Stabilization of finite element method for solving incompressible viscous flows

Czech Technical University in Prague Faculty of Mechanical Engineering Department of Technical Mathematics Diploma thesis Stabilization of finite element method for solving incompressible viscous flows Jakub Šístek Specialization Mathematical Modelling in Technology Advisor Doc. RNDr. Pavel Burda, CSc. Consultant RNDr. Jaroslav Novotný, PhD. Prague, 2004 1

Abstract The topic of the thesis is an advanced application of the finite element method (FEM to the solution of flows of viscous incompressible fluids. The aim is to extend the applicability of conventional methods to higher Reynolds numbers. The derivation and the verification of the algorithm for stabilizing the FEM are considered by the author as the main achievement of the work presented. By applying it, we are able to increase the critical Reynolds number of the flow, which can be succesfully solved by the FEM. Motivation and main goals are described in the introductory chapter, together with the recent state of the research in this field. The problem of flow of incompressible Newtonian fluid in a two-dimensional domain is introduced. Following part is devoted to the FEM solution and problems of discretization in general. Stabilized methods based on the FEM for the scalar advective-diffusive equation are described, followed by the extension to the flow problem. Next chapter is the principal part of the thesis. It contains a careful derivation of a FEM algorithm based on stabilization known as Galerkin Least-Squares (GLS method. The algorithm is applied to several problems, and results of the experiments are presented. Following chapter is devoted to the aspect of accuracy of the FEM solution. It presents work aimed on the application of a priori error estimates of FEM. Estimates are applied to the generation of the mesh, and cheap and precise computing of the solution even in domains with corner-like singularities is reached. Numerical results are included for two channels with sharp nonconvex inner corners. Conclusions are summarized, including achievements and topics for further research. 2

Acknowledgements I greatly thank Mr Pavel Burda, my advisor, proofreader, consultant, teacher, and friendly colleague in one personality. He came up with the idea of performing the following research and interested me in the finite element method. He kept his optimism when everything went wrong, was helpful anytime I needed it, and his friendship with me has been beyond the role of mere advisor. No less do I thank Mr Jaroslav Novotný. None of used program implementations would have been developed without his assistance. He spent huge amounts of his time working on my problems and helping me to realize my ideas. He also has had many useful comments which have helped to orient me in problems of computational fluid dynamics. I thank him also for providing his frontal solver for systems of linear equations. It has helped to save a lot of time. I also thank my parents who have supported me in studies. Jakub Šístek This research has been realized as a part of the project supported by the GAAV Grant No. IAA2120201/02. 3

Contents 1 Introduction 5 2 Navier-Stokes equations for incompressible viscous fluids 6 2.1 Unsteady two-dimensional flow......................... 6 2.2 Steady two-dimensional flow........................... 7 2.3 Unsteady axisymmetric flow........................... 7 2.4 Variational formulation of Navier-Stokes equations............... 8 3 Finite element methods for Navier-Stokes equations 10 3.1 Function spaces for velocity and pressure approximation........... 10 3.2 Hood-Taylor finite elements........................... 11 3.3 Discretization of steady Navier-Stokes equations by FEM........... 12 3.4 Discretization of unsteady Navier-Stokes equations.............. 14 3.5 Space semidiscretization of unsteady Navier-Stokes equations by FEM.... 14 3.6 Time discretization of unsteady Navier-Stokes equations by the Euler method 14 4 Stabilization techniques for finite element method 16 4.1 Advection-diffusion equation........................... 16 4.2 Navier-Stokes equations............................. 17 5 Algorithm for solving Navier-Stokes equations based on GLS 20 5.1 Stabilizing terms for steady Navier-Stokes equations.............. 20 5.2 Functionals for the Newton method and their differentials in steady case.. 21 5.3 Matrices for the finite element method in steady case............. 24 5.4 Stabilizing terms for unsteady Navier-Stokes equations............ 32 5.5 Functionals for the Newton method and their differentials in unsteady case. 33 5.6 Matrices for the finite element method in unsteady case............ 34 5.7 Stabilization parameters............................. 39 5.8 Numerical implementation............................ 42 6 Numerical experiments 45 6.1 Steady solution of lid driven cavity....................... 45 6.2 Unsteady solution of lid driven cavity...................... 49 6.3 Unsteady solution of flow past NACA 0012 airfoil............... 51 7 On the application of a priori error estimates for Navier-Stokes equations 57 7.1 Algorithm for generation of computational mesh................ 57 7.2 Geometry and design of the mesh........................ 58 7.3 Measuring of error................................ 60 7.4 Numerical results................................. 61 8 Conclusion 65 9 Appendix 66 4

1 Introduction Application of the finite element method (FEM in engeneering has made a rapid progress, and it is widely used in industry as well as in research centers today. Well-established commercial software based on the FEM helps with performing more complex simulations to make developments of products of desired properties cheaper. When application of the FEM in structure mechanics is now clear enough for solving common tasks, and only special problems remain to be resolved, fluid dynamics still includes amount of open and not well handled problems. One of them is reliable modelling of flows for high Reynolds numbers, which appear in engineering practice. The idea of stabilizing the FEM is not quite new in comparison to history of FEM itself. Many researchers are involved in this area and have already presented many techniques and results. Some of them have provided theroretical basis for the problem from the mathematical point of view, others presented often better results but usually with quite unclear background. Let us briefly review several publications related to the area. T.J.R. Hughes, L.P. Franca, and M. Balestra [20] presented the stable Petrov-Galerkin formulation of the Stokes problem in 1986. J. Douglas, Jr. and J. Wang [10] introduced another stabilized method for the Stokes problem in 1988. In the same year, T.J.R. Hughes, L.P. Franca, and G.M. Hulbert [21] presented SUPG and GLS stabilized finite element methods for the advective-diffusive equation. Their ideas were extended to the Navier-Stokes equations and completed by L.P. Franca and S.L. Frey [12], L.P. Franca and T.J.R. Hughes [14], L.P. Franca and A.L. Madureira [15], and L.P. Franca, S.L. Frey, and A.L. Madureira [13]. Work of T.E. Tezduyar (e.g. [27] and work of G. Lube and his coworkers (e.g. [16] can be listed as a recent research in the stabilization of the FEM for fluid dynamics. R. Glowinski investigates another approach to stabilization (e.g. [18]. It uses spliting of the Navier-Stokes equations into the Stokes problem and the advective-diffusive equation. Work of L.P. Franca and T.J.R. Hughes provides the theoretical basis for the presented research. The goal of it was to develop an algorithm based on the FEM stabilized by the technique introduced in [21] as Galerkin Least-Squares (GLS method. Let us mention the structure of the thesis. Flow of an incompressible viscous fluid described by the Navier-Stokes equations is introduced in Chapter 2. In Chapter 3, FEM formulation of the Navier-Stokes problem is described, together with difficulties accompanying numerical solution. An overview of stabilization techniques, especially GLS is presented in Chapter 4. This stabilization is applied to the advective-diffusive equation and to the Navier-Stokes equations. Chapter 5 is the principal part of the work. It contains a careful derivation of the algorithm based on the FEM. The algorithm is derived for the steady case and then extended to the unsteady case. Several numerical experiments are presented in Chapter 6. They are selected and sorted in order to show all positive as well as negative aspects of the stabilization, and of solving the Navier-Stokes equations in general. Whereas most of the thesis (Chapters 2-6 is concerned with the stability and therefore convergence of the FEM solution of fluid flow for high Reynolds numbers, Chapter 7 is devoted more to the aspect of accuracy of the FEM applied to flow in channels with sharp nonconvex inner corners. It deals with the application of a priori error estimates of the FEM to mesh generation. This approach offers quite cheap and precise computing of selected problems. Numerical results are demonstrated on two examples. Chapter 8 includes main achievements and topics for further research. 5

2 Navier-Stokes equations for incompressible viscous fluids Let be an open bounded domain in R 2 filled with a fluid, and let Γ be its boundary. The generic point of R 2 is denoted by x = (x 1, x 2 T considered in meters, and t denotes time variable considered in seconds. 2.1 Unsteady two-dimensional flow In this work, we deal with isothermal flow of Newtonian viscous fluids with constant density. Such flow is modelled by the following Navier-Stokes system of partial differential equations (nonconservative form where ρ ( u t + (u u µ u + p r = ρf in [0, T] (2.1 u = 0 in [0, T] (2.2 u = (u 1, u 2 T denotes the vector of flow velocity considered in m/s, which is a function of x and t p r denotes the pressure considered in Pa, which is a function of x and t ρ denotes the density of the fluid considered in kg/m 3 µ denotes the dynamic viscosity of the fluid considered in Pa s, which is supposed to be constant f denotes the density of volume forces per mass unit considered in N/kg, which could be a function of x and t, Let us divide both sides of the momentum equation (2.1 by ρ and leave the continuity equation (2.2 unchanged. Then we obtain where u t + (u u ν u + p = f in [0, T] (2.3 u = 0 in [0, T] (2.4 p = pr ρ denotes the pressure divided by the density considered in Pa m3 /kg ν = µ ρ denotes the kinematic viscosity of the fluid considered in m2 /s The system introduced is not sufficient to define a flow since it has an infinity of solutions. To restrict the number of solutions, we have to consider further conditions, such as the initial condition where u 0 = 0, and the boundary conditions where u = u 0 in, t = 0 (2.5 u = g on Γ g [0, T] (2.6 ν( un + pn = 0 on Γ h [0, T] (2.7 6

Γ g and Γ h are two subsets of Γ satisfying Γ = Γ g Γ h, µ R 1(Γ g Γ h = 0 n denotes an outer normal vector to the boundary Γ with unit lenght Introduced g is a given function of x and t satisfying in the case of Γ = Γ g for all t [0, T] g ndγ = 0 2.2 Steady two-dimensional flow Γ For the case of steady flow, the time derivative in (2.3 becomes zero. Then the Navier-Stokes equations are reduced to and boundary conditions to (u u ν u + p = f in (2.8 u = 0 in (2.9 u = g on Γ g (2.10 ν( un + pn = 0 on Γ h (2.11 Initial condition is not present, and u, p, f, and g are no more functions of t. 2.3 Unsteady axisymmetric flow Let us now consider the system of Navier-Stokes equations for incompressible viscous fluid in three dimensions, cf. [18]. After performing transformation of the cartesian system of coordinates {x 1, x 2, x 3 } into the cylindrical system of coordinates {r, ϕ, z} where x 1 = r cosϕ; x 2 = r sin ϕ; x 3 = z, and considering axialy symmetric flow, i.e. variables are independent of ϕ, we obtain Navier- Stokes equations in the form (cf. e.g. [5] ( u 2 t + v u r + u u z ν u r + 1 u 2 r r + 2 u + p = f z 2 z in [0, T] (2.12 z ( v 2 t + v v r + u v z ν v r + 1 v 2 r r v r + 2 v + p = f 2 z 2 r in [0, T] (2.13 r v r + v r + u = 0 in [0, T] (2.14 z where u denotes the axial component of velocity (direction of z-coordinate considered in m/s, which is a function of x and t v denotes the radial component of velocity (direction of r-coordinate considered in m/s, which is a function of x and t f = (f z, f r T denotes the density of volume forces per mass unit considered in N/m 3, which could be a function of x and t Equations (2.12-(2.14 govern the axisymmetric flow in a domain R 2, where the generic point of R 2 is now denoted by x = (z, r T for arbitrary ϕ. 7

2.4 Variational formulation of Navier-Stokes equations We need to introduce several function spaces for further derivations. Note, that all integrals are considered in the Lebesque sense. Let L 2 ( be the space of square integrable functions on, and let L 2 (/R be the space of functions in L 2 ( ignoring an additive constant. Let H 1 ( and H0 1 ( be the Sobolev spaces defined as H 1 ( = H 1 0( = { v v L 2 (, v } L 2 (, i = 1, 2 x { i } v v H 1 (,Tr v = 0 where Tr is the trace operator Tr : H 1 ( L 2 (Γ g, and derivatives are considered in the weak sense. The norm of function v in the space L 2 ( is considered as v 2 L 2 ( = v 2 d and the norm of function v in the Sobolev space H 1 ( is considered as ( 2 ( 2 v v 2 H 1 ( = v 2 + d x k Sometimes, the notation L2 ( is shortened to 0 and H 1 ( to 1. The inner product of two functions u and v in the space L 2 ( is considered as (u, v L2 ( = uv d Similarly, the notation (u, v L2 ( is shortened to (u, v 0. For more about Sobolev spaces and their properties, see e.g. [11]. Let us define vector function spaces V g and V by { } V g = v = (v 1, v 2 T v [H 1 (] 2 ;Tr v i = g i, i = 1, 2 V = {v = (v 1, v 2 T v [H 10(] } 2 Let us note, that the norm of vector function v in the space V g and V is then 2 ( 2 ( 2 v 2 [H 1 (] = v 2 vi 2 i + d x k i=1 k=1 k=1 and the norm of vector function v in the space [L 2 (] 2 is 2 v 2 [L 2 (] = 2 i=1 v 2 i d 8

Let us derive the weak formulation of the Navier-Stokes equations (2.3-(2.4 in the way of mixed methods, i.e. usage of different function spaces of test functions for the momentum equation and for the continuity equation (cf. [17]. To derive it, we suppose for a while, that the functions appearing in the system are sufficiently smooth. Then for any t [0, T], we have u t vd + (u u vd ν u vd + p vd = f vd (2.15 ψ ud = 0 (2.16 u u g V (2.17 for any v V and ψ L 2 ( where u g V g is a representation of the Dirichlet boundary condition g in (2.6. We assume g [L 2 (Γ g ] 2 and f [L 2 (] 2. Using Green s formula on the third and the fourth term of equation (2.15, we obtain u t vd + (u u vd ν ( uv ndγ + ν u : vd + Γ + pv ndγ p vd = f vd (2.18 Γ ψ ud = 0 (2.19 u u g V (2.20 The integrals over boundary in (2.18 vanish for considered boundary conditions. The operation u : v is defined in Appendix. Then the weak unsteady Navier-Stokes problem means seeking of u(t = (u 1 (t, u 2 (t T V g and p(t L 2 (/R satisfying for any t [0, T] u t vd + (u u vd + ν u : vd p vd = f vd (2.21 ψ ud = 0 (2.22 for v V and ψ L 2 (. Similarly, the weak steady Navier-Stokes problem reads: u u g V (2.23 Seek u = (u 1, u 2 T V g and p L 2 (/R satisfying (u u vd + ν u : vd p vd = f vd (2.24 ψ ud = 0 (2.25 for v V and ψ L 2 (. u u g V (2.26 9

3 Finite element methods for Navier-Stokes equations Let us divide the domain (supposed now polygonal into N elements of a triangulation T such that N = =1 µ R 2 ( T L = 0, L Let h mean the largest distance in element. 3.1 Function spaces for velocity and pressure approximation Difficult problem in solving the Navier-Stokes equations by the FEM consists in a proper choice of function spaces for velocity and pressure approximation. Appart from spectral finite element methods, polynomial approximation is commonly used in the FEM in general. To solve the Navier-Stokes equations, it is possible to choose different polynomial approximation for velocities and for pressure. Equal order approximation is easy to implement, but pressure exhibits instability. Approximation with different order is more suitable for practical computing, cf. [1]. The following properties of desired solution are linked with the variational formulation of the Navier-Stokes equations (2.21-(2.23 or (2.24-(2.26: each component of velocity is a square integrable function of x, and at least its first derivative by any coordinate (in the weak sense exists pressure is a square integrable function of x There is an effort for using higher order approximations induced by p-methods and hpmethods. But this is not straightforward for the Navier-Stokes equations. I. Babuška and F. Brezzi proved a condition (also called inf-sup condition limitting the choice of combinations of approximation CB >0,const. qh Q h vh V gh (q h, v h 0 C B q h 0 v h 1 (3.1 where Q h and V gh are the function spaces for approximation of pressure and velocity. This is an important stability result. It has been shown, that there are big difficulties with applying approximations, which do not satisfy the Babuška-Brezzi condition, cf. e.g. [1]. Chosen combination of final polynomial approximation is achieved by usage of corresponding finite elements. There are several finite elements (in 2D as well as in 3D which do satisfy the BB-condition. Following list is not complete. Finite elements satisfying the Babuška-Brezzi condition (cf. [1] P + 1 P 1 (mini element P 2 P 1 (Hood-Taylor, 1973 P + 2 P 1 P + 2 P 1 (Crouzeix-Raviart Q 2 Q 1 (Hood-Taylor Q 2 P 1 10

3.2 Hood-Taylor finite elements For their application in this work, Hood-Taylor finite elements on triangles and quadrilaterals are described more precisely. Values of velocity are aproximated in corner nodes and in midsides, and values of pressure in corner nodes (Figure 3.1. It corresponds to the following function spaces on element : triangle quadrilateral v i P 2 (, i = 1, 2, i.e. polynomial of the second order p P 1 ( i.e. linear polynomial v i Q 2 (, i = 1, 2, i.e. polynomial of the second order for each coordinate p Q 1 ( i.e. bilinear polynomial Ù ¹ Æ Ú Ü Ú Ý Ô ¹ Æ Ú Ü Ú Ý Ù ¹ Æ Ú Ü Ú Ý Ô ¹ Æ Ù Ú Ü Ú Ý Ô ¹ Æ Ú Ü Ú Ý ¹ Æ Ú Ü Ú Ý ¹ ¹ Æ Ú Ü Ú Ý ¹ Æ Ú Ü Ú Ý Ù Ú Ü Ú Ý Ô Ú Ü Ú Ý ¹ ¹ Æ Æ Ù ¹ Æ ½ ¾ Ú Ü Ú Ý Ô Ù Ú Ü Ú Ý Ô Ú Ü Ú Ý ¹ ¹ Æ Æ Ù ¹ Æ ½ ¾ Ú Ü Ú Ý Ô ¹ Figure 3.1: Hood-Taylor reference elements The approximation leads to the following shape functions written in local coordinate system {ξ, η}: triangle Space functions for approximation of velocity component N 1 = 1/2 (2 ξ η(1 ξ η N 2 = 1/2 ξ(ξ 1 N 3 = 1/2 η(η 1 N 4 = ξ(2 ξ η N 5 = ξη N 6 = η(2 ξ η 11

Space functions for approximation of pressure M 1 = 1/2 (2 ξ η M 2 M 3 = 1/2 ξ = 1/2 η quadrilateral Space functions for approximation of velocity component N 1 = 1/4 (1 ξ(1 η( ξ η 1 N 2 = 1/4 (1 + ξ(1 η(ξ η 1 N 3 = 1/4 (1 + ξ(1 + η(ξ + η 1 N 4 = 1/4 (1 ξ(1 + η( ξ + η 1 N 5 = 1/2 (1 ξ 2 (1 η N 6 = 1/2 (1 η 2 (1 + ξ N 7 = 1/2 (1 ξ 2 (1 + η N 8 = 1/2 (1 η 2 (1 ξ Space functions for approximation of pressure M 1 = 1/4 (1 ξ(1 η M 2 = 1/4 (1 + ξ(1 η M 3 = 1/4 (1 + ξ(1 + η M 4 = 1/4 (1 ξ(1 + η 3.3 Discretization of steady Navier-Stokes equations by FEM Let us consider the variational formulation of the steady Navier-Stokes equations (2.24- (2.26. For isoparametric finite elements, velocities and pressure on each element are given as v x = v y = p = N u i=1 N u i=1 N p i=1 v xi N i v yi N i p i M i where v x denotes the component of velocity in the direction of x-coordinate v y denotes the component of velocity in the direction of y-coordinate p denotes pressure v xi is the value of velocity v x in the node i 12

v yi is the value of velocity v y in the node i p i is the value of pressure in the node i N u is the number of nodes with value of velocity on element given (in the case of Hood-Taylor elements, N u = 8 for quadrilateral and N u = 6 for triangle N p is the number of nodes with value of pressure on element given (in the case of Hood-Taylor elements, N p = 4 for quadrilateral, N p = 3 for triangle Let us employ the notation { Pm (T R m ( =, if is a triangle Q m (, if is a quadrilateral and let C( denote the space of continuous functions on. Application of Hood-Taylor finite elements leads to the final approximation on the domain satisfying u h V gh and p h Q h where { V gh = v h = (v h1, v h2 T [C(] 2 ; v hi T R 2 (, = 1,..., N, i = 1, 2, v h = g } in nodes on Γ g { } Q h = ψ h C(; ψ h T R 1 (, = 1,...,N For further reasons, we introduce the space { V h = v h = (v h1, v h2 T [C(] 2 ; v hi T R 2 (, = 1,...,N, i = 1, 2, v h = 0 } in nodes on Γ g Since these function spaces satisfy V gh V g, V h V, and Q h L 2 (/R for prescribed arbitrary value of pressure (e.g. p h = 0 in one node, we can introduce approximate steady Navier-Stokes problem: Seek u h V gh and p h Q h satisfying (u h u h v h d + ν u h : v h d p h v h d = f v h d, v h V h (3.2 ψ h u h d = 0, ψ h Q h (3.3 where u gh V gh is the projection of u g onto the space V gh. u h u gh V h (3.4 Using the shape regular triangulation and refining the mesh such that h max 0 where h max = max h, the solution of the approximated problem converges to the solution of the continuous problem (for more cf. e.g. [1]. 13

3.4 Discretization of unsteady Navier-Stokes equations To solve the unsteady Navier-Stokes equations (2.21-(2.23, we need to discretize the system both in space and time. Two techniques are avaiable: Method of lines (MOL 1. step semidiscretization in space (e.g. by FEM 2. step discretization in time (e.g. by the Euler method Rothe s method 1. step semidiscretization in time 2. step discretization in space 3.5 Space semidiscretization of unsteady Navier-Stokes equations by FEM Let us perform space semidiscretization of the system (2.21-(2.23 by the FEM in the context of the MOL. Extending derivations for the steady case in Chapter 3.3, we introduce the problem: Seek u h (t V gh, t [0, T] and p h (t Q h, t [0, T] satisfying u h t v hd + (u h u h v h d + ν u h : v h d p h v h d = f v h d, v h V h (3.5 ψ h u h d = 0, ψ h Q h (3.6 u h u gh V h (3.7 3.6 Time discretization of unsteady Navier-Stokes equations by the Euler method In the unsteady case of the Navier-Stokes system of equations, the approximation of time is needed. As for discretization in space, several possibilities are avaiable. Let us restrict to those introducing discrete time layers, in which the solution is seeked. These methods can be divided into explicit, semi-imlicit, and implicit families. Employing MOL leads to solving system of ordinary differential equations (ODE after semidiscretization in space. We can choose from various numerical methods known from solving ODE systems, starting from the Euler methods to the Runge-utta methods of high order approximation. Very sophisticated way was introduced by R. Glowinski (e.g. in [18]. It is based on fractional steps methods, therefore represents the family of Rothe s methods. In this thesis, we consider partition of the time interval [0, T] into M time intervals with M +1 time layers. The time step between n-th time layer and (n+1-st time layer is assumed constant and is denoted by ϑ. 14

We employ the implicit Euler method (also known as the backward difference method for time discretization of the Navier-Stokes system (3.5-(3.7, i.e. time derivative is substituted as u h t un+1 h u n h. ϑ This leads to fully implicit method for seeking u h in (n+1-st time layer. The problem then reads: Seek u n+1 h V gh and p n+1 h Q h satisfying 1 u n+1 h v h d + (u n+1 h u n+1 h ϑ p n+1 h v h d 1 ϑ v h d + ν u n+1 h : v h d u n h v hd = f n+1 v h d, v h V h (3.8 ψ h u n+1 h d = 0, ψ h Q h (3.9 u n+1 h u n+1 gh V h (3.10 15

4 Stabilization techniques for finite element method Stability of numerical methods for solving partial differential equations is restricted, e.g. for the Navier-Stokes equations by the Reynolds number. In the effort for computing behind this line, a lot of researchers presented methods improving stability of numerical schemes. For the finite element method, let us mention work of T.J.R. Hughes, L.P. Franca, and their co-workers in [12],[13],[14],[15],[20], and [21], which gives the basis for this work. 4.1 Advection-diffusion equation Let us consider for a moment the problem of finding u = u(x, x satisfying where L σ(u = f in σ = au + κ u a denotes given flow velocity assumed solenoidal, i.e. a = 0 in κ > 0 denotes diffusivity combined with sufficient boundary conditions (see [4] for details. Variational formulation of this problem reads: Seek u V g such that where B(u, w = L(w, w V B(u, w = ( au + κ u, w + (a + n u, w Γ h L(w = (f, w + (h, w Γh = {w H 1 (, w = g on Γ g in the sense of traces} V g V = {w H 1 (, w = 0 on Γ g in the sense of traces} a + a n + a n nu = 2 Let us consider a partition of into finite elements. Let be the interior of the th element. Denote = (element interiors Let V gh V g, V h V be finite element spaces consisting of continuous piecewise polynomials of order k. 16

Classical Galerkin method means: Seek u h V gh such that B(u h, w h = L(w h, w h V h Hughes, Franca, and Hulbert presented two methods with improved stability properties for the advection-diffusion equation in [21]: Streamline Upwind Petrov-Galerkin (SUPG B SUPG (u h, w h = L SUPG (w h, B SUPG (u h, w h Galerkin Least-Squares (GLS L SUPG (w h B GLS (u h, w h = L GLS (w h, B GLS (u h, w h L GLS (w h w h V h B(u h, w h + τ(lu h,a w h L(w h + τ(f,a w h w h V h B(u h, w h + τ(lu h, Lw h L(w h + τ(f, Lw h where τ is positive parameter and is function of element Peclet number α. The authors assume the following: α = h a 2κ τ = O( h for α large a τ = O( h2 for α small κ Both SUPG and GLS methods have modifications for the Navier-Stokes equations. 4.2 Navier-Stokes equations L.P. Franca and T.J.R. Hughes [14] analyze modification of the GLS method to stabilize the steady linearized Navier-Stokes equations given by where a = 0. They define the norm on V gh Q h as ( ua + p ν u = f in (4.1 u = 0 in (4.2 u = 0 on Γ (4.3 {u, p} 2 ν u 2 0, + τ ( ua + p ν u 2 0, + δ u 2 0, (4.4 and they prove the following lemma: Lemma 4.1 (Stability The bilinear form of the linearized problem (4.1-(4.3 satisfies B GLS (u h, p h ;u h, p h = {u h, p h } 2. 17

For the Stokes problem, the following estimate is proved in [14] {u h, p h } 2 ν 2 u 2 0, + 1 τ p 2 0, + 2 δ u 2 0, (4.5 Notice, that the choice δ = 0 also gives stability. It is important in further derivations (Chapter 5. Using stability, the following convergence theorem is proved in [14]: Theorem (Convergence Assuming constant viscosity ν, the solution {u h, p h } obtained by the GLS method converges to the solution {u, p} of (4.1-(4.3 as follows: {u h, p h } {u, p} 2 C [ ( H(Re 1 sup a q h 2k+1 u 2 k+1, + sup a 1 q h2l+1 p 2 l+1, x T x + H(1 Re ( ] νh 2k u 2 k+1, + ν 1 h 2l+2 p 2 l+1, where H( is the Heaviside function given by { 0, x < y H(x y = 1, x > y and Re is local Reynolds number on element. L.P. Franca and A.L. Madureira [15] consider slightly different formulation of the steady problem than is considered in this work up to here, given by ( where ε(u ij = 1 u i 2 x j + u j x i. The stabilized problem reads: Seek u h V gh and p h Q h satisfying in ( uu 2ν ε(u + p = f in u = 0 in u = 0 on Γ B GLS (u h, p h ;v h, ψ h = L GLS (v h, ψ h, v h V h, ψ h Q h where B GLS (u h, p h ;v h, ψ h ((u h u h,v h 0 + (2νε(u h, ε(v h 0 (p h, v h 0 + + (ψ h, u h 0 + ( u h, δ v h 0 + + ( (uh u h + p h 2ν ε(u h, τ((u h v h + ψ h 2ν ε(v h L GLS (v h, ψ h (f,v h 0 + ( f, τ((uh v h + ψ h 2ν ε(v h 18

Franca and Madureira [15] suggest stabilization parameters τ and δ as δ = u(x p ξ(re (x (4.6 λ τ = ξ(re (x λ u(x p (4.7 where Re (x = u(x p 4 λ ν { Re (x, 0 Re ξ(re (x = (x < 1 1, Re (x 1 ε(v h 2 0,T λ = max 0 =v (R k ( /R N ε(v h 2 0,T ( N 1 u i (x p p, 1 p < u(x p = i=1 max u i(x, p = i=1,n Recommended way of computing λ is to find the largest eigenvalue of the problem ( ε(w h, ε(v h λ ( w h, v h = 0, v h (R k ( /R N for each element once the mesh is set up. 19

5 Algorithm for solving Navier-Stokes equations based on GLS In this chapter, the algorithm for solving Navier-Stokes equations is derived. We employ the finite element method equipped with the Galerkin Least-Squares (GLS stabilization. We assume application of the Newton method for solving resulting system of nonlinear equations. First, we derive the algorithm for the steady case of the Navier-Stokes equtions, and then we extend it to the unsteady case. Note, that vector operations are defined in Appendix. 5.1 Stabilizing terms for steady Navier-Stokes equations Let us remind the mixed FEM formulation of the steady Navier-Stokes equations (3.2-(3.4: Seek u h V gh and p h Q h satisfying (u h u h v h d + ν u h : v h d p h v h d = f v h d, v h V h (5.1 ψ h u h d = 0, ψ h Q h (5.2 u h u gh V h (5.3 We apply stabilization based on the GLS method. We combine ideas of Hughes and Franca ([21],[15] with two modifications: 1. Stabilization term with parameter δ is not considered. Since most problems are caused by the advective term, we find stabilization of it as most important. We consider as redundant to stabilize the continuity equation, which is represened by the term with δ. In spite of it, we performed several experiments with δ applied, but the results were disastrous. 2. Formulas by Franca and Madureira (Chapter 4.2 are derived for the considered formulation (5.1-(5.3 of the problem. This formulation makes the derivations much simpler, therefore less mistakes can be made and stay undetected. As long as we tried to employ the Franca s formulation, we were not able to obtain contributive results. Let us add zero to the left side of the momentum equation (5.1 as N [(u h u h ν u h + p h f h ] τ[(u h v h ν v h + ψ h ]d = =1 { N = τ(u h u h (u h v h d ν τ(u h u h v h d + =1 + τ(u h u h ψ h d ν τ u h (u h v h d + T + ν T 2 τ u h v h d ν τ u h ψ h d + T + τ p h (u h v h d ν τ p h v h d + τ p h ψ h d } τf h (u h v h d + ν τf h v h d τf h ψ h d 20

where τ denotes the positive stabilization parameter. In Chapter 5.7, we show how to determine it. 5.2 Functionals for the Newton method and their differentials in steady case Note: Since only finite element functions u h, v h, p h, and ψ h are considered in this chapter, index h is omitted in what follows. Let us introduce functionals F 1 (u, p and F 2 (u, p for Newton s minimization. F 1 (u, p is derived for the enriched momentum equation (5.1 and F 2 (u, p for the continuity equation (5.2. F 1 (u, p = (u u vd + ν u : vd p vd f vd + + { N τ(u u (u vd ν τ(u u vd + =1 + τ(u u ψd ν τ u (u vd + ν 2 T τ u vd ν τ u ψd + T τ p (u vd ν τ p vd + + τ p ψd τf (u vd + } + ν τf vd τf ψd F 2 (u, p = ψ ud 21

We need Frechet s differentials of functionals F 1 (u, p and F 2 (u, p for using the Newton method. We can find them by evaluating Gateaux s differentials since we assume that both exist. < DF 1 (u, p, [h, q] >= [ 1 = lim [(u + th ](u + th vd + ν (u + th : vd t 0 t (p + tq vd f vd + { N + τ[(u + th ](u + th [(u + th ]vd =1 ν τ[(u + th ](u + th vd + τ[(u + th ](u + th ψd T ν τ (u + th [(u + th ]vd + ν 2 τ (u + th vd T ν τ (u + th ψd + τ (p + tq [(u + th ]vd T ν τ (p + tq vd + τ (p + tq ψd } τf [(u + th ]vd + ν τf vd τf ψd (u u vd ν u : vd + p vd + f vd { N τ(u u (u vd ν τ(u u vd + =1 + τ(u u ψd ν τ u (u vd + T + ν T 2 τ u vd ν τ u ψd + T + τ p (u vd ν τ p vd + T + τ p ψd τf (u vd + }] + ν τf vd τf ψd [ ] 1 < DF 2 (u, p, [h, q] >= lim ψ (u + thd ψ ud t 0 t 22

After letting t 0, we get the following lemma. Lemma 5.1 Assume all functions sufficiently smooth. Then the Frechet s differentials of functionals F 1 (u, p and F 2 (u, p are < DF 1 (u, p, [h, q] >= = + + + + + (h u vd + (u h vd + ν h : vd q vd + { N τ(h u (u vd + τ(u h (u vd + =1 τ(u u (h vd ν τ(h u vd ν τ(u h vd + T τ(h u ψd + τ(u h ψd ν τ h (u vd T ν τ u (h vd + ν 2 τ h vd ν τ h ψd + T τ q (u vd + τ p (h vd ν τ q vd + } τ q ψd τf (h vd and < DF 2 (u, p, [h, q] >= ψ hd. Let us formally introduce the functional F(u, p = F 1 (u, p + F 2 (u, p (5.4 and its differential < DF(u, p, [h, q] >=< DF 1 (u, p, [h, q] > + < DF 2 (u, p, [h, q] >. (5.5 The algorithm consists of the following steps 1. solution of the equation system (the z th iteration of the Newton method for h, q: < DF(u, p, [h, q] >= F(u, p (5.6 2. correction of the solution u, p: u z+1 p z+1 = u z + h = p z + q 23

5.3 Matrices for the finite element method in steady case Let us derive matrices for the finite element method. We substitute h x = h y = q = N u i=1 N u i=1 N p i=1 h xi N i h yi N i q i M i and reduce all test functions to v = (N j, 0, v = (0, N j, ψ = M j where N j, j = 1,..., N u are the basis functions of space V h for each component M j, j = 1,...,N p are the basis functions of space Q h Then, we derive three equations for each node from the scalar equation (5.6 by input in turn the following combinations of test functions These equations read for the node j v = (N j, 0, ψ = 0; v = (0, N j, ψ = 0; v = (0, 0, ψ = M j. where by Lemma 5.1 we obtain following terms: < DF(u, p, [h, q] > j1 = F(u, p j1 (5.7 < DF(u, p, [h, q] > j2 = F(u, p j2 (5.8 < DF(u, p, [h, q] > j3 = F(u, p j3 (5.9 24

= N u i=1 N u < DF(u, p, [h, q] > j1 = ( u x h xi N i x + h u x y i N i N j d + N u ( Ni + ν h xi i=1 x x + N N i p d i=1 { N u ( ( u x + τ h xi N i i=1 x + h u x y i N i u x x + u y ( ( N i + τh xi u x x + u N i y u x x + u y d + ( ( u x + τ u x x + u u x y h xi N i x + h y i N i d ( ( u x ν τ h xi N i x + h u x 2 N j y i N i x + 2 N j d 2 2 ( ( N i ν τh xi u x x + u N i 2 N j y x + 2 N j d 2 2 ( ( 2 N i ν τh xi x + 2 N i u 2 2 x x + u y d ( ( 2 u x ν τ x + 2 u x h 2 2 xi N i x + h y i N i d + ( ( + ν T 2 2 N i τh xi x + 2 N i 2 N j 2 2 x + 2 N j d + 2 2 + τ p ( h xi N i x x + h y i N i d ( } τf x h xi N i x + h y i N i d + N p { ( M i + τq i u x i=1 x x + u y d ( } M i 2 N j ν τq i x x + 2 N j d 2 2 i=1 ( N i h xi u x x + u N i y N j d + q i M i x d + d + 25

= N u i=1 N u < DF(u, p, [h, q] > j2 = ( u y h xi N i x + h u y y i N i N j d + N u ( Ni + ν h yi i=1 x x + N N i p d i=1 { N u ( ( u y + τ h xi N i i=1 x + h u y y i N i u x x + u y ( ( N i + τh yi u x x + u N i y u x x + u y d + ( ( u y + τ u x x + u u y y h xi N i x + h y i N i d ( ( u y ν τ h xi N i x + h u y 2 N j y i N i x + 2 N j d 2 2 ( ( N i ν τh yi u x x + u N i 2 N j y x + 2 N j d 2 2 ( ( 2 N i ν τh yi x + 2 N i u 2 2 x x + u y d ( ( 2 u y ν τ x + 2 u y h 2 2 xi N i x + h y i N i d + ( ( + ν T 2 2 N i τh yi x + 2 N i 2 N j 2 2 x + 2 N j d + 2 2 + τ p ( h xi N i x + h y i N i d ( } τf y h xi N i x + h y i N i d + N p { ( M i + τq i u x i=1 x + u y d ( } M i 2 N j ν τq i x + 2 N j d 2 2 i=1 ( N i h yi u x x + u N i y N j d + q i M i d + d + 26

< DF(u, p, [h, q] > j3 = = + + + + N u i=1 N u i=1 τ ( h xi N i { τh xi ν τh xi N p i=1 x + h N i y i τ ( h xi N i u y ( u x N i M j d + ( u x h xi N i x + h u x y i N i Mj Mj x d + x + h u y y i N i d + x + u N i Mj y x d + τh yi ( 2 N i x + 2 N i Mj 2 2 x d ν τq i ( Mi x M j x + M i M j d τh yi x + u N i Mj y d } Mj d + ( u x N i ( 2 N i x + 2 N i 2 2 ( u x ux N j d + ν x x + u x ( u x F(u, p j1 = u x x + u y p N j x d f x N j d + + { ( ( u x τ u x x + u u x y u x x + u y ( ( u x ν τ u x x + u u x 2 N j y x + 2 N j 2 2 ( ( 2 u x ν τ x + 2 u x u 2 2 x x + u y ( ( + ν T 2 2 u x τ x + 2 u x 2 N j 2 2 x + 2 N j 2 2 + τ p ( u x x x + u y d ν τ p ( 2 N j x x + 2 N j d 2 2 ( τf x u x x + u y d + ( } 2 N j + ν τf x x + 2 N j d 2 2 d d d + d + d 27

( u y F(u, p j2 = u x x + u y p N j d f y N j d + + { ( ( u y τ u x x + u u y y u x x + u y ( ( u y ν τ u x x + u u y 2 N j y x + 2 N j 2 2 ( ( 2 u y ν τ x + 2 u y u 2 2 x x + u y ( ( + ν T 2 2 u y τ x + 2 u y 2 N j 2 2 x + 2 N j 2 2 + τ p ( u x x + u y d ν τ p ( 2 N j x + 2 N j d 2 2 ( τf y u x x + u y d + ( } 2 N j + ν τf y x + 2 N j d 2 2 ( u y uy N j d + ν x x + u y d d d + d + ( ux F(u, p j3 = x + u y M j d + + { ( u x τ u x x + u u x Mj y x d + ( u y + τ u x x + u u y Mj y d ( 2 u x ν τ x + 2 u x Mj 2 2 x d ( 2 u y ν τ x + 2 u y Mj 2 2 d + + τ p M j x x d + τ p M j d } M j τf x x d M j τf y d d 28

Now, we can peck out the elements of the ji-submatrix ji of the element stiffness matrix e (cf. Figure 5.1. ( u x ji11 (u, p = N i x N N i jd + u x x + u N i y N j d + ( Ni + ν x x + N i d + + { ( u x τn i u x x x + u y d + ( ( N i + τ u x x + u N i y u x x + u y d + ( u x + τ u x x + u u x y N i x d ( u x 2 N j ν τn i x x + 2 N j d 2 2 ( ( N i ν τ u x x + u N i 2 N j y x + 2 N j d 2 2 ( ( 2 N i ν τ x + 2 N i u 2 2 x x + u y d ( 2 u x ν τ x + 2 u x N 2 2 i x d + ( ( + ν T 2 2 N i τ x + 2 N i 2 N j 2 2 x + 2 N j d + 2 2 } + τ p x N i x d τf x N i x d u x ji12 (u, p = N i N jd + + { ( u x τn i u x x + u y d + ( u x + τ u x x + u u x y N i d ( u x 2 N j ν τn i x + 2 N j d 2 2 ( 2 u x ν τ x + 2 u x N 2 2 i d + + τ p } x N i d τf x N i d 29

ji13 (u, p = M i { + ν x d + τ M i x τ M i x ( u x x + u y ( 2 N j x + 2 N j 2 2 } d d u y ji21 (u, p = N i x N jd + + { ( u y τn i u x x x + u y d + ( u y + τ u x x + u u y y N i x d ( u y 2 N j ν τn i x x + 2 N j d 2 2 ( 2 u y ν τ x + 2 u y N 2 2 i x d + + τ p } N i x d τf y N i x d ( u y ji22 (u, p = N i N N i jd + u x x + u N i y N j d + ( Ni + ν x x + N i d + + { ( u y τn i u x x + u y d + ( ( N i + τ u x x + u N i y u x x + u y d + ( u y + τ u x x + u u y y N i d ( u y 2 N j ν τn i x + 2 N j d 2 2 ( ( N i ν τ u x x + u N i 2 N j y x + 2 N j d 2 ( ( 2 2 N i ν τ x + 2 N i u 2 2 x x + u y d ( 2 u y ν τ x + 2 u y N 2 2 i d + ( ( + ν T 2 2 N i τ x + 2 N i 2 N j 2 2 x + 2 N j d + 2 2 } + τ p N i d 30 τf y N i d

ji31 (u, p = ji32 (u, p = ji23 (u, p = M i { + + τ + + + ν N i x M jd + { u x τn i x ( N i u x x + u y N i M jd + { u x τn i ( N i u x x + u y τ d + τ M i τ M i ( u x x + u y ( 2 N j x + 2 N j 2 2 M j x d + u y τn i x N i Mj x d ν M j x d + u y τn i N i Mj d ν } d M j d + τ d ( 2 N i x + 2 N i 2 2 M j d + τ ( 2 N i x + 2 N i 2 2 } Mj x d } Mj d ji33 (u, p = τ ( Mi x M j x + M i M j d Matrix ji can be written as ji = ji11 ji12 ji13 ji21 ji22 ji23 ji31 ji32 ji33 Let us define vector of the right hand side as F(u, p j1 R j = F(u, p j2 F(u, p j3 and vector of solution as H i = h xi h yi q i This way, we obtain element stiffness matrix e and element vector of the right hand side R e (cf. Figure 5.1. After conventional assemblage procedure of stiffness matrix and the right hand side R, we solve the system of linear equations in each iteration of the Newton method. H = R 31..

Figure 5.1: Structure of element stiffnes matrix for quadrilateral element 5.4 Stabilizing terms for unsteady Navier-Stokes equations Remind the unsteady Navier-Stokes problem after space semidiscretization (3.5-(3.7: Seek u h (t V gh, t [0, T] and p h (t Q h, t [0, T] satisfying u h t v hd + (u h u h v h d + ν u h : v h d p h v h d = f v h d, v h V h (5.10 ψ h u h d = 0, ψ h Q h (5.11 u h u gh V h (5.12 Let us add stabilization terms in the same manner as for the steady case N =1 = [ ] [ ] uh t + (u vh h u h ν u h + p h f h τ t + (u h v h ν v h + ψ h d = { N =1 τ u h t (u h v h d ν τ u h t v hd + τ u h t ψ hd + + τ(u h u h (u h v h d ν T τ(u h u h v h d + + τ(u h u h ψ h d ν τ u h (u h v h d + T + ν T 2 τ u h v h d ν τ u h ψ h d + T + τ p h (u h v h d ν τ p h v h d + τ p h ψ h d } τf h (u h v h d + ν τf h v h d τf h ψ h d where we presumed v h t = 0. Note: As in the steady case, index h is omitted in the following text. 32

Let us approximate the time derivative in the (n + 1-st time layer as where ϑ is a constant time step. u t un+1 u n ϑ 5.5 Functionals for the Newton method and their differentials in unsteady case Functionals for the Newton method are defined as F 1 (u n+1, p n+1 = 1 u n+1 vd + (u n+1 u n+1 vd + ϑ + ν u n+1 : vd p n+1 vd f n+1 vd 1 u n vd + ϑ { N 1 + τu n+1 (u n+1 vd ν τu n+1 vd + ϑ =1 ϑ + 1 τu n+1 ψd + τ(u n+1 u n+1 (u n+1 vd ϑ T ν τ(u n+1 u n+1 vd + τ(u n+1 u n+1 ψd T ν τ u n+1 (u n+1 vd + ν 2 τ u n+1 vd T ν τ u n+1 ψd + τ p n+1 (u n+1 vd T ν τ p n+1 vd + τ p n+1 ψd T τf n+1 (u n+1 vd + ν τf n+1 vd T τf n+1 ψd 1 τu n (u n+1 vd + ϑ + ν τu n vd 1 } τu n ψd ϑ ϑ F 2 (u n+1, p n+1 = ψ u n+1 d Note: Another simplification of notation is employed in the following derivations. We omit index n + 1, and then u and p denote variables in the (n + 1-st time layer. Index of time layer is preserved at variables from other time layers, e.g. u n. 33

Lemma 5.2 Assume all functions sufficiently smooth. Then the Frechet s differentials of functionals F 1 (u, p and F 2 (u, p are < DF 1 (u, p, [h, q] >= = 1 h vd + (h u vd + (u h vd + ν h : vd q vd + ϑ { N 1 + τh (u vd + 1 τu (h vd ν τh vd + ϑ =1 ϑ ϑ + 1 τh ψd + τ(h u (u vd + τ(u h (u vd + ϑ T + τ(u u (h vd ν τ(h u vd ν τ(u h vd + T + τ(h u ψd + τ(u h ψd ν τ h (u vd T ν τ u (h vd + ν 2 τ h vd ν τ h ψd + T + τ q (u vd + τ p (h vd ν τ q vd + + τ q ψd τf (h vd 1 } τu n (h vd ϑ and < DF 2 (u, p, [h, q] >= ψ hd. As for the steady case, we formally introduce the functional and its differential F(u, p = F 1 (u, p + F 2 (u, p (5.13 < DF(u, p, [h, q] >=< DF 1 (u, p, [h, q] > + < DF 2 (u, p, [h, q] >. (5.14 5.6 Matrices for the finite element method in unsteady case Let us derive matrices for the finite element method. We substitute h x, h y, and q as in Chapter 5.3 and use the basis functions as test functions. 34

We obtain elements of the ji-submatrix ji of the element stiffness matrix e : ji11 (u, p = 1 ϑ + ν N i N j d + ( Ni x x + N i u x N i x N jd + d + + { ( 1 τn i u x ϑ x + u y ν ( 2 N j τn i ϑ x + 2 N j d + 2 2 ( u x + τn i u x x x + u y d + ( ( N i + τ u x x + u N i y u x x + u y ( u x + τ u x x + u u x y N i x d ( u x 2 N j ν τn i x x + 2 N j d 2 2 ( ( N i ν τ u x x + u N i 2 N j y ( ( 2 N i ν τ x + 2 N i u 2 2 x x + u y ( 2 u x ν x + 2 u x N 2 2 i x d + τ + ν 2 τ + τ p x N i ( 2 N i x + 2 N i 2 2 x d ( N i u x x + u N i y N j d + d + 1 τu x N i ϑ x d x + 2 N j 2 2 d + ( 2 N j x + 2 N j 2 2 τf x N i x d 1 ϑ u x ji12 (u, p = N i N jd + + { 1 τu x N i ϑ d + u x τn i ( u x + τ u x x + u u x y N i d ( u x 2 N j ν τn i x + 2 N j d 2 2 ( 2 u x ν τ x + 2 u x N 2 2 i d + + τ p x N i d τf x N i d 1 ϑ d d d + τu n xn i x d ( u x x + u y τu n x N i } d + d } 35

ji13 (u, p = M i ν τ M i x x d + { ( 2 N j x + 2 N j 2 2 τ M i x } d ( u x x + u y d ji21 (u, p = + + + u y N i x N jd + { 1 ϑ ( u y τn i u x x x + u y ( u x u y τu y N i x d + d + N i τ x + u u y y x d ( u y 2 N j ν τn i x x + 2 N j d 2 2 ( 2 u y ν τ x + 2 u y N 2 2 i x d + τ p N i x d τf y N i x d 1 ϑ τu n y N i x d } ji22 (u, p = 1 ( u y N i N j d + N i ϑ N N i jd + u x x + u N i y N j d + ( Ni + ν x x + N i d + + { ( 1 τn i u x ϑ x + u y d + 1 τu y N i ϑ d ν ( 2 ( N j τn i ϑ x + 2 N j u y d + τn 2 2 i u x x + u y ( ( N i + τ u x x + u N i y u x x + u y d + ( u y + τ u x x + u u y y N i d ν u y τn i ( ( N i ν τ u x x + u N i 2 N j y x + 2 N j d 2 ( ( 2 2 N i ν τ x + 2 N i u 2 2 x x + u y d ( 2 u y ν τ x + 2 u y N 2 2 i d + ( ( + ν T 2 2 N i τ x + 2 N i 2 N j 2 2 x + 2 N j d + 2 2 } + τ p N i d τf y N i d 1 ϑ 36 d + ( 2 N j x + 2 N j d 2 2 τu n yn i d

ji23 (u, p = M i ji31 (u, p = ji32 (u, p = + + ν τ M i N i x M jd + { 1 ϑ τ + + d + { ( 2 N j x + 2 N j 2 2 τn i M j x d + ( u x N i x + u y N i M jd + { 1 ϑ τ τn i M j d + ( u x N i x + u y τ M i } d N i Mj x d ν ( u x x + u y τ d u x M j τn i x x d + u y τn i x ( 2 N i x + 2 N i 2 2 u x M j τn i x d + u y τn i ( 2 N i x + 2 N i 2 2 N i Mj d ν τ M j d + } Mj x d M j d + } Mj d ji33 (u, p = τ ( Mi x M j x + M i M j d Let us remind matrix ji ji = ji11 ji12 ji13 ji21 ji22 ji23 ji31 ji32 ji33, vector of the right hand side R j = F(u, p j1 F(u, p j2 F(u, p j3, and vector of solution H i = h xi h yi q i. 37

Elements of the vector of the right hand side are F(u, p j1 = 1 ( ( u x u x N j d + u x ϑ x + u u x ux y N j d + ν x x + u x p N j x d f x N j d 1 u n ϑ xn j d + + { ( 1 τu x u x ϑ x + u y d ν τu x ϑ ( ( u x + τ u x x + u u x y u x x + u y d ( ( u x ν τ u x x + u u x 2 N j y x + 2 N j d 2 2 ( ( 2 u x ν τ x + 2 u x u 2 2 x x + u y d + ( ( + ν T 2 2 u x τ x + 2 u x 2 N j 2 2 x + 2 N j d + 2 2 + τ p ( u x x x + u y d ν τ p ( 2 N j x x + 2 N j d 2 2 ( ( τf x u x x + u 2 N j y d + ν τf x x + 2 N j d 2 2 1 ( τu n x u x ϑ x + u y d + ν ( } τu n 2 N j x ϑ x + 2 N j d 2 2 F(u, p j2 = 1 ( ( u y u y N j d + u x ϑ x + u u y uy y N j d + ν x p N j d f y N j d 1 u n ϑ yn j d + + { ( 1 τu y u x ϑ x + u y d ν τu y ϑ ( ( u y + τ u x x + u u y y u x x + u y d ( ( u y ν τ u x x + u u y 2 N j y x + 2 N j d 2 2 ( ( 2 u y ν τ x + 2 u y u 2 2 x x + u y d + ( ( + ν T 2 2 u y τ x + 2 u y 2 N j 2 2 x + 2 N j d + 2 2 + τ p ( u x x + u y d ν τ p ( 2 N j x + 2 N j 2 2 ( ( τf y u x x + u 2 N j y d + ν τf y x + 2 N j 2 2 1 ( τu n y u x ϑ x + u y d + ν ( τu n 2 N j y ϑ x + 2 N j 2 2 38 d ( 2 N j x + 2 N j d + 2 2 x + u y d ( 2 N j x + 2 N j d + 2 2 d d d }

F(u, p j3 = + + ( ux x + u y ( u x u x τ ν τ T τ p x x + u y ( 2 u x x + 2 u x 2 2 M j x d + τf x M j x d M j d + u x Mj { 1 ϑ τ x d + Mj x d ν τ p M j d τf y M j d 1 ϑ τ τ ( u x M j x + u y ( u x u y τ x + u y ( 2 u y x + 2 u y 2 2 M j d + u y Mj ( u n M j x x + un y d Mj d + } M j d As in Chapter 5.3, after we obtain element stiffness matrix e and element vector of the right hand side R e (cf. Figure 5.1 and perform the assemblage procedure of stiffness matrix and the right hand side R, we solve the system of linear equations in each iteration of the Newton method. H = R Once we obtain the solution in a particular time layer, we solve the problem in next time layer and use the previous solution as the initial value for the Newton method. This is repeated, until we reach the desired time. 5.7 Stabilization parameters It has been already mentioned, that we do not employ stabilization parameter δ. The way to obtain τ is not straightforward. Following the ideas of Franca and Madureira in [15], we compute τ as (cf. (4.7 in Chapter 4.2 where τ = ξ(re (x λ u(x 2 (5.15 Re (x = u(x 2 4 λ ν { Re (x, 0 Re ξ(re (x = (x < 1 1, Re (x 1 v 2 0,T λ = max 0 =v (R 2 ( /R 2 v 2 0, ( 2 1 2 u(x 2 = u i (x 2 i=1 Parameter λ is computed for each element as the largest eigenvalue of the problem ( w, v = λ ( w, v, v (R 2 ( /R 2 (5.16 This is done once, before entering the main computational loop of the Newton method, since λ is not a function of velocity and depends only on the computational mesh and space functions on element. 39

Let us focus on computing λ more precisely. Problem (5.16 can be written as w vd = λ w : vd, v (R 2 ( /R 2 (5.17 In the finite element dialect, similarly to Chapters 5.3 and 5.6, we substitute and input the vector basis functions w x = w y = N u i=1 N u i=1 w xi N i w yi N i v = (N j, 0; v = (0, N j as test functions, to get two equations from the scalar one (5.17. It leads to N u i=1 N u i=1 w xi w yi ( ( 2 N i x + 2 N i 2 N j 2 2 x + 2 N j N u d = λ 2 2 i=1 ( ( 2 N i x + 2 N i 2 N j 2 2 x + 2 N j N u d = λ 2 2 i=1 w xi w yi ( Ni x ( Ni x x + N i x + N i Let us create element matrices A and B for the purpose of computation of the largest eigenvalue of this problem. ( ( T 2 N i A ji = + 2 N i 2 N j + 2 N j d 0 x 2 2 x 2 2 ( ( 0 T 2 N i + 2 N i 2 N j + 2 N j d x 2 2 x 2 ( 2 N i T B ji = + N i d 0 x x ( N 0 i T + N i d x x Now, we need to find the largest eigenvalue of the generalized matrix eigenvalue problem d Aw = λ Bw (5.18 for each element. Here, λ is the desired eigenvalue and w is the corresponding eigenvector, which is not used in stabilization. Recommended method for solving this problem in [15] is the power method. But several difficulties are hidden behind it 1. The power method is designed for finding of the largest eigenvalue and the corresponding eigenvector of the problem Aw = λ w and not for the generalized problem. We need to transform problem (5.18 to the ordinary problem of eigenvalues. Possible way without necessity of inverting full matrix is sketched. We decompose matrix B by Choleski s method, i.e. find L so that B = LL T 40 d

and L is lower triangular matrix. Its inversion is simpler, and when we have it, we get L 1 Aw = λ L T w. Let us denote z = L T w or w = L T z. After substitution, we have L 1 AL T z = λ z. If we denote G = L 1 AL T, we can solve the ordinary problem of eigenvalues Gz = λ z. It is clear, that applied transformations do not take effect on eigenvalues of the generalized problem. 2. During realizing Choleski s decomposition (as for computing an inverse matrix, we need B to be nonsingular. But obtained matrix, which is similar to element stiffness matrix without application of boundary conditions, is singular. Recommended way to regularize it is to fix corresponding number of degrees of freedom. But since this is done by putting units on diagonal and zeros on relevant columns and rows of matrix B, this way could affect the largest eigenvalue, if it is less then one. We experienced, that more suitable way to regularize the matrix is to cut off two rows and columns from both matrices A and B. As far as we have tested this way, it has taken no effect on the largest eigenvalue for different omited rows and columns. The only restriction is to omit one for each component of velocity. Let us investigate the dependence of τ on local Reynolds number Re (x given by (5.15. We can observe that Re (x is a linear function of u(x 2 for constant viscosity on element, i.e. where C 1 = 1 4 λ ν. Substituting (5.19 in (5.15 we get τ(re (x,x = where C 2 = 1 4λ ν and C 3 = 1 λ, cf. Figure 5.2. Re (x = C 1 u(x 2 (5.19 { C 2, 0 Re (x < 1 C 3 u(x 2 = C 2, Re Re (x (x 1 41

Figure 5.2: Plot of τ(re An unpleasant effect of stabilization can be discovered. In Chapter 5.3 as well as in Chapter 5.6, we violated the continuity equation through the non-zero term of element matrix ji33 (u, p = τ ( Mi x M j x + M i M j d. This term introduces dependence on pressure into the continuity equation and affects the presumed incompressibility. Since derivatives of shape functions in ji33 are independent of solution, and since τ is never zero (cf. Figure 5.2, ji33 does not vanish for converged solution. But Figure 5.2 gives a hope: we can observe, that τ is decreasing for higher local Reynolds number, therefore described perturbation of the continuity equation is also decreasing for higher Re. 5.8 Numerical implementation The algorithm was implemented using Fortran programming language. Block scheme of the program follows. Main part of programmer s work consisted in writing routines for computing eigenvalues on elements and computing finite element matrices. For the solution of the system of linear equations, resulting from the Newton method, frontal solver is used. This solver was provided by the consultant of my thesis. Program was tested and benchmarks were run on Compaq AlphaServerES47 in the Institute of Thermomechanics. 42