Finite element methods for incompressible flow problems

Springer Series in Computational Mathematics 51 Volker John Finite Element Methods for Incompressible Flow Problems Springer Series in Computational Mathematics Volume 51 Editorial Board R.E Bank R.L Graham W Hackbusch J Stoer R.S Varga H Yserentant More information about this series at http://www.springer.com/series/797 Volker John Finite Element Methods for Incompressible Flow Problems 123 Volker John Weierstrass Institute for Applied Analysis and Stochastics Berlin, Germany Fachbereich Mathematik und Informatik Freie UniversitRat Berlin Berlin, Germany ISSN 0179-3632 ISSN 2198-3712 (electronic) Springer Series in Computational Mathematics ISBN 978-3-319-45749-9 ISBN 978-3-319-45750-5 (eBook) DOI 10.1007/978-3-319-45750-5 Library of Congress Control Number: 2016956572 Mathematics Subject Classification (2010): 65M60, 65N30, 35Q30, 76F65, 76D05, 76D07 © Springer International Publishing AG 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the 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have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland For Anja and Josephine Preface Incompressible flow problems appear in many models of physical processes and applications Their numerical simulation requires in particular a spatial discretization Finite element methods belong to the mathematically best understood discretization techniques This monograph is devoted mainly to the mathematical aspects of finite element methods for incompressible flow problems It addresses researchers, Ph.D students, and even students aiming for the master’s degree The presentation of the material, in particular of the mathematical arguments, is performed in detail This style was chosen in the hope to facilitate the understanding of the topic, especially for nonexperienced readers Most parts of this monograph were presented in three consecutive master’s level courses taught at the Free University of Berlin, and this monograph is based on the corresponding lecture notes First of all, I like to thank the students who attended these courses Many of them wrote finally their master’s thesis under my supervision Then, I like to thank two collaborators of mine, Julia Novo (Madrid) and Gabriel R Barrenechea (Glasgow), who read parts of this monograph and gave valuable suggestions for improvement Above all, I like to thank my beloved wife Anja and my daughter Josephine for their continual encouragement Their efforts to manage our daily life and to save me working time were an invaluable contribution for writing this monograph in the past years Colbitz, Germany July 2016 Volker John vii Contents Introduction 1.1 Contents of this Monograph 2 The Navier–Stokes Equations as Model for Incompressible Flows 2.1 The Conservation of Mass 2.2 The Conservation of Linear Momentum 2.3 The Dimensionless Navier–Stokes Equations 2.4 Initial and Boundary Conditions 7 17 19 Finite Element Spaces for Linear Saddle Point Problems 25 3.1 Existence and Uniqueness of a Solution of an Abstract Linear Saddle Point Problem 26 3.2 Appropriate Function Spaces for Continuous Incompressible Flow Problems 41 3.3 General Considerations on Appropriate Function Spaces for Finite Element Discretizations 52 3.4 Examples of Pairs of Finite Element Spaces Violating the Discrete Inf-Sup Condition 62 3.5 Techniques for Checking the Discrete Inf-Sup Condition 72 3.5.1 The Fortin Operator 72 3.5.2 Splitting the Discrete Pressure into a Piecewise Constant Part and a Remainder 76 3.5.3 An Approach for Conforming Velocity Spaces and Continuous Pressure Spaces 79 3.5.4 Macroelement Techniques 84 3.6 Inf-Sup Stable Pairs of Finite Element Spaces 93 3.6.1 The MINI Element 93 3.6.2 The Family of Taylor–Hood Finite Elements 98 3.6.3 Spaces on Simplicial Meshes with Discontinuous Pressure 111 ix x Contents 3.7 3.6.4 Spaces on Quadrilateral and Hexahedral Meshes with Discontinuous Pressure 3.6.5 Non-conforming Finite Element Spaces of Lowest Order 3.6.6 Computing the Discrete Inf-Sup Constant The Helmholtz Decomposition The Stokes Equations 4.1 The Continuous Equations 4.2 Finite Element Error Analysis 4.2.1 Conforming Inf-Sup Stable Pairs of Finite Element Spaces 4.2.2 The Stokes Projection 4.2.3 Lowest Order Non-conforming Inf-Sup Stable Pairs of Finite Element Spaces 4.3 Implementation of Finite Element Methods 4.4 Residual-Based A Posteriori Error Analysis 4.5 Stabilized Finite Element Methods Circumventing the Discrete Inf-Sup Condition 4.5.1 The Pressure Stabilization Petrov–Galerkin (PSPG) Method 4.5.2 Some Other Stabilized Methods 4.6 Improving the Conservation of Mass, Divergence-Free Finite Element Solutions 4.6.1 The Grad-Div Stabilization 4.6.2 Choosing Appropriate Test Functions 4.6.3 Constructing Divergence-Free and Inf-Sup Stable Pairs of Finite Element Spaces 115 117 124 127 137 137 144 145 163 165 180 187 198 199 213 217 218 229 237 The Oseen Equations 5.1 The Continuous Equations 5.2 The Galerkin Finite Element Method 5.3 Residual-Based Stabilizations 5.3.1 The Basic Idea 5.3.2 The SUPG/PSPG/grad-div Stabilization 5.3.3 Other Residual-Based Stabilizations 5.4 Other Stabilized Finite Element Methods 243 243 249 258 258 261 287 289 The Steady-State Navier–Stokes Equations 6.1 The Continuous Equations 6.1.1 The Strong Form and the Variational Form 6.1.2 The Nonlinear Term 6.1.3 Existence, Uniqueness, and Stability of a Solution 6.2 The Galerkin Finite Element Method 301 301 301 302 312 316 Contents 6.3 6.4 xi Iteration Schemes for Solving the Nonlinear Problem 333 A Posteriori Error Estimation with the Dual Weighted Residual (DWR) Method 342 The Time-Dependent Navier–Stokes Equations: Laminar Flows 7.1 The Continuous Equations 7.2 Finite Element Error Analysis: The Time-Continuous Case 7.3 Temporal Discretizations Leading to Coupled Problems 7.3.1 Â-Schemes as Discretization in Time 7.3.2 Other Schemes 7.4 Finite Element Error Analysis: The Fully Discrete Case 7.5 Approaches Decoupling Velocity and Pressure: Projection Methods 355 355 377 393 393 409 410 The Time-Dependent Navier–Stokes Equations: Turbulent Flows 8.1 Some Physical and Mathematical Characteristics of Turbulent Incompressible Flows 8.2 Large Eddy Simulation: The Concept of Space Averaging 8.2.1 The Basic Concept of LES, Space Averaging, Convolution with Filters 8.2.2 The Space-Averaged Navier–Stokes Equations in the Case ˝ D Rd 8.2.3 The Space-Averaged Navier–Stokes Equations in a Bounded Domain 8.2.4 Analysis of the Commutation Error for the Gaussian Filter 8.2.5 Analysis of the Commutation Error for the Box Filter 8.2.6 Summary of the Results Concerning Commutation Errors 8.3 Large Eddy Simulation: The Smagorinsky Model 8.3.1 The Model of the SGS Stress Tensor: Eddy Viscosity Models 8.3.2 Existence and Uniqueness of a Solution of the Continuous Smagorinsky Model 8.3.3 Finite Element Error Analysis for 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methods Numerical mathematics and scientific computation Oxford University Press, Oxford, pp xx+393 References 803 Winckelmans G, Jeanmart H, Carati D (2002) On the comparison of turbulence intensities from large-eddy simulation with those from experiment or direct numerical simulation Phys Fluids 14:3 Xu J, Zikatanov L (2003) Some observations on Babuška and Brezzi theories Numer Math 94:195–202 Yosida K (1995) Functional analysis Classics in mathematics Springer, Berlin, pp xii+501 Reprint of the sixth (1980) edition Zhang S (2005) A new family of stable mixed finite elements for the 3D Stokes equations Math Comp 74:543–554 Zhang S (2009) Bases for C0-P1 divergence-free elements and for C1-P2 finite elements on union jack grids http://www.math.udel.edu/~szhang/research/p/uj.pdf Accessed 20 July 2016 Zang Y, Street RL, Koseff JR (1993) A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows Phys Fluids A 5:3186–3196 Index of Subjects adjoint variable, 346 affine mapping, see mapping ASGS-VMS method, see method attractor global, 456 augmented Lagrangian-based preconditioner, see preconditioner auxiliary problem, 548 averaging radius, 459 Babuška–Brezzi condition, see inf-sup condition backscatter, 449, 539, 607, 618 backward Euler scheme, see Euler scheme barycenter, 712 barycentric coordinates, 711 barycentric-refined grid, see grid basis global, 710 local, 708 nodal, 184 BDF2, 409, 437, 439 Beltrami flow, 767 Bernardi–Raugel element, see finite element Bernoulli pressure, see pressure BiCGStab, 652 bilinear form V-elliptic, 697 bounded, 697 coercive, 697 boundary locally Lipschitz continuous, 685 boundary condition directional do-nothing, 23, 316 Dirichlet, 20 do-nothing, 21 essential, 21 natural, 21 no-slip, 20 outflow, 43 periodic, 24 slip with linear friction and no penetration, 508, 509 Boussinesq hypothesis, 484 box filter, see filter Bramble–Hilbert lemma, see lemma bubble function cell, 193, 716, 721 edge, 715 face, 193 Cauchy stress vector, 10, 466 Cauchy–Schwarz inequality, see inequality CGS, 652 checkerboard instability, 64, 69 checkpoint technique, 353 CIP method, see method coarse grid solver, 665 coercivity, 35 commutation, 622 commutation error, 466, 470 compact set, 678 compatibility condition, 20 compatibility conditions at initial time, 377 complex de Rham, 238 finite element sub-, 239 © Springer International Publishing AG 2016 V John, Finite Element Methods for Incompressible Flow Problems, Springer Series in Computational Mathematics 51, DOI 10.1007/978-3-319-45750-5 805 806 Stokes, 238 condensed sparse row format, 650 consistency error, 167 continuity equation, continuous interior penalty method, see method convective term convective form, 303 divergence form, 303, 599 divergence form with modified pressure, 303 rotational form, 303 skew-symmetric form, 308 skew-symmetry, 245 convergence weak, 696 weak , 696 convolution, 689 Crank–Nicolson scheme, 394, 397, 409, 429, 436, 437 cross term, 541 curl, 127 two dimensions, 132 cut-off wave number, 459 cycle multigrid, 655 de Rham complex, see complex defect restriction, 661 deformation tensor formulation, 138 degree of freedom, 180 density, diameter of a mesh cell, 734 differential filter, see filter Direct Numerical Simulation, 451 discrete differential filter, see filter dissipation of turbulent energy, 449 divergence distributional, 44 weak, 44 divergence of a tensor, 13 divergence operator, see operator divergence-free discretely, 56 weakly, 44, 45 divergence-free subspace sequence with optimal approximation property, 221 domain polyhedral, 148 with Lipschitz continuous boundary, 148 downwind, 297 drag coefficient, 353, 751, 761 drag force, see force Index of Subjects driven cavity problem, 753 dual problem, 346 dual weighted residual method, see method DWR method, see method dynamic Smagorinsky model, see model dynamic subgrid scale model, see model dynamical system, 455 autonomous, 455 eddy viscosity, see viscosity eddy viscosity model, see model edge, 707 energy cascade, 449 energy dissipation, 369 energy inequality, see inequality error equation, 146 essential boundary condition, see boundary condition estimate a posteriori, 190 interpolation, 736 inverse, 745 Euler formula, 735 Euler scheme backward, 394, 397, 401, 412, 433 forward, 394, 397 IMEX, 424 Eulerian description of a flow field, 576 face, 707 FGMRES, 666 field irrotational, 128 isotropic, 449 statistically homogeneous, 449 statistically stationary, 449 filter, 464 box, 462, 477 characteristic width, 459 differential, 358, 548, 554, 562, 566 discrete differential, 566 Gaussian, 460, 542, 554 scale, 459 skewed, 477 test, 537 top hat, 462 width, 459 finite element P0 , 713 P1 , 713 P1 =P0 , 64 Index of Subjects P1 =P1 , 63 PBR =P0 , 114 Pnc , 295, 717 Pnc =P0 , 165, 291 h=2 P1 =P1 , 111 P2 , 714 disc PBR =P1 , 115 bubble =Pdisc P2 , 111, 113 P3 , 715 =Pdisc Pbubble , 114 Pk =Pk , 98 Pk =Pdisc k , 112 Pdisc k , 723 Q0 , 720 Q1 , 720 Q1 =Q0 , 67 Qrot , 295, 722 Qrot =Q0 , 118, 165 Q2 , 721 8/ Q2 =Q1 , 98 Q3 , 721 Qk =Pdisc k mapped, 115 Qk =Pdisc k unmapped, 115 Qk =Qk , 98 RTk , 718 Bernardi–Raugel, 111 Crouzeix–Raviart, 117, 165, 229, 717 Hsieh–Clough–Tocher, 239 isoparametric, 329 Lagrangian, 713 MINI, 93, 153, 226, 254, 326 modified Taylor–Hood, 111 parametric, 710 Rannacher–Turek, 117, 165, 722 Raviart–Thomas, 241, 718 rotated bilinear, 118 Scott–Vogelius, 70, 111, 112, 160, 238, 239, 385 simplicial, 712 Taylor–Hood, 98, 153, 224, 254, 326 finite element method conforming, 52 continuous-in-time, 516 non-conforming, 52, 117 finite element space, see space fluctuation, 216, 292, 458, 609 force drag, 751 irrotational, 144 lift, 751 Fortin operator, see operator forward Euler scheme, see Euler scheme Fourier transform, 459, 689 807 fractional-step  -scheme, 396 friction velocity, see velocity function absolutely continuous, 690 average on a face, 61 continuous with respect to functional, 709 jump across a face, 61 function restriction, 661 functional energy, 40 global, 709 Lagrangian, 40 linear, 697 local, 709 Galerkin least squares (GLS) method, see method Galerkin method, see method Galerkin orthogonality, 348, 703, 705 Galilean invariance, 483 Gaussian filter, see filter global basis, see basis GMRES, 651 grad-div stabilization, see stabilization grad-div term, 264 gradient velocity, 15 gradient operator, see operator grid, 709 barycentric-refined, 112, 222 criss-cross, 222 Union Jack, 222 Gronwall’s lemma, see lemma Hölder coefficient, 683 Hölder’s inequality, see inequality Hagen–Poiseuille flow, 22 Hausdorff dimension, 457 Helmholtz decomposition, 130 Helmholtz projection, see projection ILU factorization, 653 image, 696 imbedding theorem, see theorem IMEX scheme, 406, 424, 429 inequality Cauchy–Schwarz, 684 Cauchy–Schwarz, sum, 679 energy, 369, 488 Hölder’s, 684 808 Hölder’s, sum, 679 Korn’s, 46, 510 Poincaré’s, 686 Poincaré–Friedrichs’, 686 Poincaré-type, 730 triangle, 678 Young’s, 680 Young’s for convolutions, 690 inertial subrange, 452 inexact solution, 341, 342, 399, 652 inf-sup condition, 34, 140 Babuška, 35, 142 discrete, 52 initial condition, 20 inner product, 679 integral length scale, see scale interpolant, 733 interpolation of Sobolev spaces, 687 interpolation estimate, see estimate inverse estimate, see estimate irrotational field, see field irrotational force, see force isotropic turbulence, see turbulence iteration Newton, 338, 398, 652 Picard, 335, 398, 652 jump across a face, 61 Kármán vortex street, 761 kernel, 696 kinetic energy, 369, 452, 607 kinetic energy spectrum, 451 Kolmogorov 5=3 spectrum, 453 Kolmogorov hypotheses, 450 Kolmogorov length scale, see scale Kolmogorov scales, see scale Korn’s inequality, see inequality Krylov subspace, 651 Lagrangian description of a flow field, 576 Lagrangian multiplier, 25, 35, 395 Lagrangian point of view, 483 laminar flow, 355 large scale advective term, 541 LBB condition, see inf-sup condition Lebesgue space, see space lemma Bramble–Hilbert, 732 Cea, 704 Index of Subjects discrete Gronwall’s, 695 Gronwall’s, 692 Gronwall’s, variation of, 694 Strang, second, 167 Leray-˛ model, see model LES basic idea, 458 lift coefficient, 353, 751, 761 lift force, see force Lipschitz constant, 683 Lipschitz continuity, 512 local basis, see basis local projection stabilization (LPS) method, see method locking phenomenon, 67 LSC preconditioner, see preconditioner macroelement, 85 macroelement condition, 88 mapping affine, 712 parametric, 724 mass, conservation of, violation of conservation of, 57 matrix Gramian, 126 mass, 126, 406 Schur complement, 666 stiffness, 704 mesh, 709 mesh cell, 707 anisotropic, 123 reference, 712, 719 method absolutely stable, 213 algebraic subgrid scale (ASGS) VMS, 608 AVM3 , 614 AVM4 , 618 CIP, 289, 430, 431 coarse space projection-based VMS, 619 diagonally implicit Runge–Kutta, 410 dual weighted residual, 343 DWR, 343 finite difference, 473 Galerkin, 358, 486, 704 GLS, 214 higher order term-by-term stabilization, 295, 590 LPS, 217, 292, 430, 431, 590, 607 of lines, 393 of Rothe, 393 pressure-robust, 229 Index of Subjects projection, 431 pseudo-compressibility, 199 PSPG, 199, 283, 430, 435, 617 residual-free bubble, 613 Rosenbrock, 409 SUPG/PSPG/grad-div, 262, 292, 608 VMS, 591 VMS with orthogonal subscales (OSS), 605 MINI element, see finite element model ADM, 553 approximate deconvolution, 553 dynamic Smagorinsky, 537, 595 dynamic subgrid scale, 537 eddy viscosity, 484 isotropic Lagrangian-averaged Navier– Stokes, 576 Ladyzhenskaya, 482 Leray-˛, 563 Leray-˛ ADM, 574 Leray-deconvolution, 574 multifractal subgrid scale, 618 Navier–Stokes-˛, 575 rational LES, 547 rational LES model with auxiliary problem, 551 rational LES model with convolution, 549, 551 Smagorinsky, 482, 486, 550, 614, 620 Smagorinsky, stationary, 535 Taylor LES, 545, 551 viscous Camassa–Holm, 576 momentum linear, momentum equation, 13 monotonicity, 499 strong, 499, 511 multi-index, 681 multigrid method algebraic, 614 coupled, 654 natural boundary condition, see boundary condition Navier–Stokes equations dual linearized, 324 stationary, 19, 301 steady-state, 19, 301 Navier–Stokes-˛ model, see model Newton’s method, see iteration Newtonian fluid, 15 809 nodal functionals, 707 non-conforming finite element method, see finite element method norm, 678 norm, induced, 679 norms equivalent, 678 operator adjoint, 701 bounded, 696 continuous, 697 discrete divergence, 56 discrete gradient, 56 discrete Laplacian, 568 divergence, 43 Fortin, 72 gradient, 43 Green’s, fine-scale, 597 interpolation, Clément, 739 linear, 696 orthogonal complement, 680 element, 680 Oseen equations, 244, 336, 398 OSS-VMS method, see method outer layer, 453 parametric finite element, see finite element patch test, 79 PCD preconditioner, see preconditioner Picard iteration, see iteration Poincaré’s inequality, 166, see inequality Poincaré–Friedrichs’ inequality, 166, see inequality Poincaré-type inequality, see inequality precompact set, 678 preconditioner, 652 augmented Lagrangian-based, 673 boundary-corrected LSC, 671 left, 652 LSC, 669, 675 PCD, 672 right, 653 pressure, 7, 14 at initial time, 376 Bernoulli, 24, 304 dynamic, 751 pressure Poisson equation, 375 pressure stabilization Petrov–Galerkin term, see PSPG term 810 pressure-correction scheme non-incremental, 433 rotational incremental, 441 standard incremental, 436 pressure-projection equation, 434 projection L2 ˝/, 743 elliptic, 743 Galerkin, 656 Helmholtz, 131, 139, 433 local, L2 !i /, 740 local, L2 K/, 661 orthogonal, 743 Riesz, 743 projection method, see method prolongation, 656 PSPG method, see method PSPG term, 264 range, 696 rational LES model, see model rational LES model with auxiliary problem, see model rational LES model with convolution, see model Rayleigh quotient, 680 reactive term, 243 residual, 259 dual, 347 primal, 346 residual vector, 651 residual-free bubble method, see method Reynolds equation, 465 Reynolds number, 17 Reynolds stress tensor, see tensor Richardson energy cascade, 449 Ritz approximation, 702 rms turbulence intensity, 770 rotation, 127 saddle point problem, 26, 40 generalized, 41 scalar product, 679 scale integral length, 484 Kolmogorov, 450 Kolmogorov length, 450 large, 458, 621 small, 458 small resolved, 621 subgrid, 458 unresolved, 458 Index of Subjects viscous dissipation length, 565 viscous length, 453 Schur complement matrix, see matrix Scott–Vogelius element, see finite element seminorm, 678 sequence Cauchy, 677 convergent, 678 sgs term, 541, 549 SIMPLE, 666 simplex, 711 unit, 712 simulation robust, 482 Smagorinsky coefficient, 485 Smagorinsky model, see model Sobolev spaces, see space interpolation, 687 solution local, 375 Smagorinsky model, weak, 487 strong, 373 variational, 357 weak, 356, 357 weak in the sense of Leray–Hopf, 358 space C1 ˝/, 682 C1 ˝/, 682 Cm;˛ ˝/, 683 Cm ˝/, 681 Cm ˝/, 682 C0m ˝/, 683 C01 ˝/, 683 CBm ˝/, 682 ˝/, 683 C0;div H div; ˝/, 44 H m ˝/, 685 Hdiv ˝/, 45, 129 Lp ˝/, 683 Lp t0 ; t1 I X/, 689 Pk , 713 Qk , 720 W 1;p ˝/, 685 Banach, 678 complete, 678 discretely divergence-free, 56 finite element, 710 finite element, unisolvent, 708 for tensor-valued function, 42 for vector-valued function, 42 Hilbert, 679 isometric, 677 Lebesgue, 683 metric, 677 Index of Subjects null, 696 Sobolev, 684 wave number, 452 weakly divergence-free, 45 space-averaged momentum equation, 469 space-averaged Navier–Stokes equations, 463, 465 sparse direct solver, 650 spurious pressure mode, 63 stabilization grad-div, 218, 332, 430, 436, 510, 599, 612, 624, 673 residual-based, 258 Stokes complex, 238 equations, 19, 137, 335 operator, 579 operator, discrete, 389 problem, dual, 150 projection, 163 stopping criterion, 403 stream function, 132, 133, 752 streamline diffusion term, 264 Streamline-Upwind Petrov–Galerkin term, 264 stress normal, 16 shear, 16 wall shear, 453 stress tensor, see tensor Strouhal number, 17, 762 subgrid scale term, 541, 598 subscale static, 607 SUPG term, 264, 599 support, 682 Taylor LES model, see model Taylor–Hood finite element, see finite element tensor residual-stress, 465 Reynolds stress, 465, 770 stress, 12, 16 subgrid, 465 subgrid-scale (sgs) stress, 465 velocity deformation, 14 viscous stress, 14 theorem Carathéodory, 691 closed range, 698 Lax–Milgram, 701 Reynolds transport, Riesz, 699 Sobolev imbedding, 687 811 trace, 686 time-continuous Galerkin formulation, 377 top hat filter, see filter torque, 13 torsion vector, 10 trapezoidal rule composite, 415 triangulation, 709 admissible, 709 quasi-uniform family of, 734 regular family of, 735 turbulence isotropic, 449 turbulent channel flow, 539, 552, 638 turbulent viscosity, see viscosity unisolvence, 708 unit cube, 719 upwind, 297 Samarskij, 298 simple, 298 van Cittert family of approximate deconvolutions, 556, 590 Vanka smoother, 663 mesh-cell-oriented, 663 pressure-node-oriented, 664 variational multiscale method, see method variational solution, see solution vector potential, 133 velocity, friction, 453 mean, 770 subgrid scale, 602 velocity deformation tensor, see tensor vertex, 707 viscosity dimensionless, 18 dynamic, 15 eddy, 484 kinematic, 16 shear, 15 turbulent, 484 viscous length scale, see scale, 618 viscous sublayer, 453 viscous term deformation tensor formulation, 183 gradient formulation, 184, 340 Laplacian formulation, 184 viscous wall region, 453 VMS method, see method 812 vortex stretching, 454 vorticity, 454 vorticity equation, 454 wall law, 481 wall shear stress, 453 wall unit, 453 wave number, 459 wave number space, see space Index of Subjects weak convergence, see convergence weak divergence, 44 weak solution, see solution weakly convergence, see convergence Yosida approximation, 358, 548 Young’s inequality, see inequality Young’s inequality for convolutions, see inequality ... Finite Element Methods for Incompressible Flow Problems, Springer Series in Computational Mathematics 51, DOI 10.1007/978-3-319-45750-5_3 25 26 Finite Element Spaces for Linear Saddle Point Problems. .. for numerical simulations, which are divided into three groups: • examples for steady-state flow problems, • examples for laminar time-dependent flow problems, • examples for turbulent flow problems. .. considers finite element methods Finite element methods are very popular and they are understood quite well from the mathematical point of view First applications of finite element methods for the