Finite Element Methods and Navier-Stokes Equations

In present-day technology the engineer is faced with complicated problems which can be solved only by advanced numerical methods using digital computers. He should have mathematical models at his disposal in order to simulate the behavior of physical systems. In many cases, including problems in physics, mechanics, chemistry, biology, civil engineering, electrodynamics, solid mechanics, space engineering, petroleum engineering, weather-forecasting, mass transport, multiphase flow and, last but not least, hydrodynamics, such a physical system can be described mathematically by one or more partial differential equations. Some of these problems require extreme accuracy, while for other problems only the qualitative behavior of the solution need to be studied. The finite element method (FEM) is one of the most commonly used methods for solving partial differential equations (PDEs). It makes use of the computer and is very general in the sense that it can be applied to both steady-state and transient, linear and nonlinear problems in geometries of arbitrary space dimension. The FEM is in fact a method which transforms a PDE into a system of linear (algebraic) equations. The following aspects may be identified in the study of a physical phenomenon: (i) engineering-mathematical sciences to formulate the problem correctly in terms of PDEs (ii) numerical methods to construct and to solve the system of algebraic equations; applied numerical functional analysis to give error estimates and convergence proofs (iii) informatics and programming to perform the calculations efficiently on the computer.

I Introduction to the Finite Element Method.- 1 Examples of partial differential equations.- 1.1 Classification of PDEs.- 1.2 Laplace and Poisson equation.- 1.3 Steady state convection-diffusion equation.- 1.4 Time dependent convection-diffusion equation.- 1.5 Reynolds equation.- 1.6 Equations of fluid dynamics; Navier-Stokes equations.- 1.7 Equations of linear elasticity.- 1.8 Comments.- 2 Finite difference schemes for Poisson equation and convection-diffusion equation.- 2.1 1D Poisson equation with Dirichlet boundary conditions.- 2.2 1D Poisson equation with other type of boundary conditions.- 2.2.1 Mixed homogeneous Dirichlet-Neumann boundary conditions.- 2.2.2 Non-homogeneous Dirichlet boundary conditions.- 2.2.3 Non-homogeneous Neumann boundary conditions.- 2.2.4 Non-homogeneous Robbins boundary conditions.- 2.3 2D Poisson equation with Dirichlet boundary conditions.- 2.4 Boundary conditions, geometry and variable coefficients in 2D.- 2.4.1 Boundary conditions.- 2.4.2 Geometry.- 2.4.3 Variable coefficients.- 2.5 Comments.- 2.6 Convection-diffusion equation.- 3 The finite element method.- 3.1 Extremal problem; Euler-Lagrange equation.- 3.2 Extremal formulation of the Poisson equation.- 3.2.1 1D case.- 3.2.2 2D case.- 3.2.3 Various types of boundary conditions.- 3.3 Comments.- 3.4 The Ritz method.- 3.5 The FEM.- 3.5.1 Definition.- 3.5.2 1D Poisson equation.- 3.6 The Galerkin method.- 3.6.1 General procedure.- 3.6.2 1D Poisson equation; homogeneous boundary conditions.- 3.6.3 1D Poisson equation; non-homogeneous boundary conditions.- 3.6.4 2D problem.- 3.7 Comments.- 4 Construction of finite elements.- 4.1 Linear, quadratic and cubic basis functions in 1D.- 4.2 Triangular basis functions in 2D.- 4.2.1 Barycentric coordinates.- 4.2.2 Linear finite element.- 4.2.3 Linear finite element (with reduced continuity).- 4.2.4 Quadratic finite element.- 4.2.5 Extended quadratic finite element.- 4.3 Triangular basis functions in 3D.- 4.3.1 Barycentric coordinates.- 4.3.2 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.