Finite Element Methods and Navier-Stokes Equations

Linear finite element.- 4.3.3 Linear finite element (with reduced continuity).- 4.4 Coordinate and element transformation.- 4.5 Quadrilateral finite element.- 4.5.1 Bilinear finite element.- 4.5.2 Biquadratic finite element.- 4.6 Hexahedral finite elements.- 4.6.1 Trilinear finite element.- 4.6.2 Triquadratic finite element.- 5 Practical aspects of the finite element method.- 5.1 Finite element assembly algorithm.- 5.2 1D Poisson equation; quadratic finite elements.- 5.3 2D Poisson equation; linear and quadratic triangular elements.- 5.3.1 Linear finite element.- 5.3.2 Quadratic finite element.- 5.4 Numerical integration formulas.- 5.4.1 Numerical integration on intervals, triangles and tetrahedra.- 5.4.2 Numerical integration on quadrilaterals and hexahedra.- 5.5 Accuracy aspects of the FEM.- 5.6 Solution methods for systems of (non-)linear equations.- 5.6.1 Direct methods to solve systems of linear equations.- 5.6.2 Iterative methods for the solution of systems of linear equations.- 5.6.3 Linearization techniques for systems of nonlinear equations.- 5.6.3.1 Picard iteration.- 5.6.3.2 Newton's method.- 5.6.3.3 The quasi-Newton method.- II Application of the Finite Element Method to the Navier-Stokes Equations.- 6 Alternative formulations of Navier-Stokes equations.- 6.1 The basic equations of fluid dynamics.- 6.2 Alternative formulations.- 6.3 Initial and boundary conditions.- 6.3.1 Introduction.- 6.3.2 Velocity-pressure formulation.- 6.3.3 Stream function-vorticity formulation.- 6.3.4 Some practical remarks concerning the boundary conditions.- 6.4 Evaluation of the various formulations.- 7 The integrated method.- 7.1 General approach.- 7.2 Practical elaboration.- 7.2.1 Complicated boundary conditions.- 7.2.2 Necessary conditions for the elements.- 7.2.3 Examples of admissible elements.- 7.2.3.1 Introduction.- 7.2.3.2 Triangular elements (Taylor-Hood (1973)).- 7.2.3.3 Triangular elements (Crouzeix-Raviart (1973)).- 7.2.3.4 Quadrilateral elements.- 7.2.3.5 3D elements.- 7.2.4 The structure of the equations.- 7.3 The Navier-Stokes equations.- 7.3.1 Introduction.- 7.3.2 The Picard iteration.- 7.3.3 Newton and quasi-Newton methods.- 7.3.4 The structure of the system of equations.- 7.4 Evaluation of the integrated method.- 8 The penalty function method.- 8.1 General approach.- 8.2 Alternative formulations of the penalty function method.- 8.2.1 Minimization formulation.- 8.2.2 The continuous penalty function method.- 8.2.3 Iterative penalty function method.- 8.3 Practical consequences.- 8.3.1 Element conditions.- 8.3.2 The modified P2+-P1 Crouzeix-Raviart element.- 8.3.2.1 Introduction.- 8.3.2.2 Elimination of the velocities in the centroid.- 8.3.2.3 Elimination of the pressure derivatives.- 8.3.2.4 Construction of matrices.- 8.3.2.5 Concluding remarks.- 8.3.3 The structure of the equations.- 8.4 Evaluation of the penalty function method.- 9 Divergence-free elements.- 9.1 General approach.- 9.2 The construction of divergence-free basis functions for 2D elements.- 9.2.1 Introduction.- 9.2.2 The non-conforming Crouzeix-Raviart element.- 9.2.3 The modified P2+-P1 Crouzeix-Raviart element.- 9.2.4 The Q2-P1 9-node quadrilateral.- 9.2.5 Boundary conditions.- 9.2.6 The structure of the system of equations.- 9.2.7 The implementation of boundary conditions of the type ? equals unknown constant.- 9.3 The construction of divergence-free basis functions for 3D elements.- 9.3.1 Introduction.- 9.3.2 A non-conforming Crouzeix-Raviart element in IR3.- 9.3.3 The construction of a divergence-free basis.- 9.4 Evaluation of the solenoidal method.- 10 The instationary Navier-Stokes equations.- 10.1 General approach.- 10.2 The numerical solution of systems of ordinary differential equations.- 10.2.1 Introduction.- 10.2.2 Stability of the ?-method.- 10.3 The solution of the systems of ordinary differential equations resulting from the Galerkin method applied to the Navier-Stokes equations.- 10.3.1 The penalty function method and artificial compressibility methods.- 10.3.2 Divergence-free elements.- 10.3.3 The pressure-correction method.- 10.4 Streamline upwinding.- III Theoretical Aspects of the Finite Element Method.- 11 Second order elliptic PDEs.- 11.1 Dirichlet problem for the Laplace operator.- 11.2 Neumann problem for the Laplace operator.- 11.3 General variational formulation; existence, uniqueness.- 11.4 Examples.- 11.5 (Navier-) Stokes equations.- 11.6 Regularity of the solution of the variational problem.- 12 Finite element approximations of variational problems.- 12.1 Internal approximation of Hilbert spaces.- 12.2 Discretized variational problem.- 12.3 Finite element approximations of Sobolev spaces.- 12.3.1 Definition of finite element.- 12.3.2 Linear finite element approximation of L2(?).- 12.3.3 Linear finite element approximation of H01(?).- 12.3.4 Quadratic finite element approximation of H01(?).- 12.3.5 Linear finite element approximation of H1(?).- 12.3.6 Finite element approximation of V.- 12.4 Interpretation of the discretized variational problem.- 12.4.1 Dirichlet-Neumann problem for the Laplace operator.- 12.4.2 Stokes problem.- 13 Error analysis of the FEM.- 13.1 H1 and L2 error estimates.- 13.2 Numerical integration.- 14 Mixed Finite Element Methods.- IV Current Research Topics.- 15 Capillary free boundaries governed by the Navier-Stokes equations.- 15.1 Mathematical model.- 15.2 Normal stress iterative method.- 15.3 Newton's method for free boundaries.- 16 Non-Isothermal flows.- 16.1 Mathematical model.- 16.2 Numerical treatment.- 17 Turbulence.- 17.1 Mathematical models.- 17.2 Numerical treatment.- 18 Non-Newtonian fluids.- 18.1 Mathematical models.- 18.2 Numerical treatment.