : v h dx ( K T h + E K {{ v h }} E E E h [ u h ] E ds σ h E E E {{ ε(u h ) }} E [ u h ] E [ ] E jump across face {{ }} E average on face {{ε }} E average of tangential component on face σ parameter reminds on DG [ v h ] E ds [ v h ] E ds ) School 2016, Prague, August 29 - September 2 Page 71 (102)

88 3 H (div, Ω)-Conforming Finite Element Methods modify H (div, Ω) to impose tangential continuity in a weak sense local space (2d) ˆV h (K) = V h (K) + curl (b K S(K)) b K bubble function S(K) auxiliary space example: V h (K) = RT 0 (K), S(K) = P 1 (K) global space possesses correct order of consistency error optimal error estimates can be proved School 2016, Prague, August 29 - September 2 Page 72 (102)

89 3 H (div, Ω)-Conforming Finite Element Methods modify H (div, Ω) to impose tangential continuity in a weak sense local space (2d) ˆV h (K) = V h (K) + curl (b K S(K)) b K bubble function S(K) auxiliary space example: V h (K) = RT 0 (K), S(K) = P 1 (K) global space possesses correct order of consistency error optimal error estimates can be proved summary use of H (div, Ω)-conforming methods not (yet) popular needs more studies School 2016, Prague, August 29 - September 2 Page 72 (102)

90 3 Pressure-Robust FEM idea: satisfy discrete version of fundamental invariance principle by using appropriate reconstructions of the test functions [1] finite element Stokes problem with reconstruction: Given f L 2 (Ω), find ( u h, p h) V h Q h such that ν ( u h, v h) ( v h, p h) = ( f, Π h v h) v h V h ( u h, q h) = 0 q h Q h with Π h : V h R h (R h H (div, Ω)-conforming fe space) [1] Linke; Comput. Methods Appl. Mech. Engrg. 268, , 2014 School 2016, Prague, August 29 - September 2 Page 73 (102)

91 3 Pressure-Robust FEM consider V h Q h = P2 bubble P1 disc discontinuous pressure of importance R h = RT 1 general requirements on Π h : projection and interpolation properties (v Π h v) dx = 0, v V, K T h K (v Π h v) n E q h ds = 0, v V, q h P 1 (E) E Π h v v L 2 (K) cell-wise computation of Π h v h possible Ch m K v Hm (K), m = 0, 1, 2 School 2016, Prague, August 29 - September 2 Page 74 (102)

92 3 Pressure-Robust FEM V h Q h = P2 bubble P1 disc (cont.) it holds for all K T h, v V, and q h Q h K vq h dx prod. rule = int. by parts = prop. 1 & 2 = int. by parts = K T T K (vq h ) dx K q h v dx q h v n T ds q h v dx K q h (Π h v) n T ds q h (Π h v) dx (Π h v)q h dx = Q h projection of the divergence = Q h projection of the divergence of the projection K School 2016, Prague, August 29 - September 2 Page 75 (102)

93 3 Pressure-Robust FEM V h Q h = P2 bubble P1 disc (cont.) fundamental invariance principle for modified discretization: for all v h Vdiv h (= consider only velocity) ( f + ( PQ hψ ), Π h v h) int. by parts = div. prop = disc. div-free = ( f, Π h v h) ( (Π h v h), P Q hψ ) ( f, Π h v h) ( v h, P Q hψ ) ( f, Π h v h) discrete counterpart of fundamental invariance principle satisfied: ( P Q hψ ) possesses no impact on finite element velocity School 2016, Prague, August 29 - September 2 Page 76 (102)

94 3 Pressure-Robust FEM finite element error analysis consistency error is introduced, is of optimal order estimate for velocity error (u u h ) L2 (Ω) 2(1+C PF) inf (u v h ) v h V h L2 (Ω) +Ch2 u H3 (Ω) does not depend on pressure School 2016, Prague, August 29 - September 2 Page 77 (102)

95 3 Pressure-Robust FEM Example 1 Crouzeix Raviart finite element with reconstruction in RT Crouzeix-Raviart Crouzeix-Raviart with modified test function (u u h ) L 2 (Ω) Ra =1 Ra =100 Ra =10 4 Ra =10 6 (u u h ) L 2 (Ω) Ra =1 Ra =100 Ra =10 4 Ra = level level School 2016, Prague, August 29 - September 2 Page 78 (102)

96 3 Pressure-Robust FEM Example 2 Crouzeix Raviart finite element with reconstruction in RT Crouzeix-Raviart Crouzeix-Raviart with modified test function (u u h ) L 2 (Ω) (u u h ) L 2 (Ω) Re = Re =1 Re = Re =10 Re =10 Re =100 Re =100 Re = level level School 2016, Prague, August 29 - September 2 Page 79 (102)

97 3 Pressure-Robust FEM summary extensions to almost arbitrary finite element pairs with discontinuous pressure in [1] (simplicial and brick-shaped meshes) extension to finite element pairs with continuous pressure [2] extension to Navier Stokes equations possible (and partly research in progress) modifications of nonlinear convective term and of term with temporal derivative necessary leads to modifications of the system matrix divergence-free finite element velocity in R h by applying reconstruction operator to u h [1] Linke, Matthies, Tobiska; ESAIM: M2AN 50, , 2016 [2] Lederer, Linke, Merdon Schöberl; WIAS Preprint 2288, 2016 School 2016, Prague, August 29 - September 2 Page 80 (102)

98 3 Summary of this Part most standard pairs of finite element spaces do not preserve mass grad-div stabilization does not solve the problem only divergence-free stable pair that is of some importance is Scott Vogelius pair H (div, Ω)-conforming finite element methods of interest, need more studies currently most interesting approach: pressure robust methods with appropriate test functions by reconstructions in H (div, Ω)-conforming finite element spaces survey of Part 3 in [1,2] [1] J., Linke, Merdon, Neilan, Rebholz; SIAM Review, in press, 2016 [2] J.; Finite Element Methods for Incompressible Flow Problems, Springer, in press, 2016 School 2016, Prague, August 29 - September 2 Page 81 (102)

99 4. Finite Element Methods that Circumvent the Discrete Inf-Sup Condition School 2016, Prague, August 29 - September 2 Page 82 (102)

100 4 Difficulties of Inf-Sup Stable FEMs need implementation of different finite element spaces for velocity and pressure one obtains a linear algebraic saddle point problem ( A B T B 0 ) ( ) u = p ( ) f 0 sparse direct solvers only efficient in 2d and for small and medium sized systems ( d.o.f.s) construction of special preconditioners for iterative methods necessary, because of zero on main diagonal School 2016, Prague, August 29 - September 2 Page 83 (102)

101 4 Difficulties of Inf-Sup Stable FEMs need implementation of different finite element spaces for velocity and pressure one obtains a linear algebraic saddle point problem ( A B T B 0 ) ( ) u = p ( ) f 0 sparse direct solvers only efficient in 2d and for small and medium sized systems ( d.o.f.s) construction of special preconditioners for iterative methods necessary, because of zero on main diagonal goal: remove saddle point character by removing the zero block introduce pressure-pressure coupling in mass balance several proposals in literature here: PSPG method leads automatically to violation of mass conservation enables use of same finite element spaces for velocity and pressure School 2016, Prague, August 29 - September 2 Page 83 (102)

102 4 The PSPG Method Pressure Stabilization Petrov Galerkin (PSPG) method [1] most popular method given f L 2 (Ω), find ( u h, p h) V h Q h such that A pspg (( u h, p h), ( v h, q h)) = L pspg (( v h, q h)) ( v h, q h) V h Q h with A pspg ((u, p), (v, q)) = ν ( u, v) ( v, p) + ( u, q) + γ E ([ p ] E, [ q ] E ) E + ( ν u + p, δ p K q) K E E h K T h and L pspg ((v, q)) = (f, v)+ K T h (f, δ p K q) K [1] Hughes, Franca, Balestra; Comput. Methods Appl. Mech. Engrg. 59, 85 99, 1986 School 2016, Prague, August 29 - September 2 Page 84 (102)

103 4 The PSPG Method meaning of the terms γ E ([ p ] E, [ q ] E ) E + ( ν u + p, δ p K q) K E E h K T h K T h (f, δ p K q) K E h set of faces of mesh cells E face T h triangulation, set of mesh cells K mesh cell γ E, δ p K stabilization parameters [ ] E jump across a face goal: appropriate choice of stabilization parameters by finite element error analysis School 2016, Prague, August 29 - September 2 Page 85 (102)

104 4 Norm for Finite Element Error Analysis finite element error analysis for stabilized methods is usually performed for norms that include stabilization norm for PSPG method ( v h, q h) pspg = ( ν v h 2 L 2 (Ω) + + K T h δ p K E E h γ E q h 2 L 2 (K) ) 1/2 [ q h ] E 2 L 2 (E) sum of seminorms to check that from ( v h, q h) pspg = 0 it follows that v h = 0 and q h = 0: direct calculation School 2016, Prague, August 29 - September 2 Page 86 (102)

105 4 Properties of the Solution existence and uniqueness of a solution if apply Theorem of Lax Milgram 0 < δ p K h2 K νc 2 inv show coercivity of bilinear form (board) A pspg (( v h, q h), ( v h, q h)) 1 2 ( v h, q h) 2 pspg show boundedness of right-hand side School 2016, Prague, August 29 - September 2 Page 87 (102)

106 4 Properties of the Solution stability of solution 1/2 ( u h, p h) pspg 2 2 C ν f 1/2 L 2 (Ω) + δ p K f 2 L 2 (K) K T h use unique solution as test function School 2016, Prague, August 29 - September 2 Page 88 (102)

107 4 Properties of the Solution stability of solution 1/2 ( u h, p h) pspg 2 2 C ν f 1/2 L 2 (Ω) + δ p K f 2 L 2 (K) K T h use unique solution as test function Galerkin orthogonality A pspg (( u u h, p p h), ( v h, q h)) = 0 ( v h, q h) V h Q h direct calculation using definition of PSPG method School 2016, Prague, August 29 - September 2 Page 88 (102)

108 4 Finite Element Error Analysis finite element error analysis: let and δ p K = C h 2 K 0 ν, γ E = C 1 h E ν then P k or Q k V h V, k 1, P l or Q l Q h Q, l 0, ( u u h, p p h) pspg C ) (ν 1/2 h k hl+1 u Hk+1(Ω) + ν p 1/2 H l+1 (Ω) triangle inequality = interpolation error + discrete term coercivity for discrete term estimate of individual terms on right-hand side by standard estimates (Cauchy Schwarz, Young, interpolation error estimates) using bounds for stabilization parameters School 2016, Prague, August 29 - September 2 Page 89 (102)

109 4 Finite Element Error Analysis error estimate ( u u h, p p h) pspg C ) (ν 1/2 h k hl+1 u Hk+1(Ω) + ν p 1/2 H l+1 (Ω) convergence in pspg is at least min{k, l + 1} neglect all terms but ( u u h) L 2 (Ω) shows that ν 1 appears in front of the pressure term for continuous pressure finite element spaces ( p p h) ( ) C L2 νh k 1 u (Ω) H k+1 (Ω) + hl p H l+1 (Ω) estimates for u u h L2 (Ω) and p p h L2 in [1] (Ω) [1] Brezzi, Douglas; Numer. Math. 53, , 1988 School 2016, Prague, August 29 - September 2 Page 90 (102)

110 4 Numerical Example analytical example which supports the error estimates prescribed solution u = ( u1 p = 10 ) ( ) y ψ = x ψ ( ( x 1 2 u 2 ( x 2 (1 x) 2 y(1 y)(1 2y) = 200 x(1 x)(1 2x)y 2 (1 y) 2 ) 3 y 2 + (1 x) 3 ( y 1 2 ) 3 ) ) School 2016, Prague, August 29 - September 2 Page 91 (102)

111 4 Numerical Example P 1 /P 1, δ p K = 0.5h2 K /ν (u u h ) L 2 (Ω) h ν =10 6 ν =10 2 ν =1 ν =10 2 ν =10 6 ν =10 10 p p h L 2 (Ω) h 3/2 ν =10 6 ν =10 2 ν =1 ν =10 2 ν = ν = levels levels u u h L 2 (Ω) 10 8 h ν = ν = ν = ν = ν = ν = levels u h L 2 (Ω) 10 9 h 10 8 ν = ν = ν =1 ν = ν = ν = levels larger velocity errors for small ν, but higher order of convergence School 2016, Prague, August 29 - September 2 Page 92 (102)