CONFORMING STOKES ELEMENTS YIELDING DIVERGENCE FREE APPROXIMATIONS ON QUADRILATERAL MESHES

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1 CONFORMING STOKES ELEMENTS YIELDING DIVERGENCE FREE APPROXIMATIONS ON UADRILATERAL MESHES by Duygu Sap B. S. in Engineering Mathematics, Istanbul Technical University, 2008 M. S. in Electronics & Communication Engineering, Istanbul Technical University, 2011 M. S. in Mathematics, University of Pittsburgh, 2014 Submitted to the Graduate Faculty of the Kenneth P. Dietrich School of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2017

2 UNIVERSITY OF PITTSBURGH KENNETH P. DIETRICH SCHOOL OF ARTS AND SCIENCES This dissertation was presented by Duygu Sap It was defended on July 6, 2017 and approved by Prof. Michael J. Neilan, Dept. of Mathematics, University of Pittsburgh Prof. William J. Layton, Dept. of Mathematics, University of Pittsburgh Prof. Catalin Trenchea, Dept. of Mathematics, University of Pittsburgh Prof. Noel J. Walkington, Dept. of Mathematical Sciences, Carnegie Mellon University Dissertation Director: Prof. Michael J. Neilan, Dept. of Mathematics, University of Pittsburgh ii

3 CONFORMING STOKES ELEMENTS YIELDING DIVERGENCE FREE APPROXIMATIONS ON UADRILATERAL MESHES Duygu Sap, PhD University of Pittsburgh, 2017 In this dissertation, we propose conforming finite element methods that yield divergence free velocity approximations for the steady Stokes problem on cubical and quadrilateral meshes. In the first part, we construct the finite element spaces for the two-dimensional problem on rectangular grids. Then in the second part, we extend these spaces to n-dimensional spaces. We use discrete differential forms and smooth de Rham complexes to verify the stability and the conformity of the proposed methods, and the solenoidality of the velocity approximations. In the third part, we shift our focus from a dimensionwise extension to a meshwise improvement by introducing macro elements on general shape regular quadrilateral meshes. By utilizing a smooth de Rham complex, we prove that the macro finite element method yields divergence free velocity solutions, and with the construction of a Fortin operator, we validate the stability of the method. To improve the pressure approximation properties, we compute a post processed pressure solution locally. In addition, we describe the implementation process of the (velocity) macro finite element. We show that the methods developed in this dissertation yield optimal convergence rates and present numerical experiments which are supportive of the theoretical results. Moreover, we provide experimental results of our method for the Navier-Stokes equations and show that the convergence rates are preserved. iii

4 TABLE OF CONTENTS PREFACE ix 1.0 INTRODUCTION Finite Element Method Applied to the Stokes Problem Previous Works and the Advantages of Divergence Free Approximations Methodology Outline FEM FOR THE STOKES PROBLEM ON RECTANGULAR GRIDS The Local Finite Element Spaces The C 1 Bogner-Fox-Schmidt Finite Element Space The Velocity Space The Pressure Space A Local Characterization of the Divergence Operator The Global Finite Element Spaces The Global Finite Element Spaces with Imposed Boundary Conditions Nitsche s Method and the Convergence Analysis Numerical Experiments Experiment Experiment Discussion FEM FOR THE STOKES PROBLEM ON CUBICAL GRIDS The Local Finite Element Spaces The Velocity Space The Pressure Space A Local Characterization of the Divergence Operator iv

5 3.2 The Global Finite Element Spaces The Global Finite Element Spaces with Imposed Boundary Conditions Inf sup Stability The Reduced Elements with Continuous Pressure Approximations Convergence Analysis FEM FOR THE STOKES PROBLEM ON UADRILATERAL MESHES The uadrilateral Mesh The Local Finite Element Spaces The Global Finite Elements with Imposed Boundary Conditions Inf-sup Stability Convergence Analysis Convergence Analysis for the Post-Processed Pressure Solution Implementation Construction of a Canonical Basis Derivation of the Reference Basis Functions Numerical Experiments Experiment 1: Stokes Problem Discussion on the Experimental Results for the Stokes Problem Experiment 2: NSE with Homogeneous Boundary Conditions Experiment 3: NSE with Nonhomogeneous Boundary Conditions Discussion on the Experimental Results for the NSE CONCLUSION BIBLIOGRAPHY v

6 LIST OF TABLES 2.1 Experiment 1, Convergence results on rectangular meshes Experiment 2, Convergence results on rectangular meshes Mesh statistics for h = Node statistics for the tensor product mesh with h = Node statistics for the quadrilateral mesh with h = Experiment 1, Convergence results for the Div Free Macro Stokes elements on tensor product meshes Experiment 1, Convergence results for the Div Free Macro Stokes elements on tensor product meshes Experiment 1, Convergence results for the Div Free Macro Stokes elements on quadrilateral meshes Experiment 1, Convergence results for the Div Free Macro Stokes elements on quadrilateral meshes Experiment 1, Convergence results for the Serendipity elements on tensor product meshes Experiment 1, Convergence results for the Serendipity elements on quadrilateral meshes Experiment 2, Convergence results for the Div Free Macro elements on tensor product meshes Experiment 2, Convergence results for the Div Free Macro elements on quadrilateral meshes Experiment 2, Convergence results for the Serendipity elements on tensor product meshes vi

7 4.13 Experiment 2, Convergence results for the Serendipity elements on quadrilateral meshes Experiment 3, Convergence results for the Div Free Macro elements on tensor product meshes Experiment 3, Convergence results for the Div Free Macro elements on quadrilateral meshes Experiment 3, Convergence results for the Serendipity elements on tensor product meshes Experiment 3, Convergence results for the Serendipity elements on quadrilateral meshes vii

8 LIST OF FIGURES 1.1 Stokes flow around the unit sphere, [30] Degrees of freedom on a rectangular element Velocity solution, u Velocity solution, u Pressure solution Degrees of freedom on the cubical mesh Degrees of freedom of the macro element Σ h () Degrees of freedom of the macro velocity element V h () Construction of the skewed reference element Labeling of the skewed reference element Mesh with h = viii

9 PREFACE I would like to express my deepest gratitude to my supervisor, Michael Neilan, for his guidance, insightful comments and support throughout my graduate studies at the University of Pittsburgh. I thank him for introducing me to the unconventional and inspiring interplay between different fields of mathematics. I also thank him for always being available, and willing to discuss different viewpoints. I believe working on a classical problem would not be as much fun as it was if I did not work with him. In addition, I appreciate the freedom he has provided me with in my research which helped me become an independent researcher. I would like to thank all my committee members for their support. I would particularly like to thank William Layton for encouraging me to continue scientific research. I would also like to thank Ayse H. Bilge, who had introduced me to mathematical research when I was a sophomore at Istanbul Technical University, for encouraging me to pursue a Ph.D. in mathematics in the United States. I would like to thank my dear friends for their support and encouragement. My graduate school experience abroad would be really hard without them by my side. Most of all, I would like to thank my dear parents, Kadriye Sap and Ali Asker Sap, my siblings, Filiz and Hakan, and my uncle, Zeki Sap, for all their support and encouragement. I am truly grateful for having such a loving, understanding and generous family. I hope that some readers will be inspired by the concepts I shed light on and the methods I introduce, and some readers will enjoy the scenery I paint with symbols and figures as I exhibit the hidden beauty of mathematics in nature. ix

10 1.0 INTRODUCTION Stokes equations describe the motion of the Stokes flow which is a kind of fluid flow where the inertial forces are smaller than the viscous forces. In fluid dynamics, a dimensionless quantity known as the Reynolds number (Re) is used in scaling the relative importance of the inertial (or convective) terms and viscous (or diffusion) terms in a flow equation. By definition, Re = UL/ν where ν denotes the fluid dynamic viscosity and the constants U and L denote the respective reference length and velocity of the simulated flow. Stokes flow occurs when Re is low, that is, where the fluid velocities are very low, or the viscosities are very large, or the length-scales of the flow are very small [27]. In nature, this kind of flow occurs in the locomotion of microorganisms, and the flow of lava, and in technology, it often occurs in paint, MEMS devices, and in the flow of viscous polymers [17]. The steady Stokes system with no slip boundary conditions is given by ν u + p = f, in Ω, u = 0, in Ω, (1.0.1) u = 0, on Ω, where u is the fluid velocity, p is the fluid pressure and f is the external force applied to the fluid. We assume that Ω R n is an open, bounded, simply connected domain with a piecewise smooth boundary. For simplicity, we also assume that ν is constant. In Figure 1.1, an example of the Stokes flow where a solid sphere is moving at a constant speed through a viscous fluid is illustrated [30]. 1

11 Figure 1.1: Stokes flow around the unit sphere, [30]. In this dissertation, we construct conforming and stable finite element spaces that yield pointwise divergence free velocity approximations and optimal convergence rates for the discrete version of the Stokes system (1.0.1). However, the methods developed here can be extended for the steady, incompressible Navier-Stokes Equations (NSE) given by ν u + (u )u + p = f, in Ω, (1.0.2a) u = 0, in Ω, (1.0.2b) u = 0, on Ω, (1.0.2c) where equation (1.0.2a) represents the conservation of momentum, and equation (1.0.2b), known as the continuity equation, represents the conservation of mass. The NSE play an essential role in modeling fluid flows such as the flows in pipes and channels and the flows around wings of a plane [31]. Solving the NSE for specified boundary conditions provides approximations for the fluid velocity and the fluid pressure in a given geometry. However, due to their complexity, the NSE admit only a limited number of analytical solutions. For example, the NSE for a laminar, steady, two dimensional flow of an incompressible fluid between two parallel plates or in a circular pipe can be solved analytically [29]. 2

12 1.1 FINITE ELEMENT METHOD APPLIED TO THE STOKES PROBLEM Finite element methods are numerical techniques developed for approximating solutions for partial differential equations by piecewise polynomials. A finite element method is based on the division of a mathematical model into disjoint components of simple geometry. These disjoint components are called finite elements. Each element contributes to the model a finite number of degrees of freedom evaluated at a set of points, and then the model is approximated via the discrete model obtained by connecting or assembling these elements. Consider a partition T h of an open, bounded and simply connected domain Ω for a fixed discretization parameter h. A finite element space is a triple (T, P T, Σ T ) that satisfies [10] T R n is closed and bounded with a piecewise smooth boundary T, (element domain). P T is a finite dimensional space of real valued functions over T, (basis functions, a.k.a., shape functions). Σ T consists of linear forms over C (T ), (degrees of freedom, a.k.a, nodal points). In finite element theory, it is assumed that Σ T is P T unisolvent, that is, if Σ T = {φ i } N i=1, then for scalars {α i } N i=1 there exists a unique function p P T that satisfies φ i (p) = α i for 1 i N. Therefore, dim(p T ) = N = card(σ T ) and there exists a set of functions {p j } N j=1 that forms a basis of P T and satisfies φ i (p j ) = δ i,j for 1 i, j N. As a result, any p P T can be written as p = N φ i (p)p i. i=1 To approximate (u, p) that solves the Stokes system given by (1.0.1), we need to construct velocity and pressure finite element spaces. Let V h H 1 0(Ω) denote the velocity finite element space and W h L 2 0(Ω) denote the pressure finite element space. The finite element discretization of (1.0.1) reads: Find (u h, p h ) V h W h that satisfies ν u h : v p h div(v) = f v, v V h, Ω Ω Ω (1.1.1) q div(u h ) = 0, q W h. Ω A necessary criterion for the well posedness and the stability of the discrete problem (1.1.1) is the discrete inf sup condition, also known as the Ladyzenskaja-Babuska-Brezzi (LBB) 3

13 condition, which is given by [9]: α q L 2 (Ω) sup v V h \{0} q div(v) Ω, q W h, (1.1.2) v H 1 (Ω) where α > 0 is a constant independent of the discretization parameter h. The discrete inf sup stability condition is equivalent to the property W h P Wh div(v h ) with a bounded right-inverse, where P Wh : L 2 0(Ω) W h denotes the L 2 projection onto W h. The inf sup condition is invalid if the space S p = {q W h : Ω q div(v) = 0, v V h} is non trivial. A non zero element s S p is known as a spurious pressure mode. Every spurious pressure mode s yields a pressure solution (s + p h ) for (1.1.1). Therefore, the existence of spurious pressure modes violates the uniqueness of the pressure solution [9]. Proposition 1 (Proposition 2.1 in [9]). Let (u, p) be the solution of (1.0.1) and suppose that the inf sup condition (1.1.2) holds. Then, there exists a unique pair (u h, p h ) V h W h that solves the discretized system given by (1.1.1) and the errors satisfy u u h H 1 (Ω) + p p h L 2 (Ω) c inf {v h V h, q h W h } ( u v h H 1 (Ω) + p q h L 2 (Ω) where c > 0 is a constant independent of the discretization parameter h. Remark 1. In the rest of this dissertation, we denote by c > 0 a generic constant that may take different values at different instances. ), 1.2 PREVIOUS WORKS AND THE ADVANTAGES OF DIVERGENCE FREE APPROXIMATIONS Over the past years, several mixed finite element methods for the Stokes problem on triangular meshes have been developed. Although conforming and stable approximations have been derived by most of these methods, the incompressibility condition is usually only weakly satisfied. Taylor-Hood elements, the MINI element [3], the Crouzeix-Raviart elements [15] and the (P 2 P dc 0 ) pair in [9] are among the methods that belong to this class. The discrete velocity solution is divergence free if the image of the divergence of the discrete velocity finite element space is a subset of the discrete pressure finite element space. On the other hand, the cases where the image of the discrete divergence operator is strictly smaller than the discrete pressure finite element space invalidate both the uniqueness of the 4

14 discrete pressure solution and the inf sup stability condition on general meshes. For example, the ( 1 P dc 0 ) element does not satisfy the inf sup stability condition [9]. The stability of this element depends highly on the mesh and global spurious modes, which can not be eliminated easily, are observed on some regular meshes. Another example is the (P 1 P 0 ) element. In this case, the dimension of the spurious modes grows as the mesh size tends to zero [9]. It is known that the finite element spaces that only satisfy the incompressibility condition weakly may lead to instabilities in nonlinear problems [9, 10]. Additionally, since the divergence free condition models the conservation of mass, the inexact satisfaction of the incompressibility condition also leads to the lack of mass conservation [22]. Additional advantages of divergence free finite element methods are as follows: The invariance f f + φ = p p + φ is preserved numerically, that is, the velocity approximation is only influenced by the divergence free part of the source function. If the velocity is coupled with transport, then the enforcement of mass conservation is paramount to ensure accurate approximations [8]. The construction of divergence free yielding elements results in explicit characterizations of the divergence free subspaces, possibly leading to efficient iterative solvers [40]. The velocity error estimates for divergence free, stable and conforming finite element pairs are decoupled from the pressure error estimates, i.e., [22]: u u h H 1 (Ω) c inf v Z h u v H 1 (Ω) where Z h denotes the kernel of the divergence operator acting on V h. On simplicial meshes, the first conforming, divergence free finite elements were introduced by Scott-Vogelius [41]. They showed that the P k Pk 1 dc pair is stable on simplicial triangulations of two dimensional polygonal domains if the polynomial degree k satisfies k 4 and the triangulation does not contain singular vertices. Their results were extended in [22, 25]. In [6, 38, 45, 47], it was shown that the spaces P k Pk 1 dc provide mass conservation for smaller values of k if the meshes are Hsieh-Clough-Tocher or Powell-Sabin triangulations. Arnold and in [6] showed that P 2 P dc 1 pair yields an unstable method for some meshes even after the spurious pressure modes are removed. However, for some meshes the method is stable and gives optimal order convergence for velocity once the local spurious pressure modes are 5

15 removed, and yet for other meshes, the method is stable and yields optimal convergence rates for both the pressure and the velocity approximations. Falk and Neilan [22] developed conforming and stable finite elements that satisfy mass conservation on general triangular meshes by using discrete smooth de Rham complexes. Unlike Scott and Vogelius, they do not require quasiuniform meshes in their analysis and their elements have significantly fewer global degrees of freedom than the Scott-Vogelius elements. The methods developed in the first two chapters of this dissertation are extensions of the Falk-Neilan elements to rectangular grids. Guzmán and Neilan [25] constructed divergence free, conforming and stable finite element pairs for the three dimensional Stokes problem on general simplicial triangulations. They used a pressure space consisting of piecewise constants and a velocity space consisting of cubic polynomials augmented with rational functions. Christiansen and Hu [12] presented a general framework for the discretization of de Rham sequences of differential forms with high regularity and some examples of finite element spaces that fit in the framework. Similar to the simplicial case, the construction of Stokes pairs yielding divergence free approximations on tensor product meshes is mostly limited to the two dimensional case [26, 46, 7]. The first conforming, divergence free element on a rectangular mesh was proposed by Austin, Manteuffel and McCormick [7]. The finite element space they introduced is a continuous space that is based on a Raviart-Thomas finite element space. The authors constructed a P 3,2 P 2,3 finite element pair as a direct sum of two L 2 orthogonal spaces and proved that the optimal convergence is obtained in the energy norm for tensor product grids. Another conforming, divergence free element on rectangular grids was proposed by Zhang [45]. He showed that the (P k+1,k P k,k+1 P k ) mixed element, where P k denotes the discontinuous polynomials of separated degree k or less with spurious modes filtered, is stable and yields an optimal order of approximation for the Stokes problem for all k 2. Moreover, Buffa, de Falco and Sangalli [11] proposed several choices of spline spaces which can be perceived as extensions of the Taylor-Hood, Nédélec and Raviart-Thomas pairs of finite element spaces for the approximations of the velocity and pressure fields. They studied the stability and convergence of each method and discussed the exact mass conservation of the discrete velocity fields. Evans and Hughes [21, 20] developed B spline discretizations that produce pointwise divergence free velocity approximations. Their method is motivated 6

16 by isogeometric discrete differential forms and can be interpreted as a smooth generalization of Raviart-Thomas elements. 1.3 METHODOLOGY The construction of the finite element pairs in this dissertation is motivated by smooth de Rham complexes (Stokes complexes) [22]. The Stokes complex we use as a guiding tool in developing our methods is a Hilbert complex used for the Stokes flow model that constitutes Hilbert spaces as scalar and vector potentials, the flow velocity and the flow pressure. The Stokes complex employs the standard Sobolev spaces: L 2 0(Ω) = {v L 2 (Ω) : v = 0}, H k (Ω) = {v L 2 (Ω) : D α v L 2 (Ω), α k}, H0 k (Ω) = {v H k (Ω) : D α v = 0 on Ω, α k 1}. The two dimensional Stokes complex is given by the sequence: Ω R H 2 (Ω) curl H 1 (Ω) div L 2 (Ω) 0, (1.3.1) where the curl operator acting on a scalar function z H 2 (Ω) is defined as curl(z) = ( z y, z x )t. Provided Ω is simply connected, the complex (1.3.1) is exact, that is, the range of each map in the complex is the null space of the succeeding map. Our main goal is to construct finite element spaces (Σ h, V h, W h ) which by the sequence (1.3.2) forms an exact subcomplex of the complex (1.3.1). R Σ h curl V h div W h 0. (1.3.2) Since (1.3.2) is a subcomplex of (1.3.1), there holds Σ h H 2 (Ω), V h H 1 (Ω) and W h L 2 (Ω), that is, the finite element spaces are conforming. To verify the exactness of (1.3.2), we need to show that for every q W h, there exists v V h satisfying div(v) = q, and if v V h with div(v) = 0, then v = curl(z) for some z Σ h. Along with an estimate of the right inverse, this result also yields inf sup stability. In n dimensions, we view the functions in H 1 (Ω) and L 2 (Ω) as (n 1)-forms and n-forms, 7

17 respectively. In particular, if v H 1 (Ω) with v = (v (1), v (2),..., v (n) ) t and q L 2 (Ω), then n v v (j) dx 1 dx j... dx n, q q dx 1... dx n, j=1 where hat indicates a suppressed argument. Let d denote the exterior differentiation operator and set HΛ l (Ω) as the space of L 2 (Ω) l forms with exterior derivatives in L 2 (Ω). Then the n dimensional de Rham complex with minimal L 2 (Ω) smoothness is given by [4, 5] R d HΛ 0 (Ω) d... d HΛ n 2 (Ω) d HΛ n 1 (Ω) d HΛ n (Ω) 0. (1.3.3) The n dimensional Stokes complex is obtained by imposing additional regularity in the second to last and third to last spaces in the sequence as follows: R d HΛ 0 (Ω) d... d ĤΛn 2 (Ω) d H 1 Λ n 1 (Ω) d HΛ n (Ω) 0, (1.3.4) where H 1 Λ n 1 (Ω) denotes the space of (n 1)-forms with coefficients in H 1 (Ω) and ĤΛ n 2 (Ω) := {ω HΛ n 2 (Ω) : dω H 1 Λ n 1 (Ω)}. Starting with a HΛ 0 (Ω) conforming finite element space, we follow the sequence (1.3.4) to derive a finite element pair V h W h with the desired properties. Throughout this dissertation, we refer to the following theorem in obtaining the stability results. Theorem 1 (Theorem in [19]). For every q L 2 0(Ω), there exists v H 1 0(Ω) such that div(v) = q, v H 1 (Ω) c q L 2 (Ω), where c > 0 is a constant independent of the discretization parameter h. 1.4 OUTLINE In the next chapter, we describe our method for constructing a pair of conforming, stable and divergence free finite element spaces for the Stokes problem on rectangular meshes. We discuss the stability and the approximation properties of the element pairs we propose. In the third chapter, we explain the extension of the two dimensional finite element spaces constructed on rectangular meshes to n dimensions, and state the stability and the convergence characteristics of the new element pairs. Additionally, we derive reduced elements 8

18 with less global degrees of freedom and carry out the convergence analysis for the reduced elements. In the fourth chapter, we introduce a macro finite element method on convex quadrilateral meshes. We construct a Fortin operator to prove the inf sup stability on general shape regular quadrilateral meshes and derive convergence results. Furthermore, we compute a post processed pressure solution locally to improve the rate of convergence of our pressure approximations. In addition, we describe the implementation process of the (velocity) macro finite element, and provide numerical experiments that support the theoretical results. Moreover, we provide experimental results of our method for the Navier-Stokes equations. 9

19 2.0 FEM FOR THE STOKES PROBLEM ON RECTANGULAR GRIDS In this chapter, we construct a pair of conforming and stable finite elements that yield divergence free velocity approximations on rectangular grids. We assume that Ω R 2 is an open, bounded, simply connected, polygonal domain with edges parallel to the coordinate axes. We denote by h the shape regular rectangular mesh of Ω. To each element h, we assign h = diam(), and then define the global mesh size h := max h h. Additionally, we let V := {a i } 4 i=1 and E := {L i } 4 i=1 denote the vertices and the edges of a rectangular element h. Moreover, we assume that the elements of V are ordered in a counterclockwise fashion starting at a 1 = (x 0, y 0 ) and ending at a 4 = (x 0, y 1 ) (See Figure 2.1). The space of polynomials of degree less than or equal to m in x and n in y is denoted by P m,n. The vector valued functions and variables are written in bold face. For example, H 1 (Ω) = (H 1 (Ω)) THE LOCAL FINITE ELEMENT SPACES The C 1 Bogner-Fox-Schmidt Finite Element Space We utilize the Bogner-Fox-Schmidt(BFS) rectangular element in building the finite element space Σ h () = P 3,3 () [13]. The degrees of freedom of Σ h () shown in Figure 2.1 are given by 2 z S Σ = {z(a i ), z(a i ), x y (a i) : a i V }. Lemma 1. The degrees of freedom stated in S Σ are unisolvent on Σ h (). Proof. Note that the cardinality of the set S Σ equals the dimension of the polynomial space P 3,3 (). Therefore, it suffices to prove that if z nullifies S Σ, then z = 0 in. 10

20 If z Σ h (), then we may write z(x, y) = s 1 (x)s 2 (y), where s 1 and s 2 are univariate cubic polynomials. Then, if z nullifies S Σ, we have s 1 (x 0 )s 2 (y 0 ) =s 1 (x 1 )s 2 (y 0 ) = s 1 (x 0 )s 2 (y 1 ) = s 1 (x 1 )s 2 (y 1 ) = 0, s 1(x 0 )s 2 (y 0 ) =s 1(x 1 )s 2 (y 0 ) = s 1(x 0 )s 2 (y 1 ) = s 1(x 1 )s 2 (y 1 ) = 0, s 1 (x 0 )s 2(y 0 ) =s 1 (x 1 )s 2(y 0 ) = s 1 (x 0 )s 2(y 1 ) = s 1 (x 1 )s 2(y 1 ) = 0. On L 4, z(x, y) = s 1 (x 0 )s 2 (y), and if s 1 (x 0 ) 0, then s 2(y 0 ) = s 2(y 1 ) = s 2 (y 0 ) = s 2 (y 1 ) = 0. This implies that s 2 L4 = 0 since s 2 P 3 (). Therefore, z L4 = 0. Similar computations show that z = 0. Thus, z = α b, where α P 1,1 () and b is a bubble function. More precisely, b is a biquadratic polynomial that vanishes on, takes the value one at the center of and is nonnegative in. If we denote by b i a non trivial linear function that vanishes on L i for every 1 i 4, then we may write b = b 1 b 2 b 3 b 4. Moreover, we have b 2 P 4,4, this implies that z = 0 in. z x y (a i) = 0 for all 1 i 4, therefore, z has b 2 as a factor. Since The Velocity Space We define the local velocity space as V h () = P 3,2 () P 2,3 (). The degrees of freedom of V h () illustrated in Figure 2.1 are given by the set v 1 S v = {v(a i ), x (a v 2 i), y (a i) : v = (v 1, v 2 ), a i V, v κ : κ P 1,0 () P 0,1 (), v n : L i E }, L i where n denotes the (outward) unit normal vector. Lemma 2. The degrees of freedom stated in S v are unisolvent on V h (). Proof. Since dim(v h ()) = 24 equals the cardinality of the set S v, it suffices to show that if v nullifies S v, then v = 0 in. We may write v 1 (x, y) = s 1 (x)s 2 (y), where s 1 and s 2 are cubic and quadratic polynomials, respectively. Then, if v nullifies S v, the following holds v 1 (x 0, y 0 ) = s 1 (x 0 )s 2 (y 0 ) = v 1 (x 0, y 1 ) = s 1 (x 0 )s 2 (y 1 ) = 0, v 1 (x 1, y 0 ) = s 1 (x 1 )s 2 (y 0 ) = v 1 (x 1, y 1 ) = s 1 (x 1 )s 2 (y 1 ) = 0, 11

21 v 1 x (x 0, y 0 ) = s 1(x 0 )s 2 (y 0 ) = v 1 x (x 0, y 1 ) = s 1(x 0 )s 2 (y 1 ) = 0, v 1 x (x 1, y 0 ) = s 1(x 1 )s 2 (y 0 ) = v 1 x (x 1, y 1 ) = s 1(x 1 )s 2 (y 1 ) = 0, v 1 = v 1 = 0, L 3 L 4 v 1 κ 1 + v 2 κ 2 = 0, for all κ = (κ 1, κ 2 ) P 1,0 () P 0,1 (). On L 1, v 1 (x, y) = s 1 (x)s 2 (y 0 ), and if s 2 (y 0 ) 0, then s 1 (x 0 ) = s 1 (x 1 ) = s 1(x 0 ) = s 1(x 1 ) = 0. This implies that s 1 L1 = 0 since s 1 P 3 (). Thus, v 1 L1 = 0. Similar arguments show that v 1 L2 = 0. On L 4, v 1 (x, y) = s 1 (x 0 )s 2 (y), and if s 1 (x 0 ) 0, then s 2 (y 0 ) = s 2 (y 1 ) = 0. Moreover, L 4 v 1 = 0 implies that s 2 (y ) = 0 for some y (y 0, y 1 ). As a result, we have s 2 L4 = 0 since s 2 P 2 (). Therefore, v 1 L4 = 0. Similar arguments show that v 1 L3 = 0. Hence, v 1 H 1 0() P 3,2 (). Therefore, we may write v 1 = q 1 b, where b P 2,2 () is a bubble function and q 1 P 1,0 (). Similar computations show that v 2 H 1 0() P 2,3 (). Hence, we may write v 2 = q 2 b where q 2 P 0,1 (). Thus, we have v = 0. In order to show that v = 0 in, we need to verify that q 1 = q 2 = 0 in. Note that v κ = 0 for all κ = (κ 1, κ 2 ) P 1,0 () P 0,1 (). Letting κ = (q 1, q 2 ) gives v κ = (q2 1 + q 2 2) b = 0. Since b > 0 in, this implies that (q q 2 2) = 0 on, therefore, q 1 = q 2 = 0 in. As a result, v = 0 in The Pressure Space We define the local pressure space as W h () = P 2,2 (). The degrees of freedom of W h () shown in Figure 2.1 are as follows: S q = {q(a i ), q r : r P 2,2 (), r(a i ) = 0, a i V }. 12

22 a 4 L 2 a 3 L 4 L 3 a 1 a 2 L 1 The BFS Element, Σ h (). a 4 a 3 a 1 a 2 a 4 a 3 a 1 a 2 The Velocity Elements, V h (). a 4 a 3 a 1 a 2 The Pressure Element, W h (). Figure 2.1: Degrees of freedom on a rectangular element. Solid circles indicate function evaluations, larger circles indicate gradient evaluations and arrows indicate derivative evaluations. 13

23 Lemma 3. The degrees of freedom stated in S q are unisolvent on W h (). Proof. Note that the cardinality of the set S q equals dim(w h ()) = 9. If q nullifies S q, then q(a i ) = 0 and q r = 0 for all r P 2,2() that satisfies r(a i ) = 0. Choosing r = q yields = 0, and this implies q = 0 in. As a result, the given set of degrees of freedom q2 determines W h () A Local Characterization of the Divergence Operator In this section, we impose boundary conditions on the local finite element spaces and prove a local inf sup stability condition. To this end, on each h, we define the spaces Σ h,0 () = H0() 2 Σ h (), V h,0 () = H 1 0() V h (), W h,0 () = {q W h () : q = 0, q(a i) = 0 at a i V }. Lemma 4. The space Σ h,0 () is the trivial set, i.e., Σ h,0 () = {0}. Proof. If z Σ h,0 (), then z = b 2 w where b H0() 2 is a biquadratic bubble function and w P 1,1 (). Since b 2 P 4,4(), we conclude that w = 0 in, therefore, z = 0 in. Lemma 5. There holds div(v h,0 ()) W h,0 (). Proof. Let r div(v h,0 ()). Then, there exists v V h,0 () such that div(v) = r. Since v H 1 0(), the divergence theorem yields r = div(v) = v n = 0, where n denotes the unit (outward) normal vector to. Moreover, at each vertex a i V, r(a i ) = div(v(a i )) = v 1 x (a i) + v 2 y (a i) = 0, since v = 0 implies that v i (a x i) = v i (a y i) = 0. Thus, r W h,0 (). Lemma 6. The kernel of the divergence operator acting on V h,0 () satisfies Ker(div(V h,0 ())) = curl(σ h,0 ()) = {0}. 14

24 Proof. By Lemma 4, we have curl(σ h,0 ()) = {0}. Therefore, it suffices to show that Ker(div(V h,0 ())) curl(σ h,0 ()). Let v Ker(div(V h,0 ())), then v V h,0 () and div(v) = 0. This implies that v = curl(z) for some z H0() 2 [24]. In addition, the definitions of the spaces V h () and Σ h () imply that z Σ h (). Thus, z Σ h,0 (). This yields the desired inclusion. Theorem 2. The mapping div : V h,0 () W h,0 () is bijective, i.e., div(v h,0 ()) = W h,0 () and Ker(div(V h,0 ())) = {0}. Proof. Lemmas 5-6 imply that it suffices to show that dim(div(v h,0 ())) = dim(w h,0 ()) to verify the bijectivity of the divergence map. Since there are 5 linearly independent constraints imposed on the space W h () in the definition of W h,0 (), we have dim(w h,0 ) = dim(w h ) 5 = 4. Furthermore, by the rank-nullity theorem, Lemma 6 and the definition of V h,0, we have dim(div(v h,0 ()) = dim(v h,0 ()) dim(ker(div(v h,0 ())) = dim(v h,0 ()) 0 = dim(p 1,0 P 0,1 ) 0 = 4. (2.1.1) Hence, div(v h,0 ()) = W h,0 (), therefore, for any q W h,0 (), there exists v V h,0 () such that div(v) = q. Moreover, Ker(div(V h,0 ())) = {0} implies that v is unique. Thus, div : V h,0 () W h,0 is bijective. We define a reference element ˆ and an affine transformation that maps ˆ to a physical element. We utilize the unit square as the reference element ˆ and consider the transformation F : ˆ defined as F (ˆx) = Bˆx + b, where B R 2 2, b R 2 and ˆx = (ˆx, ŷ) R 2. It is easy to see that DF = B. An affine transformation maps points (resp. edges) to points (resp. edges), and preserves parallelism, convexity and colinearity [32]. Moreover, the inverse of an affine transformation is also an affine transformation. Therefore, we may write F as follows: F (ˆx, ŷ) = (x 0 + ˆx(x 1 x 0 ), y 0 + ŷ(y 1 y 0 )) = (x, y). Setting h x := x 1 x 0 and h y := y 1 y 0 and using these in the definition of F, we write B = h x 0, b = x 0. 0 h y y 0 15

25 The shape regularity condition implies that h h x h y. We note that B ch, B 1 ch 1, det(df ) ch 2 1 and det(df ) ch 2. Then, we define the velocity space V h,0 ( ˆ) on the reference element. Since B is diagonal, v V h,0 () implies that ˆv := v F V h,0 ( ˆ). Lemma 7. ˆv = div(ˆv) L 2 ( ˆ) defines a norm on V h,0( ˆ). Proof. The following arguments show that defines a norm on V h,0 ( ˆ). 1. (Positivity) Theorem 2 implies that ˆv = div(ˆv) L 2 ( ˆ) = 0 if and only if ˆv = (Scalar multiplication) ĉv = div(ĉv) L 2 ( ˆ) = c div(ˆv) L 2 ( ˆ) = c ˆv. 3. (Triangle inequality) ˆv + ˆω div(ˆv) L 2 ( ˆ) + div(ˆω) L 2 ( ˆ) = ˆv + ˆω. Lemma 8. For q W h,0 (), define ˆq through the relation ˆq = q F. Then, ˆq W h,0 ( ˆ). Proof. Since F is affine, q P 2,2 () implies that ˆq P 2,2 ( ˆ) and q(a i ) = 0 yields ˆq(â i ) = 0. Furthermore, by a change of variables, we have 0 = q(x) = ˆq(ˆx) det(df (ˆx)). Since det(df (ˆx)) is constant, this implies that ˆq(ˆx) = 0. Hence, ˆq W h,0 ( ˆ). ˆ ˆ Theorem 3. For every q W h,0 (), there exists v V h,0 () such that div(v) = q and v H 1 () c q L 2 (), where c > 0 is a constant independent of the mesh size. condition inf sup q W h,0 ()\{0} v V h,0 ()\{0} q div(v) Ω v H 1 () q L 2 () Therefore, the (local) inf-sup c > 0 holds. This implies that the V h,0 () W h,0 () pair is locally stable. Proof. By Theorem 2, for q W h,0 (), there exists v V h,0 () that satisfies div(v) = q. In addition, by scaling, we have div(ˆv) 2 L 2 () = div(ˆv) 2 dˆx ˆ 16 h 2 div(v) 2 det(df 1 ) dx

26 c div(v) 2 dx = c div(v) 2 L 2 (), where c > 0 is an h -independent constant. Moreover, by Lemma 7 and the equivalence of norms in finite dimensions, we have Then, again by scaling, we obtain ˆv H 1 ( ˆ) c div(ˆv) L 2 ( ˆ). v H 1 () c ˆv H 1 ( ˆ) c div(ˆv) c div(v) = c q L 2 (). 2.2 THE GLOBAL FINITE ELEMENT SPACES In this section, we introduce the global finite element spaces. With the help of the Scott- Zhang interpolant [42], trace and inverse inequalities and the Nitsche s method, we verify the global conformity and stability constraints. We denote by, V and E the sets of faces, vertices and edges in the mesh. In addition, we let stand for the cardinality measure of a set. For instance, V = card(v). The local finite element spaces introduced in Section 2.1 induce the global spaces Σ h ={z H 2 (Ω): z Σ h (), 2 z x y (a) is C0, a V, h }, V h ={v H 1 (Ω): v V h (), v i x i (a) is C 0, a V, h }, W h = {q L 2 (Ω) : q W h (), q(a) is C 0, a V, h }. Lemma 9. There holds Ker(div(V h )) = curl(σ h ). (2.2.1) Proof. If v curl(σ h ), then there exists z Σ h that satisfies curl(z) = v. It is easy to see that v Ker(div(V h )). On the other hand, if v Ker(div(V h )), then v V h and div(v) = 0. Moreover, since Ω is simply-connected, there exists z H 2 (Ω) such that v = curl(z) [22]. From the definition of the space V h, it follows that v P 3,2 () P 2,3 () and v i x i is C 0 at the vertices. As a result, z P 3,3 () and 2 z x y therefore, z Σ h. Thus, Ker(div(V h )) curl(σ h ). is C0 at the vertices, Theorem 4. The Stokes complex (1.3.2) is exact provided Ω is simply connected. 17

27 Proof. By Lemma 9, it suffices to show that div : V h W h is surjective. Clearly, div(v h ) W h, therefore, we need to show that dim(div(v h )) = dim(w h ) to complete the proof. By the rank-nullity theorem, and Lemmas 1 and 2, we have dim(div(v h )) = dim(v h ) dim(curl(σ h )) = dim(v h ) (dim(σ h ) 1) = (2(2 + 2 V ) + E ) 4 V + 1 = V + E 4 V + 1 = 4 + E + 1. Thus, by Lemma 3 and the Euler s identity on simply connected domains (Lemma 4.41 in [28]), we obtain dim(w h ) dim(div(v h )) = ( V + 5 ) 4 E 1 = V + E 1 = 0. This implies that div(v h ) = W h, and the result follows. Lemma 10 (Lemma 2.2 in [22]). For any simply connected domain S with piecewise smooth boundary S, there exists a constant c > 0 such that v L 2 ( S) c v 1/2 L 2 (S) v 1/2 H 1 (S) v H 1 (S). By scaling, this implies that for every v H 1 (), the estimate holds on every element h. v 2 L 2 ( ) c(h 1 v 2 L 2 () + h v 2 H 1 () ), for all Lemma 11. For any q W h, there exists v (1) V h that satisfies (q div(v (1) )) W h,0 () for all h. Moreover, v (1) H 1 (Ω) c q L 2 (Ω). Proof. For q W h L 2 (Ω), there exists ω H 1 (Ω) such that div(ω) = q and ω H 1 (Ω) c q L 2 (Ω) (Lemma 2.2 in [9]). Let I h ω : H 1 (Ω) P 2,2 (Ω) be the Scott-Zhang interpolant of ω [42]. Define v (1) = (v (1) 1, v (1) 2 ) V h that satisfies the following conditions: (i) v (1) (a) = I h ω(a), a V. (ii) v(1) 1 x (a) = v(1) 2 y (a) = q(a), a V. 2 (iii) v(1) κ = ω κ, κ P 1,0 P 0,1, h. (iv) e v(1) n = e ω n, e E. 18

28 Note that condition (iv) yields v(1) n implies that div(v (1) (a)) = q(a) at all a V. = ω n for all h and condition (ii) Then, by the divergence theorem and condition (iv), we deduce that q div(v (1) ) = q div(v (1) ) = q v (1) n = q ω n = q div(ω) = 0. Thus, (q div(v (1) )) W h,0 (). Moreover, by the triangle inequality, we have v (1) H 1 () v (1) I h ω H 1 () + I h ω H 1 (). (2.2.2) Since (v (1) I h ω) V h (), by Lemma 2 and scaling, we obtain v (1) I h ω 2 H 1 () ( (v (1) I h ω)(a j ) 2 + h 2 (v ) (1) I h ω)(a j ) 2 a j V + L i E h 1 e (v (1) I h ω) n 2 + h 2 L i where D = {r : r P 1,0 () P 0,1 (), r 2 L 2 () = 1}. sup (v (1) I h ω) r 2, r D Combining this with the conditions stated in the construction of v (1), using scaling arguments and the Cauchy Schwarz inequality, we obtain v (1) I h ω 2 H 1 () c ( h 2 ( q(a j ) 2 + I h ω(a j ) 2 ) + a j V ) + h 2 sup (ω I h ω) r 2 r D ( c h 2 ( q(a j ) 2 + I h ω 2 ) + h 1 ω I h ω 2 L 2 ( ) a j V ) + h 2 ω I h ω 2 L 2 () L i E h 1 e (ω I h ω) n 2 L i. (2.2.3) Again by scaling, we have h 2 I h ω(a j ) 2 c I h ω 2 H 1 (), and h 2 q(a j ) 2 c q 2 L 2 (). (2.2.4) a j V a j V Moreover, by Lemma 10 and the approximation properties of the Scott Zhang interpolant [42], we obtain ( ) h 1 ω I h ω 2 L 2 ( ) c h 2 ω I h ω 2 L 2 () + ω I h ω 2 H 1 () c ω 2 H 1 (ω()) (2.2.5) where ω() denotes the patch of rectangles that touch. Using (2.2.4), (2.2.5) and the approximation and the stability properties of the Scott-Zhang interpolant [42] in (2.2.3), we 19

29 derive the estimate v (1) I h ω 2 H 1 () ( I c h ω 2 H 1 () + q 2 L 2 () + ω 2 H 1 (ω()) ( ) c q 2 L 2 () + ω 2 H 1 (ω()) ) (2.2.6) Combining (2.2.2) and (2.2.6), and summing over h yields v (1) H 1 (Ω) ω H 1 (Ω) c q L 2 (Ω). Corollary 1. Let v (1) and q be defined as in the proof of Lemma 11. Then, there exists v (2) V h that satisfies v (2) = (q div(v (1) )) and v (2) H 1 (Ω) c( q L 2 (Ω) + v (1) H 1 (Ω)). Proof. Since (q div(v (1) )) W h,0 (), by Theorem 2, there exists v (2) satisfies div(v (2) ) = (q div(v(1) )) and it is easy to see that V h,0() that v (2) H 1 () c q div(v (1) ) L 2 () c( q L 2 () + v (1) H 1 ()). (2.2.7) Let v (2) be such that v (2) = v (2). Since v(2) at a V. V h,0() H 1 0(), v(2) 1 x This implies that the partial derivatives v(2) 1 x and v(2) 2 y and v(2) 2 y vanish are continuous at a V. As a result, v (2) V h. Finally, summing (2.2.7) over h yields v (2) H 1 (Ω) c( q L 2 (Ω) + v (1) H 1 (Ω)). Theorem 5. For any q W h, there exists v V h such that div(v) = q and v H 1 (Ω) c q L 2 (Ω). where c is a constant independent of the discretization parameter h. This implies the global inf sup stability condition inf sup q W h \{0} v V h \{0} q div(v) Ω v H 1 (Ω) q L 2 (Ω) c > 0. Proof. For q W h, let v (1) and v (2) be defined as in Lemma 11 and Corollary 1, respectively, and set v := v (1) + v (2). Then, we note that v V h and div(v) = q. Combining the results of Lemma 11 and Corollary 1, we derive the desired estimate v H 1 (Ω) c( v (1) H 1 (Ω) + v (2) H 1 (Ω)) c( v (1) H 1 (Ω) + q L 2 (Ω)) q L 2 (Ω). 20

30 2.3 THE GLOBAL FINITE ELEMENT SPACES WITH IMPOSED BOUNDARY CONDITIONS In this section, we impose homogeneous boundary conditions on the finite element spaces defined in Section 2.2. We let V b V denote the boundary vertices and V c denote the corner vertices of the mesh. In addition, we denote by E b the boundary edges, and we let E b x E b and E b y E b stand for the vertical and horizontal boundary edges of the mesh, respectively. Moreover, we denote by V b,x (resp. V b,y ) the non corner boundary vertices where x (resp. y) is constant. Hence, we may decompose the boundary vertices and the boundary edges as follows: V b = V b,x V b,y V c and E b = E b x E b y. A candidate list of global finite element spaces with boundary conditions is: Σ 0 h = Σ h H 2 0(Ω), V 0 h = V h H 1 0(Ω), W 0 h = W h L 2 0(Ω). We note that z Σ h is in Σ 0 h if and only if: (i) z(a) = 0, a V b, (ii) z(a) = 0, a V b, (iii) 2 z (a) = 0, a V x y b. The number of constraints imposed on Σ h in (i)-(iii) is 4 V b. Thus, we have dim(σ 0 h) = dim(σ h ) 4 V b = 4 V 4 V b. Simiarly, we note that v V h is in Vh 0 if and only if: (i) v(a) = 0, a V b, (ii) v 1 x (a) = 0, e V b,y V c, (iii) v 2 y (a) = 0, e V b,x V c, (iv) e v 1 = 0, e E b x, (v) e v 2 = 0, e E b y. The number of constraints imposed on V h in (i)-(v) is 3 V b + V c + E b. We then have dim(v 0 h) = dim(v h ) 3 V b V c E b = (4 + 4 V + E) ) 3 V b V c E b. On the other hand, there is only one constraint imposed on W h in the definition of the space W 0 h. Therefore, dim(w 0 h ) = dim(w h) 1 = (4 + E + 1) 1 = 4 + E. 21

31 Lemma 12. There holds Ker(div(V 0 h )) = curl(σ0 h ). Proof. The inclusion curl(σ 0 h ) Ker(div(V0 h )) is trivial, thus, it suffices to prove the inclusion Ker(div(V 0 h )) curl(σ0 h ). If v Ker(div(V0 h )), then v V0 h and div(v) = 0. Moreover, there exists z H 2 (Ω) such that v = curl(z). Since v V h (), we have z Σ h (). In addition, the conditions (i)-(v) imposed on V h in the definition of the space V 0 h imply that z satisfies the conditions (i)-(iii) imposed on Σ h in the definition of the space Σ 0 h. Hence, z Σ0 h, and this implies that Ker(div(V0 h )) curl(σ0 h ). By Lemma 12 and the rank-nullity theorem, we have dim(w 0 h) dim(div(v 0 h)) = dim(w 0 h) (dim(v 0 h) dim(ker(div(v 0 h)))) = dim(w 0 h) (dim(v 0 h) dim(curl(σ 0 h))) = dim(w 0 h) (dim(v 0 h) dim(σ 0 h)). (2.3.1) From the dimension analysis, it is easy to see that (2.3.1) implies dim(σ 0 h) + dim(w 0 h) dim(v 0 h) =(4 V 4 V b ) + (4 + E ) 4 4 V E + 3 V b + V c + E b = V b + V c + E b = V c > 0, since E b = V b. As a result, (Σ 0 h, V0 h, W h 0 ) does not form an exact sequence. Moreover, this result implies that the pressure space is larger than desired. Therefore, we define the global finite element spaces as follows: (i) Σ h,0 = Σ h H 1 0(Ω), (ii) V h,0 = {v h V h : (v h n) Ω = 0}, (iii) W h,0 = W h L 2 0(Ω). We note that z Σ h is in Σ h,0 if and only if: (i) z(a) = 0, a V b, (ii) z x (a) = 0, a V b,y V c, (iii) z y (a) = 0, a V b,x V c. Thus, the number of constraints imposed on Σ h is: 2 V b + V c. As a result, dim(σ h,0 ) = dim(σ h ) 2 V b V c = 4 V 2 V b V c. Similarly, we note that v h V h is in V h,0 if and only if: 22

32 (i) v 1 (a) = 0, a V b,x V c, (ii) v e 1 = 0, e Ex, b (iii) v 2 (a) = 0, a V b,y V c, (iv) v e 2 = 0, e Ey. b The number of constraints imposed on V h,0 is: V b + V c + E b. Thus, dim(v h,0 ) = dim(v h ) V b V c E b = V + E V b V c E b. Lemma 13. There holds Ker(div(V h,0 )) = curl(σ h,0 ). Proof. As in the proof of Lemma 12, the inclusion curl(σ h,0 ) Ker(div(V h,0 )) is trivial. Thus, it suffices to show that Ker(div(V h,0 )) curl(σ h,0 ). Suppose v Ker(div(V h,0 )), then v V h,0 such that div(v) = 0. Moreover, there exists z H 1 (Ω) such that v = curl(z). Since v V h (), we have z Σ h (). Furthermore, the continuity of the partial derivatives of v V h at the vertices implies that 2 z x y is continuous at the vertices. The conditions (i) and (iii) imposed on V h in the definition of V h,0 yield the conditions (ii) and (iii) imposed on Σ h in the definition of Σ h,0. In addition, the conditions (ii) (iv) imposed on V h imply that the condition (i) imposed on Σ h holds. Therefore, z Σ h,0, and this implies that Ker(div(V h,0 )) curl(σ h,0 ). Remark 2. By Lemma 3 and the Euler s formula [28], dim(w h,0 ) = dim(w h ) 1 = V = V + 5 ( V + E ) (2.3.2) = 4 + E. Theorem 6. div : V h,0 W h,0 is a surjective map. Proof. Note that div(v h,0 ) = {r : r P 2,2 (), div(v) = r for v V h,0 }. Let r div(v h,0 ). Then, there exists v V h,0 such that div(v) = r. Then, by the divergence theorem, r = div(v) = v n = 0. Ω Ω Ω Moreover, r is continuous at the vertices since v V h, therefore, r W h,0. As a result, div(v h,0 ) W h,0. Furthermore, by the rank nullity theorem and Lemma 13, we have dim(div(v h,0 )) = dim(v h,0 ) dim(ker(div(v h,0 )) = dim(v h,0 ) dim(σ h,0 ) 23

33 = (4 + 4 V + E V b V c E b ) (4 V 2 V b V c ) = 4 + V b E b + E. Thus, dim(div(v h,0 )) = 4 + E. From (2.3.2), it follows that div(v h,0 ) = W h,0. Now, we define a discrete H 1 type norm on V h,0 by v 2 h := v 2 L 2 (Ω) + e E b h e v n e 2 L 2 (e) + e E b 1 h e v 2 L 2 (e). (2.3.3) Lemma 14. The finite element pair V h,0 W h,0 satisfies the inf-sup stability condition inf sup q div(v) Ω c > 0, q W h,0 \{0} v h q L 2 (Ω) v V h,0 \{0} where c > 0 is an h-independent constant. Proof. Let v, v (1), v (2), ω and I h ω be defined as in the proof of Lemma 11, Corollary 1 and Theorem 5. Then, the proof is similar to that of Lemma (11). By the divergence theorem, we note that v n = Ω = Ω Ω v (1) n + div(v (1) ) + Ω Ω v (2) n = div(v (1) ) + Ω (q div(v (1) )) = q = 0, Ω Ω div(v (2) ) since q W h,0. Therefore, v V h,0. By the triangle inequality and scaling, we have Moreover, Corollary 1 and the equivalence of norms imply v 2 h c( v (1) 2 h + v (2) 2 h). (2.3.4) v (2) h c v (2) H 1 (Ω) c( q L 2 (Ω) + v (1) H 1 (Ω)) (2.3.5) By using the following result of the trace inequality: h e v 2 L n 2 (e) c v 2 L 2 (Ω) (2.3.6) e e E b in the definition of the h norm given by (2.3.3), we derive a bound on the h norm v 2 h c( v 2 H 1 (Ω) + 1 v 2 L h 2 (e) ). e e E b This implies that v (1) I h ω 2 h c( v(1) I h ω 2 H 1 (Ω) + e E b 1 h e v (1) I h ω 2 L 2 (e) ). Then, by the same arguments used in the proof of Lemma 11, we deduce that v (1) 2 h c ω 2 H 1 (Ω) c q 2 L 2 (Ω). (2.3.7) Using (2.3.7) and (2.3.5) in (2.3.4) yields v h c q L 2 (Ω). As a result, the inf sup condition 24

34 inf sup q W h,0 \{0} v V h,0 \{0} holds for an h-independent constant c > 0. q div(v) Ω v h q L 2 (Ω) c > NITSCHE S METHOD AND THE CONVERGENCE ANALYSIS In this section, we apply Nitsche s method [37] to prove the existence and the uniqueness of the numerical solution (u h, p h ) for the discrete problem and analyze the approximation properties. Recall the Stokes system ν u + p = f, in Ω, (2.4.1a) div(u) = 0, in Ω, (2.4.1b) u = 0, on Ω. (2.4.1c) Multiplying (2.4.1a) by v h V h,0 and integrating over Ω, we obtain u ν u : v h ν Ω Ω n v h p div(v h ) + p (v h n) = f v h, (2.4.2) Ω Ω Ω where n denotes the (outward) unit normal vector of Ω and : denotes the Frobenius inner product which provides u : v h = i,j ( u) ij ( v h ) ij. Since v h V h,0, we have (v h n) Ω = 0, therefore, Ω p (v h n) = 0. Let t denote the unit tangent vector to Ω. Then, we may write v h = (v h n) n+(v h t) t = (v h t) t. Moreover, we may write ( u n v h) Ω = u n (v h t)t = (( u n)n + ( u n n t)t)(v h t)t = ( u n t)(v h t). Thus, (2.4.2) becomes: ν u : v h ν ( u Ω Ω n t)(v h t) p div(v h ) = f v h. (2.4.3) Ω Ω Let b : H 1 (Ω) L 2 (Ω) R denote the bilinear form and F : H 1 (Ω) R denote the bounded linear functional defined by the following expressions: b(v h, p) = p div(v h ), Ω F (v h ) = f v h. Ω 25

35 Then (2.4.3) can be written as: ν u : v h ν ( u Ω Ω n t)(v h t) + b(v h, p) = F (v h ). (2.4.4) Since u Ω=0, we may symmetrize and stabilize (2.4.4) as follows: ν u : v h ν ( u ( t)(v h t) + ( v h t)(u t) σ ) u v h + b(vh, q) = F (v h ), n e n e h e e E b Ω e where σ is an h-independent penalization parameter. Let a h : (H 2 (Ω) + V h,0 ) (H 2 (Ω) + V h,0 ) R denote the bilinear form defined by ( a h (v, ω) = ν v : ω (( v t)(ω t) + ( ω t)(v t) σ ) v ω). n e n e h e e E b Ω The weak formulation of (2.4.1) is given by The finite element method for (2.4.6) reads: e (2.4.5) a h (u, v h ) + b(v h, p) = F (v h ), v h H 1 0(Ω), (2.4.6a) Find (u h, p h ) V h,0 W h,0 that satisfies b(u, q h ) = 0, q h L 2 0(Ω). (2.4.6b) a h (u h, v h ) + b(v h, p h ) = F (v h ), v h V h,0, (2.4.7a) b(u h, q h ) = 0, q h W h,0. (2.4.7b) To prove the existence and uniqueness of the solution to (2.4.7), we need to show that a h (, ) is continuous and coercive on V h,0. Lemma 15. With respect to the H 1 type norm given by (2.3.3), a h (, ) is a continuous bilinear form on (H 2 (Ω) + V h,0 ). Moreover, a h (, ) is coercive on V h,0 provided σ is sufficiently large. Proof. To prove the coercivity of a h (, ) on V h,0, we need to show that a h (v, v) cν v 2 h holds for every v V h,0. By the definition of a h (, ), we have ( a h (v, v) = ν v 2 2 v v + Ω e E b e n e e E b Using (2.3.6), for any ɛ > 0, we have 2 v v 2 n e e E b e E b e (he 1/2 v)(he 1/2 e v ) n e e σ ) v 2. h e 2( h 1 e v 2 L 2 (e) )1/2 ( h e v 2 L n 2 (e) )1/2 e e E b e E b 26

36 2c( h 1 e v 2 L 2 (e) )1/2 v L 2 (Ω) e E b 2c( 1 2ɛ = c ɛ e E b h 1 e v 2 L 2 (e) + ɛ 2 v 2 L 2 (Ω) ) e E b h 1 e v 2 L 2 (e) + cɛ v 2 L 2 (Ω). Applying this estimate to the definition of a h (, ) yields ( a h (v, v) ν v 2 L 2 (Ω) + σ 1 v 2 L h 2 (e) 2 e e E b e E ( b ν (1 cɛ) v 2 L 2 (Ω) + (σ c ɛ ) 1 v 2 L h 2 (e) e e E b If we choose ɛ = 1, then we have σ c = σ 2c ɛ 2c2. Thus, for σ 4c 2, ( 1 a h (v, v) ν 2 v 2 L 2 (Ω) + 1 ) 2c2 v 2 L h 2 (e) e e E b v ) vds n e ν min{ 1 2, 2c2 }( v 2 L 2 (Ω) + e E b 1 h e v 2 L 2 (e) ) ) cν v 2 h. As a result, a h (, ) is coercive on V h,0 provided σ is sufficiently large. To prove the continuity of a h (, ) on V h,0 + H 2 (Ω), we need to show that a h (v, ω) cν v h ω h for all v, ω V h,0 + H 2 (Ω). By the Cauchy-Schwarz inequality, for v, ω V h,0 + H 2 (Ω), we have a h (v, ω) 2 cν ( 2 ω : v 2 + ( ω v + v ω + σ ) ω v) 2 Ω e E b e n e n e h e cν ( v 2 2 L 2 (Ω) ω 2 L 2 (Ω) + ω v 2 + n e e E b e E b + e E b e σ ) ω v) 2 h e e ). e v n e ω 2 ( cν 2 ( v 2 L 2 (Ω) ω 2 L 2 (Ω) + (h e ω 2 L n 2 (e) )( 1 v 2 L e h e 2 (e) ) e E b + (h e v 2 L n 2 (e) )( 1 ω 2 L e h e 2 (e) ) + σ ) 2 w 2 L h 2 (e) v 2 L 2 (e) ) e e E b e E b cν 2 ( v 2 L 2 (Ω) + e E b h e v n e 2 L 2 (e) + e E b 1 h e v 2 L 2 (e) )( ω 2 L 2 (Ω) + e E b h e ω n e 2 L 2 (e) + e E b 1 h e ω 2 L 2 (e) ) = cν 2 v 2 h ω 2 h. 27

37 Thus, a h (, ) is continuous on V h,0 + H 2 (Ω). The equations (2.4.6b) and (2.4.7b) suggest that the systems of equations given by (2.4.6) and (2.4.7) can be reduced by utilizing the continuous and discrete kernels of the bilinear form b(, ). The continuous and discrete kernels of b(, ) are given by the following sets Z and Z h : Z = {v H 1 (Ω) : b(v, q) = 0, (v n) Ω = 0, q L 2 0(Ω)}, Z h = {v h V h,0 : b(v h, q h ) = 0, q h W h,0 }. Therefore, the system given by (2.4.7) reduces to the system: a h (u h, v h ) = F (v h ), v h Z h. (2.4.8) Since a h (, ) is a symmetric, coercive and continuous bilinear form on Z h, and F is a continuous linear form, by the Lax-Milgram theorem [10], there exists a unique solution u h Z h satisfying (2.4.8). Suppose u is the velocity solution of the system given by (2.4.1). Then, since Z h Z, for all v h Z h, we have a h (u u h, v h ) = 0. Furthermore, by the coercivity and the continuity of a h (, ) on Z h, for arbitrary v h Z h, we have u u h 2 h a h (u u h, u u h ) = a h (u u h, u v h ) + a h (u u h, v h u h ) = a h (u u h, u v h ) c u u h h u v h h. This implies that u u h h c u v h h for all v h Z h. Therefore, From Lemma 14, it follows that [13], u u h h c inf v h Z h u v h h. (2.4.9) u u h h c inf u v h h c inf u v h h. (2.4.10) v h Z h v h V h,0 Lemma 16. Suppose that the pair (u, p) that solves (2.4.6) satisfies the regularity condition u H 3 (Ω) and p H 2 (Ω), and let (u h, p h ) be the solution of the discrete problem (2.4.7). Then, there holds u u h h ch 2 u H 3 (Ω), p p h L 2 (Ω) ch 2 ( p H 2 (Ω) + ν u H 3 (Ω)), where h is the H 1 type norm given by (2.3.3). Proof. Let Π h : H 3 (Ω) V h,0 (Ω) be the nodal interpolant and P h : L 2 (Ω) W h,0 be the 28

38 L 2 projection operator. Then, the following holds [10]. u Π h u H l (Ω) ch 3 l u H 3 (Ω), 0 l 3, (2.4.11) P h p q = p q, q W h,0, (2.4.12) Ω Ω p P h p L 2 (Ω) ch s p H s (Ω), 1 s 2. (2.4.13) By the definition of the H 1 type norm given by (2.3.3), (2.3.6), the trace inequalities and (2.4.11), we obtain u Π h u 2 h = (u Π h u) 2 L 2 (Ω) + e E b h e (u Π hu) n e 2 L 2 (e) + h 1 e u Π h u 2 L 2 (e) e E b (u Π h u) 2 L 2 (Ω) + c (u Π hu) L 2 (Ω) + e E b h 1 e (h 1 (u Π h u) 2 L 2 (Ω) + h u Π hu 2 H 1 (Ω) ) Thus, we have ch 4 u 2 H 3 (Ω), u Π h u h ch 2 u H 3 (Ω). Since Π h u V h,0, (2.4.10) implies that the error in the velocity approximation of this scheme satisfies u u h h ch 2 u H 3 (Ω). (2.4.14) On the other hand, by the triangle inequality and (2.4.13), we have Then, Lemma 14 yields p p h L 2 (Ω) p P h p L 2 (Ω) + P h p p h L 2 (Ω) ch s p H s (Ω) + P h p p h L 2 (Ω), 1 s 2. c p h P h p L 2 (Ω) Since div(v h ) W h,0, (2.4.12) and (2.4.6a) imply (2.4.15) b(v h, p h P h p) sup. (2.4.16) v h V h,0 \{0} v h h a h (u, v h ) + b(v h, P h p) = F (v h ). (2.4.17) Subtracting (2.4.7a) from (2.4.17) yields a h (u u h, v h ) + b(v h, P h p p h ) = 0. Thus, b(v h, P h p p h ) = a h (u u h, v h ). By the continuity of a h (, ) in the H 1 type norm, we obtain b(v h, P h p p h ) = a h (u u h, v h ) ν u u h h v h h. (2.4.18) 29

39 Using (2.4.18) in (2.4.16), by (2.4.14), we derive the following result: P h p p h L 2 (Ω) cν u u h h v h h cν u u h h cνh 2 u H v h 3 (Ω) h Consequently, (2.4.15) with s = 2 yields p p h L 2 (Ω) ch 2 ( p H 2 (Ω) + ν u H 3 (Ω)). 2.5 NUMERICAL EXPERIMENTS In this section, we perform some numerical experiments on the domain Ω = (0, 1) 2. We assume that the viscosity constant ν = Experiment 1 In this experiment, the exact solutions are given by u(x, y) = 2x2 y(x 1) 2 (y 1) 2 + x 2 y 2 (2y 2)(x 1) 2, 2xy 2 (x 1) 2 (y 1) 2 x 2 y 2 (2x 2)(y 1) 2 p = Experiment 2 This experiment is a modified version of Experiment 1. The exact solutions are given by u(x, y) = 2x2 y(x 1) 2 (y 1) 2 + x 2 y 2 (2y 2)(x 1) 2, 2xy 2 (x 1) 2 (y 1) 2 x 2 y 2 (2x 2)(y 1) 2 p = x y. 30

40 Table 2.1: Experiment 1, Convergence results on rectangular meshes. h u u h L 2 (Ω) Rate u u h H 1 (Ω) Rate p p h L 2 (Ω) Rate 1/8 3.44e e e / e e e / e e e / e e e Table 2.2: Experiment 2, Convergence results on rectangular meshes. h u u h L 2 (Ω) Rate u u h H 1 (Ω) Rate p p h L 2 (Ω) Rate 1/ e e e 03 1/ e e e / e e e / e e e Discussion Numerical experiments show that the velocity solution has second-order convergence in H 1 norm as the theory suggests (See Lemma 16). However, the pressure solutions exhibit superconvergence (See Table 2.1, Table 2.2). For the case with mesh size h = 1, the discrete solutions for the Experiment 2 are illustrated 32 in Figures

41 Figure 2.2: Velocity solution, u 1. Figure 2.3: Velocity solution, u 2. Figure 2.4: Pressure solution. 32

42 3.0 FEM FOR THE STOKES PROBLEM ON CUBICAL GRIDS In this chapter, we extend the method we introduced in Chapter 2 to n dimensions. We assume that Ω R n is an open, bounded, simply connected domain with boundary parallel to the coordinate axes, and define a conforming cubical partition h of Ω. We further assume that every element h has boundaries parallel to the coordinate axes. For h, h denotes its diameter and s () stands for the set of its s-dimensional faces with (n s) coordinates equal to one of two constant values. We denote the subset of s () with x i constant by i s() and set ˆ i s() := s () \ i s(). We write the space of polynomials on a domain D R n of degree less than or equal to k i in x i as P k (D) := P k1,k 2,...,k n (D) and set Λ k (D) := P k (D) with k i = k for all i. Then, we define the vector valued space Λ k () := {v (Λ k()) n : v (i) P k (), k i = k, k j = k 1 for i j}. (3.0.1) For instance, for n = 3, Λ k () = P k,k 1,k 1() P k 1,k,k 1 () P k 1,k 1,k (). Note that for n = 2 and k = 3, Λ 3 () = P 3,2 () P 2,3 () which is the polynomial space forming the local velocity space defined in Section 2.1. For h with face S s () for 1 s n, b S denotes the bubble function with respect to S. We note that b S 0 on S, however, b S vanishes on (s 2) dimensional subfaces of S. Lemma 17 ([2]). A function q Λ k () is uniquely determined by { q λ : λ Λ k 2 (S), S s (), s = 0, 1,..., n}, S where S q with S 0() stands for the value of q at the vertex S. 33

43 n = 2 n = 3 n = 2 n = 3 Figure 3.1: Degrees of freedom on the cubical mesh. Degrees of freedom of the velocity (left) and pressure (right) elements in two and three dimensions (Top half). Degrees of freedom of the reduced velocity (left) and reduced pressure (right) elements in two and three dimensions (Bottom half). Solid circles indicate function evaluations and the lines indicate directional derivatives. 34

44 3.1 THE LOCAL FINITE ELEMENT SPACES In this section, we define the local velocity and pressure finite elements. Additionally, we derive a characterization of the divergence operator acting on the local velocity space which is an essential tool in the stability analysis of the global spaces. Lemma 18. Suppose v = (v (1), v (2),..., v (n) ) Λ 3 () satisfies v (i) v (i) = 0, = 0, S (i) s (), s = 0, 1,..., m, (3.1.1) x i S S for all 1 i n and for some 0 m n 2. Then, v = 0, v(i) x i 1 i n and 0 s m. Also, if then v = 0 on m+1 (). S v (i) = 0, S (i) m+1(), = 0 on s () for all Proof. The proof is by mathematical induction on m. Note that (i) 0 () = 0 () for all 1 i n, therefore, the case m = 0 is trivial. Assume that v = 0 and v(i) x i = 0 on all S m () for some m {0, 1,..., n 3} and all 1 i n. Let S m+1 (). Then, by the induction hypothesis, we have v = 0 and v (i) x i = 0 on S for all 1 i n. If x j is constant on S, then by definition, S (j) m+1() and v (j) S, v(j) x j S Λ 2 (S). Therefore, we may write v (j) S = b S q (j) and v(j) x j S = b S p (j) for some q (j), p (j) R, and (3.1.1) implies that v (j) = v(j) x j = 0 on S. If S (j) m+1(), then there exist exactly two m-dimensional faces S (1), S (2) S on which x j is constant. We denote by S and S (i) the surface gradient of S and S (i), respectively. Then, we have S ( ) = ( S (i)( ), ( ) x j ). Therefore, since v (j) and v(j) x j vanish on S (1) and S (2), we have S v (j) = 0 on S (1) and S (2). In addition, since v (j) S Λ 3 (S), and v (j) S = 0, we may write v (j) S = b S q for some q Λ 1 (S). As a result, 0 = S v (j) S (i) = q S b S S (i). (3.1.2) Therefore, q = 0 on S (i), and since q Λ 1 (S), q = 0 in S. Thus, v (j) = 0 and v(j) x j = 0 on S. Thus, the proof of the first assertion follows. The same arguments can be used to prove the second assertion. 35

45 3.1.1 The Velocity Space Lemma 19. A unisolvent set of degrees of freedom of the local velocity space Λ 3 () are given by S v = { v (i) v (i), : S (i) s (), S S x i S where 0 s n 2 and 1 i n (See Figure 3.1). v (i) : S (i) n 1(), v κ : κ Λ 1 ()}, Proof. The cardinality of S v is n 2 n 2 ( ) 2n (i) s () + n (i) n 1 n 1() + dim(λ 1 ()) = 2n 2 n s + 2n + 2n s s=0 s=0 n 1 ( ) n 1 = 4n + 4n 2 n s 1 + 4n s s=0 = 4n3 n 1 = Λ 3 (), by the binomial formula. Thus, it suffices to show that v nullifies S v if and only if v = 0. Suppose that v vanishes on S v. Then, v(i) x i = 0 on n 2 () and v = 0 on n 1 () by Lemma 18. If we denote by b the bubble function of n () = {}, then we may write v = b q for some q Λ 1 (). After we set κ = q in v κ = 0, we deduce that v = 0 in. Remark 3. For S s (), we define an orthonormal set of vectors {n (j) S }n s j=1 orthogonal to the tangent space of S. Then, we may redefine the set of degrees of freedom as follows: Sv = { v n (j) S, v n (j) S S n (j) S : S s(), (3.1.3a) S v n S : S n 1 (), (3.1.3b) S v κ : κ Λ 1 ()}, (3.1.3c) where 0 s n 2 and 1 j n s The Pressure Space We form the local pressure space by the tensor product quadratic polynomials Λ 2 (). By Lemma 17, any function q Λ 2 () is uniquely determined by the set { q : S s (), s = 0, 1,..., n}. S 36

46 We define a subspace of Λ 2 () by Λ 2 () := {q Λ 2 () : q = 0 on all (n 2) dimensional faces of }. (3.1.4) Remark 4. (3.1.4) implies that a function q Λ 2 () is uniquely determined by its average over each (n 1) dimensional face and its average over, therefore, dim( Λ 2 ()) = (2n+1). Lemma 20 (Lemma 3.4 in [35]). Any function q Λ 2 () is uniquely determined by the set of values { q : S s (), s = 0, 1,..., (n 2), S q λ : λ Λ 2 ()} A Local Characterization of the Divergence Operator Lemma 21. Let v Λ 3 (). If div(v) = 0 and v = 0, then v = 0 in. Proof. Let v denote the (n 1)-form with the vector proxy v = (v (1), v (2),..., v (n) ), i.e., n v = v (i) dx 1 dx i... dx n, i=1 where the hat indicates a suppressed argument. Moreover, for a vector-valued function κ, denote by κ the one-form given by κ = n κ (i) dx i. i=1 We may state the divergence free condition on v as d v = 0, where d denotes the exterior differentiation operator. Additionally, the boundary condition v = 0 implies that v = 0. Moreover, by the exactness of the complex (1.3.3), we deduce that there exists φ HΛ n 2 () such that v = dφ [4, 5], where HΛ n 2 () denotes the space of L 2 () (n 2)- forms with exterior derivative in L 2 () and vanishing trace. By the Stokes Theorem, for any κ Λ 1 (), we have v κ = v κ = dφ κ = ( 1) n 1 since dκ = 0 for κ Λ 1 () as shown by the following: n n κ (j) n n dκ = dx k dx j = x k k=1 j=1 k=1 Thus, (3.1.5) and Lemma 19 imply that v = 0 in. Theorem 7. Define j=1 j k φ dκ = 0. (3.1.5) κ (j) x k dx k dx j = 0. V h,0 () = Λ 3 () H 1 0(), W h,0 () = Λ 2 () L 2 0() 37

47 Then, div : V h,0 () W h,0 () is bijective. Moreover, v H 1 () c q L 2 (), where c > 0 is a constant independent of h. Therefore, V h,0 () W h,0 () forms a locally inf sup stable pair. Proof. The definitions of V h,0 () and W h,0 () imply that div(v h,0 ()) W h,0 (). Therefore, it suffices to show that dim(div(v h,0 ())) = dim(w h,0 ()) to prove the bijectivity of the divergence operator. By Lemma 21, dim(ker(div(v h,0 ()))) = 0. Then, by the rank nullity theorem, we have dim(div(v h,0 ())) = dim(v h,0 ()) = dim(λ 1 ()) = 2n On the other hand, by Remark 4 we have dim(w h,0 ()) = dim( Λ 2 ()) 1 = 2n. Thus, dim(div(v h,0 ())) = dim(w h,0 ()). Therefore, for every v V h,0, there exists a unique q W h,0 that satisfies div(v) = q. Now, we define an affine transformation F : ˆ by F(ˆx) = Bˆx + b, where ˆ = (0, 1) n is the reference element, b R n and B R n n is a diagonal matrix with entries proportional to h. We define the velocity functions ˆv : ˆ R n on the reference element by the Piola transform ˆv := B 1 (v F)det(B) [33]. This definition yields div(ˆv) = div(v F)det(B). Then, by Lemma 21, scaling and the equivalence of norms in finite dimensions, we deduce that v H 1 () ch n 2 ˆv H 1 ( ˆ) n ch 2 div(ˆv) L 2 ( ˆ) c div(v) L 2 () = c q L 2 (). 3.2 THE GLOBAL FINITE ELEMENT SPACES In this section, we define the global finite element spaces and derive the inf sup stability condition needed for the well posedness of the discrete problem. By Lemma 19 and Lemma 20, the induced global finite element spaces are given by V h = {v H 1 (Ω) : v Λ v i 3 () : h, C 0 across (n 2) dimensional faces}, x i W h = {q L 2 (Ω) : q Λ 2 () : h, q C 0 across (n 2) dimensional faces}. 38

48 Lemma 22. For any q W h, there exists v 1 V h such that (q div(v 1 )) W h,0 () for all h. Moreover, v 1 H 1 (Ω) c q L 2 (Ω). Proof. Let ω H 1 (Ω) satisfy div(ω) = q and ω H 1 (Ω) c q L 2 (Ω)[24]. We denote by I h ω the Scott-Zhang interpolant [42] of ω such that I h ω Λ 1 (). Then, the error of the interpolant in H m -norm satisfies ω I h ω H m () ch 2 m ω H 1 (w ), m = 0, 1, where w denotes the patch of elements that touch. Then, we define v 1 V h uniquely by the following iterative process. At the vertices, we set S v 1 (S) = I h ω(s), Then, we build v 1 that satisfies v 1 n (j) S = I h ω n (j) S, S where 1 j n m 1 and t (i) S S v (i) 1 x i (S) = q(s) n, for S 0, 1 i n. (3.2.1) v 1 n (j) n (j) S = 1 (q n m 1 S S m+1 i=1 v 1 t (i) t (i) S ), S m+1, S denotes the unit tangent vector to the surface S with 1 i m + 1. Repeating this process up to m = n 3, we deduce that v 1 n (j) S = I h ω n (j) S, div(v 1 ) = q, S S, 0 s n 2. (3.2.2) S S S After that, we impose the last set of conditions on v 1 : v 1 n S = ω n S, S n 1, (3.2.3) S S v 1 κ = I h ω κ, κ Λ 1 (), h. (3.2.4) By Lemma 19, (3.2.1) (3.2.4) uniquely define v 1 V h. Moreover, by the second identity in (3.2.2), we have div(v 1 ) = q on (n 2)-dimensional faces. Also, by the Stokes theorem and (3.2.3), we have div(v 1 ) = S div(ω) = q, h. (3.2.5) As a result, (q div(v 1 )) W h,0 () for all h. It remains to derive the stability estimate v 1 H 1 (Ω) q L 2 (Ω). Since Sv unisolves Λ 3 () and (v 1 I h ω) Λ 3 (), we may write n 2 v 1 I h ω 2 H 1 () n s h n 2s ( v 1 n (j) s=0 S s() j=1 S n (j) S S + (v 1 I h ω) n S 2 S n 1 () h n S I hω n (j) n (j) S ) 2 S 39

49 Then, by scaling arguments, we have n v 1 I h ω 2 H 1 () h n 2s n 2 q 2 + s=0 S S s() n 2 s + h n 2s S s=0 i=1 n s s=0 S s() j=1 v 1 t (i) t (i) S 2 + S S n 1 h n h n 2s I h ω n (j) S n (j) S 2 S For a face S s () with 1 s n 2 and unit tangent vector t (i) S S (ω I h ω) n S 2. (3.2.6) with 1 i s, let S 1, S 2 s 1 () be the unique (s 1) dimensional faces such that t (i) S = ±n(j 1) S 1 t (i) S = ±n(j 2) S 2 for some 1 j 1, j 2 n s+1. Then, by the Fundamental Theorem of Calculus and (3.2.2), we have v 1 t (i) S t (i) S = ± v 1 n (j 1) S 1 ± v 1 n (j 2) S 2 S S 1 S 2 = ± I h ω n (j 1) S 1 ± I h ω n (j 2) S 2 = S 1 S 2 By applying (3.2.7) to (3.2.6) and scaling, we obtain n v 1 I h w 2 H 1 () h n 2s n 2 n s q 2 + s=0 S S s() s=0 S s() j=1 n 2 s + h n 2s I h ω t (i) s=0 i=1 S t (i) S 2 + S S n 1 () S I h ω t (i) S and t (i) S. (3.2.7) h n 2s I h ω n (j) S n (j) S 2 S h n S (ω I h ω) n S 2 c( q 2 L 2 () + I hω 2 H 1 () + h 1 ω I h ω 2 L 2 ( ) c ω 2 H 1 (ω ). (3.2.8) Finally, by the triangle inequality, the stability of the interpolant and summing over h, we deduce that v 1 H 1 (Ω) c( I h ω H 1 (Ω) + ω H 1 (Ω)) c ω H 1 (Ω) c q L 2 (Ω). Theorem 8. For any q W h, there exists v V h such that div(v) = q and v H 1 (Ω) c q L 2 (Ω). As a result, we derive the inf sup stability condition q div(v) Ω v H 1 (Ω) sup v V h \{0} c q L 2 (Ω), q W h. Proof. By Lemma 22, for any q W h, there exists v 1 V h that satisfies (q div(v 1 )) W h,0 () for all h and v 1 H 1 () c q L 2 (). Therefore, by Theorem 7, for every h, there exists v 2, V h,0 () that satisfies div(v 2, ) = (q div(v 1 )) and v 2, H 1 () 40

50 c q div(v 1 ) L 2 (). We define v 2 such that v 2 = v 2,, and set v := v 1 +v 2. Note that since v 2 V h,0 () for all h, v 2 H 1 0() and this implies that v 2 S = 0 on all S s with 0 s n 2. Thus, v V h with div(v) = div(v 1 )+div(v 2 ) = q. As a result, we obtain the stability estimate v H 1 (Ω) v 1 H 1 (Ω)+ v 2 H 1 (Ω) c( v 1 H 1 (Ω)+ q div(v 1 ) H 1 (Ω)) c q L 2 (Ω). 3.3 THE GLOBAL FINITE ELEMENT SPACES WITH IMPOSED BOUNDARY CONDITIONS Imposing boundary conditions on finite element spaces while preserving the surjectivity of the divergence is a significant matter [22]. For instance, if v is a globally continuous function on Ω and vanishes on Ω, then the derivatives of v vanish at the corners (if n = 2) and the edges (if n = 3) of Ω. Thus, the divergence operator is not surjective from V h H 1 0(Ω) to W h L 2 0(Ω), and this violates the inf sup condition. On simplicial meshes, by imposing mesh and regularity conditions locally on the boundary, this issue may be eased [22]. However, on cubical meshes, such procedures are not applicable. For instance, in two dimensions, there will always be elements in h that have at least two boundary edges. Therefore, we impose the weaker boundary condition v n = 0 on the velocity finite element space and impose the tangential boundary condition weakly in the finite element method via the Nitsche s method [37] as we did in Chapter 2. As a result, we specify the boundary conditions on the global finite element spaces as follows: V h,0 = {v V h : (v n) Ω = 0}, W h,0 = W h L 2 0(Ω). 3.4 INF SUP STABILITY In this section, we define a discrete norm on V h,0 and prove the inf-sup stability condition. We denote by l s the set of open s-dimensional faces that do not intersect with Ω and set 41

51 b s := s \ l s. Then, we define the discrete H 1 type norm as follows: v 2 h := v 2 L 2 (Ω) + (h S v 2 L n 2 (S) + 1 v 2 L S h 2 (S)). (3.4.1) S Theorem 9. The inf sup condition c q L 2 (Ω) S b n 1 sup v V h,0 \{0} holds for a constant c > 0 independent of h. Ω div(v)q v h, q W h,0, (3.4.2) Proof. As in the proof of Theorem 8, for any q W h,0 given, we use the degrees of freedom of V h to construct v V h,0 that satisfies div(v) = q and v H 1 (Ω) c q L 2 (Ω). By Theorem 1, for any q W h,0 L 2 0(Ω), there exists ω H 1 0(Ω) that satisfies div(ω) = q and ω H 1 (Ω) c q L 2 (Ω). We set I h ω as the Scott-Zhang interpolant [41] of ω that satisfies I h ω Λ 1 () on each h. Note that I h ω H 1 0(Ω), in particular, I h ω S = 0 for all boundary faces S b s where 0 s n 1. Then, by repeating the procedure followed in the proof of Lemma 22, we determine v 1 V h uniquely via the conditions (3.2.1)-(3.2.4). This construction yields (q div(v 1 )) W h,0 () for all h and v 1 H 1 (Ω) c q L 2 (Ω). Furthermore, we let S b n 1, and denote by s (S) with 0 s n 1 the set of s dimensional subfaces of S. Then, for a subface S s (S), we have S b s. Moreover, the (outward) unit normal of S is an (outward) unit normal of S (up-to-sign), i.e., n S = ±n (j) S, for some j {1, 2,..., n s}. Therefore, by (3.2.1)-(3.2.4), we have v 1 n S = I h ω n S = 0, S s (S), 0 s n 1. S S Since (v 1 n S ) Λ 2 (S) on S, Lemma 17 implies that v 1 n S = 0 on S b n 1. Thus, v 1 V h,0. To validate the stability estimate v 1 h c q L 2 (Ω), we set ˆv(ˆx) = v(x), where x = F(ˆx) and F : ˆ denotes an affine transformation for h. Then, by scaling, we have (v 1 I h ω) 2 L 2 () + + Ŝ n 1 ( ˆ) S b n 1 n 1() ˆv 1 Îhω 2 L 2 (Ŝ) ) 1 ( v 1 I h ω L h 2 (S) Ch n 2 ˆ (ˆv 1 Îhω) 2 L 2 ( ˆ) S Ch n 2 ˆv 1 Îhω 2 H 1 ( ˆ) C v 1 I h ω 2 H 1 () (3.4.3) 42

52 Therefore, since I h ω Ω = 0, (3.2.8) yields v 1 2 L 2 () + S n 1 () b n 1 1 h S v 1 2 L 2 (S) c ω 2 H 1 (ω ). Finally, summing over h, we obtain v 1 2 L 2 () + 1 v 1 2 L h 2 (S) c ω 2 H 1 (Ω) c q L 2 (Ω). S h S b n 1 By scaling and the equivalence of norms in finite dimensions, this implies that v 1 h c q L 2 (Ω). Then, we consider v 2, V h,0 () that satisfies div(v 2, ) = (q div(v 1 )) and define v 2 V h such that v 2 = v 2, on each h. Note that v 2 V h H 1 0(Ω), therefore, v 2 V h,0. Then, setting v := v 1 + v 2 V h,0 and using the properties of v 1 and v 2, we deduce that div(v) = q and v H 1 (Ω) c q L 2 (Ω). 3.5 THE REDUCED ELEMENTS WITH CONTINUOUS PRESSURE APPROXIMATIONS In this section, we reduce the global degrees of freedom of our finite element pair by restricting the range of the divergence operator. In forming the reduced finite element spaces, we denote by B s () Λ 2 () the space spanned by the bubble functions associated with s (), i.e., B s () = < b S >, S s() where < b S > denotes the span of b S. By Lemma 17, we have n Λ 2 () = Λ 1 () B s (). Then, we define the reduced local pressure space W r,h () by removing the (n 1) dimensional face bubbles of Λ 2 () as follows: W r,h () := Λ 1 () ( n 2 s=1 s=1 ) B s () B n (). Note that in two dimensions, W r.h () is the space of bilinear polynomials enriched with face bubbles, whereas in three dimensions, W r,h () is the space of trilinear polynomials enriched with edge and volume bubbles. By Lemma 17, the dimension of W r,h () is (3 n 2n), and a function q W r,h () is uniquely 43

53 determined by the set of values { q : S s (), s = 0, 1,..., n 2, n}. We define the reduced local velocity element as S V r,h () := {v Λ 3 () : div(v) W r,h ()}. (3.5.1) Lemma 23. A function v V r,h () is uniquely determined by the set of values (See Figure 3.1). Sv r = { S where 1 i n. v (i) v (i), S x i : S (i) s (), 0 s n 2, S v (i) : S (i) n 1()}, Proof. Note that the number of constraints imposed on Λ 3 () in the definition of the space V r,h () is 2n, therefore, dim(v r,h ()) = dim(v r,h ()) 2n = 4n3 n 1 2n = S r v. Therefore, it suffices to show that if v V r,h () nullifies S r v, then v = 0 in. If v V r,h () Λ 3 () nullifies the functionals in S r v, then v H 1 0() by Lemma 19. Moreover, Lemma 19 also implies that div(v) S = 0 for all S s () with 0 s n 2. Since W r,h () Λ 2 (), this implies that div(v) = c b for some c R n. Then, integration by parts yields c b = div(v) = v n = 0 Thus, div(v) = 0 and therefore by Lemma 21, we have v = 0 in. Let H 1 (div; Ω) := {v H 1 (Ω) : div(v) H 1 (Ω)}. spaces as follows: Then, we define the reduced global V r,h := {v H 1 (div; Ω) V h : v V r,h (), (v n) Ω = 0}, W r,h := {q H 1 (Ω) L 2 0(Ω) : q W r,h ()}. Theorem 10. For every q W r,h there exists v V r,h that satisfies div(v) = q and v H 1 (Ω) C q L 2 (Ω). Moreover, div(v r,h ) = W r,h. Therefore, the reduced pair V r,h W r,h is inf sup stable. Proof. By Theorem 9, for q W r,h W h,0 given, there exists v V h,0 such that div(v) = q and v H 1 (Ω) C q L 2 (Ω). Since q W r,h, v V r,h () on all h, and div(v) H 1 (Ω). Therefore, v V r,h. This implies that W r,h div(v r,h ). Thus, the second assertion follows from the inclusion div(v r,h ) W r,h. 44

54 3.6 CONVERGENCE ANALYSIS Let X h,0 Y h,0 denote either the pair V h,0 W h,0 or the reduced pair V r,h W r,h. Then, the finite element method reads: Find (u h, p h ) X h,0 Y h,0 that satisfies a h (u h, v) div(v) p h = f v, v X h,0 Ω Ω div(u h ) q = 0, q Y h,0, Ω where the bilinear form a h (, ) : H 1 (Ω) H 1 (Ω) R is defined as a h (v, ω) = ν v : ω ν ( v ω + ω v σ v ω), n S n S h S Ω S b n 1 where σ > 0 is an h independent penalty parameter. S (3.6.1) Theorem 11. There exists a unique pair (u h, p h ) X h,0 Y h,0 satisfying (3.6.1). Moreover, for 1 s 2, there holds u u h h ch s u H s+1 (Ω), p p h L 2 (Ω) ch s ( p H s+1 (Ω) + ν u H s+1 (Ω)). where the constant C > 0 is independent of h or the viscosity ν. Proof. Following the same procedure described in Section 2.4, it is easy to show that if σ is sufficiently large, then a h (, ) is coercive on X h,0 with respect to the discrete H 1 type norm given by (3.4.1). In addition, a h (, ) is continuous on (H 2 (Ω) + V h,0 ). Thus, there exists a unique (u h, p h ) pair that satisfies (3.6.1) by Theorem 9, Theorem 10 and standard theory [9, 24]. Moreover, the velocity approximation u h is independent of the choice of the finite element space X h,0 = V h,0 or X h,0 = V r,h since the kernel of the divergence operator acting on each of these spaces is Z h = {v X h,0 : div(v) = 0}. Restricting (3.6.1) to the divergence free space Z h, and using the consistency of the bilinear form a h (, ), we deduce by Cea s Lemma, u u h h c inf v Z h u v h provided u H s (Ω) for s > 3/2. As in Chapter 2, we estimate the approximation properties of Z h by following the arguments given in Theorem in [10]. To this end, for an arbitrary function v X h,0, we let ω X h,0 satisfy div(ω) = div(v) Y h,0, therefore, ω + v Z h. Moreover, ω h c div(v) L 2 (Ω) = div(u v) L 2 (Ω) c u v h. This yields u (ω + v) h u v h + ω h c u v h. As a result, u u h h c inf u v h c inf u v h. v Z h v X h,0 45

55 Thus, the velocity error estimate is decoupled from the pressure error estimate and independent of the viscosity. By standard approximation theory and scaling arguments, we derive the estimate u u h h c inf u v h ch s u H v X s+1 (Ω) h,0 for 1 s 2. Note that since u h is independent of the choice of the global velocity finite element space, u u h h satisfies the same estimate in both of the cases X h,0 = V r,h and X h,0 = V h,0. To derive an error estimate on the pressure solution, we define the L 2 projection P h : L 2 (Ω) Y h,0. Since div(x h,0 ) = Y h,0, it follows from Theorem 9, Theorem 10 and the properties of P h that c p h P h p L 2 (Ω) Therefore, we have for 1 s 2. sup v V h,0 \{0} Ω div(v)(p h P h p) v h = sup v V h,0 \{0} a h (u u h, v) = sup cν u u h h. v V h,0 \{0} v h Ω div(v)(p h p) v h p p h L 2 (Ω) p P h p L 2 (Ω) + cν u u h h ch s ( p H s (Ω) + ν u H s+1 (Ω)) Remark 5. The velocity error estimate has optimal order of convergence whereas the pressure error estimate is of optimal order provided Y h,0 = W r,h. 46

56 4.0 FEM FOR THE STOKES PROBLEM ON UADRILATERAL MESHES In Chapter 2, we propose a finite element method that is based on building solution spaces on rectangular meshes. However, the domains that could be considered in that framework are limited since we assume that the boundaries of both the domain and the mesh elements are parallel to the coordinate axes. In this chapter, we propose a method that gives a pair of stable and conforming finite element spaces yielding pointwise divergence free velocity approximations on general shape regular quadrilateral meshes [36]. Conforming finite element pairs that yield divergence free approximations tend to be high-order or conforming and stable only on certain meshes. Moreover, the construction of conforming finite elements that yield divergence free approximations are not extended to general convex quadrilaterals defined by bilinear mappings. In this chapter, we address some of these issues by introducing a low-order, conforming and stable finite element pair yielding divergence free approximations on general shape regular quadrilateral meshes. First, we define the H 2 finite element space Σ h as the de Veubeke Sanders macro element, which is a globally C 1 piecewise cubic spline [16, 14, 28, 39]. Then, via the subcomplex given by (1.3.2), we induce a piecewise quadratic macro velocity space and a piecewise constant pressure space. The dimension of our global spaces is comparable to the lowest order serendipity Taylor Hood spaces [43]. Our analysis shows that the velocity error is decoupled from the pressure error and a locally computed, post processed pressure solution has the same rate of convergence as the velocity solution. A nonconforming finite element method that enforce the divergence free constraint pointwise on each quadrilateral element has recently been done in [48]. The method proposed there is low-order and applicable to convex quadrilaterals. However, the error estimates are 47

57 still coupled with a negative scaling of the viscosity as a consequence of its nonconformity. Macro elements have recently been used on simplicial partitions in [1, 12]. Here, we present a study that complements and extends these results to quadrilateral meshes. 4.1 THE UADRILATERAL MESH In this chapter, h denotes a shape-regular quadrilateral partition of Ω R 2 consisting of convex quadrilateral elements. For each quadrilateral element h, Kr = {Ki }4 i=1 denotes the triangular partition of obtained by drawing in the two diagonals of. In addition, c denotes the point of intersection of these two diagonals. We assume that the triangular elements in the partition Kr are ordered in a counterclockwise fashion (See Figure 4.1). As in the previous chapters, the sets of vertices and boundary vertices are given by V and V b, respectively, and the set of vertices (resp. edges) of an element is given by V (resp. E ). Moreover, for a vertex a V, a (resp. E a ) stands for the set of quadrilaterals (resp. edges) that share a. Furthermore, we let Ea b denote the set of boundary edges in E a. The space of piecewise polynomials with respect to the triangular partition of is given by P k (Kr ). For example, p P k (Kr ) if and only if p K i P k (Ki ) for every K i Kr. We set P k (Kr ) := 4 P k (Ki ) and P k( h ) := P k (). i=1 h Let ± h be two elements in the mesh sharing an edge e = +. For a function v, we set v ± = v ±. Assuming that the global labeling number of + is smaller than that of, we define the jump of a scalar or a vector-valued function v as [v] e := v + v. For a boundary edge e = + Ω, we set [v] e = v +. Additionally, we define the local L 2 projection P h, : L 2 () P 1 () and introduce P h : L 2 (Ω) P 1 ( h ) via the relation P h = P h, for every h. By definition, P h, satisfies P h, v ω = v ω, for all ω P 1 (). For v H s () and l = min{2, s}, the L 2 projection P h, satisfies [18] v P h, v L 2 () ch l v H l (). (4.1.1) 48

58 Lemma 24. The local L 2 projection P h, : L 2 () P 1 () satisfies (v P h, v) L 2 () Ch s 1 v H s (), v H s (), s = 1, 2. (4.1.2) Proof. By the triangle inequality and inverse estimates, for an arbitrary ω P 1 (), there holds (v P h, v) L 2 () (v ω) L 2 () + Ch 1 ω P h, v L 2 () (v ω) L 2 () + Ch 1 ( ) v ω L 2 () + v P h, v L 2 (). Let w be an interpolant of v. Then, standard approximation theory [18, 13] and (4.1.1) yield (4.1.2). Lemma 25. P h is piecewise H 1 stable on H 1 (Ω). Proof. By the definition of the H 1 norm and the L 2 projection, we have P h, v 2 H 1 () = P h,v 2 L 2 () + P h,v 2 H 1 (), (4.1.3) P h, v 2 L 2 () v 2 L 2 (). (4.1.4) In addition, using the triangle inequality and Lemma 24 with s = 1, we obtain P h, v 2 H 1 () ( P h,v v H 1 () + v H 1 ()) 2 c( P h, v v 2 H 1 () + v 2 H 1 () ), c v 2 H 1 (). (4.1.5) Combining (4.1.3), (4.1.4) with (4.1.5) yields P h, v H 1 () c v H 1 (). (4.1.6) Summing over h, we deduce the H 1 stability condition P h v H 1 (Ω) c v H 1 (Ω). 4.2 THE LOCAL FINITE ELEMENT SPACES In building a divergence free conforming finite element pair, we utilize the local C 1 macro element constructed by de Veubeke and Sander [16, 39] Σ h () = {ψ P 3 (K r ) H 2 ()} = {ψ P 3 (K r ) C 1 ()}. Lemma 26. The dimension of Σ h () is 16, and its degrees of freedom are given by S Σ = {D α ψ ψ(a) : α 1, a V, : e E }, (4.2.1) e n e where n e denotes the (outward) unit normal of e (See Figure 4.1). 49

59 a 4 a 3 K 3 K 2 c K 1 K 4 a 1 a 2 Figure 4.1: Degrees of freedom of the macro element Σ h (). Solid circles indicate function evaluations, larger circles indicate gradient evaluations and the lines indicate the means of the normal derivative. Proof. First, we note that the cardinality of S Σ is 28 and the dimension of P 3 (Ki ) is 10 for every 1 i 4, therefore, the dimension of P 3 (Kr ) is 40. Since the point c is a singular vertex with respect to the partition Kr, C 1 continuity at this point imposes 8 constraints on P 3 (Kr ) [34]. In addition, C 1 continuity imposed at the interior edge midpoints yield 4 more constraints on P 3 (Kr ). As a result, S Σ and the C 1 continuity constraints provide 40 equations in total. Hence, it suffices to show that φ Σ h () nullifies S Σ if and only if φ = 0. Let µ P 1 (Kr ) H0() 1 be the unique continuous, piecewise linear polynomial that takes the value one at c. If φ Σ h () vanishes on S Σ, then φ = µ 2 p for some p P 1 (Kr ) H 1 (). Let µ i, p i P 1 (Ki ) denote the restrictions of µ and p to K i. Denote by l i = K i K i+1 the interior edge shared by the triangles K i and K i+1. Then, the C 1 continuity of φ yields φ li = 2µp µ i + µ 2 p i = 2µp µ i+1 + µ 2 p i+1. Note that µ = 0 at the vertices of and µ i is parallel to the normal direction of the edge K i, especially, µ i µ i+1. Then, ( 2p (µ i µ i+1 ) + µ (p i p i+1 ) ) li = 0 implies that p vanishes at the vertices of, therefore, p = cµ for some c R. Thus, φ = cµ 3. 50

60 However, since µ 3 C 1 (), we have c = 0. As a result, φ = 0. Alternative proofs of Lemma 26 are provided in [16, 39]. Corollary 2. There holds Σ h () H0() 2 = {0}. Proof. If φ Σ h () H0(), 2 then φ vanishes on the degrees of freedom. By Lemma 26, we deduce that φ = 0. The smooth de Rham complex and the local space Σ h () suggests that a natural candidate for the local Stokes pair is (P 2 (Kr ) H 1 ()) P 1 (Kr ). Indeed, it is clear that div(p 2 (Kr ) H 1 ()) P 1 (Kr ), and curl(σ h ()) P 2 (Kr ) H 1 (). However, the dimension arguments we state in Lemma 27 below show that (P 2 (Kr ) H 1 ()) P 1 (Kr ) is not inf-sup stable, therefore, the global pair induced by this local pair is also not inf-sup stable. Lemma 27. The space div(p 2 (Kr ) H 1 ()) has dimension 11.Therefore, div : P 2 (Kr ) H 1 () P 1 (Kr ) is not surjective. Proof. Let Z denote the kernel of the divergence operator acting on P 2 (Kr ) H 1 (), i.e., Z = {v P 2 (Kr ) H 1 () : div(v) = 0}. If v Z, then v = curl φ for some φ H 2 (). Since v is a piecewise quadratic polynomial, φ is a piecewise cubic polynomial. As a result, φ Σ h (). Therefore, Z = curl(σ h ()). Then, the rank nullity theorem and Lemma 26 yield dim(div(p 2 (Kr ) H 1 ()) = dim(p 2 (Kr ) H 1 ()) dim(z ) = dim(p 2 (Kr ) H 1 ()) dim(curl(σ h ())) = dim(p 2 (Kr ) H 1 ()) dim(σ h ()) + 1 = 2 dim(p 2 (Kr ) H 1 ()) = 11. However, dim(p 1 (Kr )) = 12 > dim(div(p 2 (Kr ) H 1 ()). This implies that there exists q P 1 (Kr ) \ {0} such that div(v) q = 0 for all v P K 2(Kr ) H 1 (). Therefore, the divergence operator mapping P 2 (Kr ) H 1 () to P 1 (Kr ) is not surjective and the pressure solution is not unique. Remark 6. (Proposition 2.1 in [44]) The range of the divergence operator is characterized as div(p 2 (Kr ) H 1 ()) = {q P 1 (Kr ) : 51 4 ( 1) i q Ki (c K ) = 0}. i=1

61 Lemma 27 implies that the dimension of the candidate pressure space P 1 (Kr ) is larger than the dimension of the range of the divergence operator acting on the candidate velocity space P 2 (Kr ) H 1 (). Therefore, we restrict the range of the divergence operator by defining the velocity space as V h () := {v P 2 (Kr ) H 1 () : div(v) P 0 ()}. Lemma 28. The dimension of V h () is 16, and a function v V h () is uniquely determined by the values (See Figure 4.2) S v = {v(a) : a V, v : e E }. (4.2.2) Proof. It is easy to see that the kernel of the divergence operator acting on V h () satisfies Ker(div(V h ()) = curl(σ h ()). Therefore, by the rank nullity theorem, we have dim V h () = dim(curl(σ h ())) + dim P 0 () = dim Σ h () 1 + dim P 0 () = 16. Since the cardinality of S v is also 16, it suffices to show that if v V h () vanishes on S v, then v = 0 in. If v V h () vanishes on S v, then v V h () H 1 0() since v is a piecewise quadratic polynomial. Therefore, by the divergence theorem, we have div(v) = v n = 0. Since div(v) P 0 (), this result implies that v is divergence free. Thus, v = curl φ for some φ Σ h (), and since v H 1 0(), we may assume that φ Σ h () H0(). 2 However, Corollary 2 implies that φ = 0 in, therefore, v = 0 in. We define the local pressure space as W h () := P 0 () to ensure that the velocity approximations are pointwise divergence free. e 4.3 THE GLOBAL FINITE ELEMENTS WITH IMPOSED BOUNDARY CONDITIONS The local spaces defined in Section 4.2 induce the following global finite element spaces: Σ h := {z H0(Ω) 2 : z Σ h (), h }, V h := {v H 1 0(Ω) : v V h (), h }, W h := {q L 2 0(Ω) : q K W h (), h }. 52

62 Figure 4.2: Degrees of freedom of the macro velocity element V h (). Solid circles indicate function evaluations. Lemma 29. The sequence stated in (1.3.2) is exact provided Ω is a simply connected domain. Proof. Clearly, the definitions of the finite element spaces imply that Ker(div(V h )) = curl(σ h ). We denote by V, E and the number of interior vertices, interior edges and quadrilaterals in the mesh. Then, by Lemma 26, dim(σ h ) = 3 V + E, and by Lemma 28, dim(v h ) = 2 V + 2 E. Therefore, the rank nullity theorem yields dim(div(v h )) = dim(v h ) dim(curl(σ h )) = dim(v h ) dim(σ h ) = E V. Moreover, by the Euler identity [28], we have E = + V 1, and the definition of W h implies that dim(w h ) = 1. Thus, we obtain dim(div(v h )) = dim(w h ). Since div(v h ) W h, this implies that div(v h ) = W h. As a result, the complex (1.3.2) is exact. 4.4 INF-SUP STABILITY In this section, we carry out the stability analysis and verify the well posedness of the discrete problem and the uniqueness of the numerical solution. We show that the finite element pair V h W h forms an inf sup stable finite element pair, therefore, the proposed method yields 53

63 a unique solution. For h, we define v 2 h, := v(a) 2 + a V Lemma 30. h, is a H 1 type norm on V h (). e E h 2 e e v 2. (4.4.1) Proof. The result follows by the unisolvency of S v over V h () (See Lemma 28). Lemma 31. Let P W : L 2 0(Ω) W h denote the L 2 projection onto W h. Then, there exists a Fortin operator Π h : H 1 0(Ω) V h such that div(π h v) = P W div(v) and Π h v H 1 (Ω) C v H 1 (Ω) for all v H 1 0(Ω). Consequently, V h W h forms an inf sup stable finite element pair. Moreover, the error of the Fortin interpolant satisfies Π h v v H 1 (Ω) Ch v H 2 (Ω). (4.4.2) Proof. Define Π h : H 1 0(Ω) V h such that it satisfies the following conditions: Π h v = v, e E, (4.4.3a) e Π h (v)(a) = 1 a e a P h, v(a), a V \ V b, (4.4.3b) Π h (v)(a) = 0, a V b, (4.4.3c) where a denotes the cardinality of the set a. To prove the inf sup stability, it suffices to show that Π h is a Fortin operator, that is, it suffices to verify that Π h satisfies [9]: div(v) = Ω Ω div(π h v), v H 1 0(Ω), (4.4.4a) Π h v H 1 (Ω) c v H 1 (Ω), v H 1 0(Ω). (4.4.4b) By the divergence theorem and (4.4.3a), div(π h v) = (Π h v n) = (v n) = Thus, (4.4.4a) holds. This implies that div(π h v) = P W div(v). By the triangle inequality and (4.1.6), we have Π h v 2 H 1 (Ω) c div(v), h. ( Π h v P h, v 2 H 1 () + P h,v 2 H 1 () ) h c ( Π h v P h, v 2 H 1 () + v 2 H 1 ()). (4.4.5) h 54

64 Since P 1 () V h (), Lemma 30 implies Π h v P h v 2 H 1 () Π hv P h v 2 h, = Π h v(a) P h, v(a) 2 + h 2 e (Π h v P h, v) 2. (4.4.6) a V e E e For a V \ V b, applying the triangle inequality to the first expression in (4.4.6) and using (4.4.3b), we obtain Π h v(a) P h, v(a) 2 1 = (P h, v(a) P h, v(a)) 2 a a C P a 2 h, v(a) P h, v(a) 2, (4.4.7) a where C > 0 depends only on the shape regularity of the mesh. Then, for, a, there exists { i } m i=0 a such that i and i+1 share a common edge. Setting 0 =, m =, and using the inverse inequality, we obtain m P h, v(a) P h, v(a) 2 P h,i+1 v(a) P h,i v(a) 2 i=0 e E a [P h v] 2 L (e) e E a h 1 e [P h v] 2 L 2 (e), (4.4.8) where [P h v] e = (P h,i+1 v) e (P h,i v) e for e E a with e = i+1 i. We note that [P h v] 2 L 2 (e) h 1 e [P h v v] 2 L 2 (e). (4.4.9) e E a h 1 e e E a Finally, combining (4.4.7)-(4.4.9), we obtain Π h v(a) P h, v(a) 2 C e E a h 1 e v P h, v 2 L 2 (e). (4.4.10) For a V b, since Π h v(a) = 0 at all a V b, (4.4.8) and (4.4.9) yield Π h v(a) P h, v(a) 2 = P h, v(a) 2 e E b a h 1 e P h, v 2 L 2 (e), h 1 e v P h, v 2 L 2 (e). (4.4.11) e Ea b Combining (4.4.10)-(4.4.11) and using the trace inequality, for a V, we obtain Π h v(a) P h, v(a) 2 C ( v P h, v 2 H 1 () + h 2 v P h, v 2 L 2 ()). (4.4.12) a 55

65 Then, (4.1.1) and (4.1.2) imply Π h v(a) P h, v(a) 2 C v 2 H 1 (ω ), (4.4.13) a V where ω =. Now consider the second expression in (4.4.6). a V a By (4.4.3a), the Cauchy-Schwarz, the trace inequality, (4.1.1) and (4.1.2), respectively, we have e E a h 2 e e (Π h v P h v) 2 = e E a h 2 e (v P h v) 2 h 1 e v P h v 2 L 2 (e) e E a e c a ( (v P h, v) 2 L 2 () + h 2 v P h, v 2 L 2 () ) c v 2 H 1 (ω ). (4.4.14) Using (4.4.13) and (4.4.14) in (4.4.6) yields Π h v P h v 2 H 1 () C v 2 H 1 (ω ). (4.4.15) Consequently, summing over h, then using (4.4.5), we deduce that Π h v H 1 (Ω) c v H 1 (Ω). Thus, (4.4.4b) holds. As a result, Π h is a Fortin operator. It suffices to prove the estimate (4.4.2) to complete the proof. For v H 2 (), (4.1.2) and (4.1.1) yield v P h, v H 1 () ch v H 2 () (4.4.16a) v P h, v L 2 () ch 2 v H 2 () (4.4.16b) Using (4.4.16) in (4.4.12) and (4.4.14), then summing (4.4.6) over h, we obtain Π h v P h v H 1 (Ω) ch v H 2 (Ω). (4.4.17) Then, by the triangle inequality and scaling, we have Π h v v 2 H 1 (Ω) c( Π hv P h v 2 H 1 (Ω) + P hv v 2 H 1 (Ω) ) c( Π h v P h v 2 H 1 (Ω) + P hv v 2 H 1 (Ω) ) (4.4.18) Therefore, using (4.1.2) and (4.4.17) in (4.4.18) yield Π h v v H 1 (Ω) ch v H 2 (Ω). 56

66 4.5 CONVERGENCE ANALYSIS Recall the weak formulation of the Stokes system (1.0.1) ν u : v div(v) p = f v, div(u) q = 0, v H 1 0(Ω), q L 2 0(Ω), (4.5.1) and the discretized version of (4.5.1) given by ν u h : v h div(v h ) p h = f v h, v h V h, div(u h ) q h = 0, q h W h, (4.5.2) As a consequence of Lemma 31, the discrete problem (4.5.2) is well-posed. We define the continuous and the discrete divergence free spaces Z = {v H 1 0(Ω) : div(v) = 0}. Z h = {v V h : div(v) = 0}. (4.5.3a) (4.5.3b) Restricting the domains of the velocity spaces in (4.5.1) and (4.5.2) to Z and Z h reduces the systems of equations given by (4.5.1) and (4.5.2) to the equations (4.5.4a) and (4.5.4b) as follows: ν u : v = f v, v Z, (4.5.4a) ν u h : v h = f v h, v h Z h. (4.5.4b) Theorem 12. Suppose that (u, p) solves (4.5.4a) and (u h, p h ) solves (4.5.4b), then for a convex polygonal domain Ω, the error estimates satisfy the following: u u h H 1 (Ω) ch s u H s+1 (Ω), u u h L 2 (Ω) ch s+1 u H s+1 (Ω), p p h L 2 (Ω) ch( p H 1 (Ω) + νh s 1 u H s+1), for s = 1, 2, where the constant c > 0 is independent of the discretization parameter and the viscosity. Proof. Since Z h is divergence free conforming, by Cea s Lemma, we have (u u h ) L 2 (Ω) inf (u v h ) L v h Z 2 (Ω). (4.5.5) h Additionally, since Z h = curl(σ h ), there exists σ h Σ h for every v h Z h such that curl(σ h ) = v h. Thus, (4.5.5) is equivalent to (u u h ) L 2 (Ω) inf σ h Σ h (u curl(σ h )) L 2 (Ω). (4.5.6) 57

67 Writing u in terms of its stream function ψ h H 2 0(Ω), we have u = curl(ψ h ). Then, (4.5.6) becomes (u u h ) L 2 (Ω) inf (curl(ψ h ) curl(σ h )) L σ h Σ 2 (Ω) = inf ψ h σ h H h σ h Σ 2 (Ω). (4.5.7) h By Theorem 6.18 in [28], for every ψ h H m+1 (Ω), there exists σ h Σ h that satisfies D α (σ h ψ h ) L 2 (Ω) ch m+1 α ψ h H m+1 (Ω), 0 m 3, 0 α m, (4.5.8) where c is an h independent constant. Since (4.5.7) implies that α = 2, m 2 in (4.5.8), and we may rewrite (4.5.7) as (u u h ) L 2 (Ω) inf ψ h σ h H σ h Σ 2 (Ω) ch s ψ h H s+2 (Ω) h (4.5.9) = ch s u H s+1 (Ω), s = 1, 2. To derive an error estimate for the velocity solution in the L 2 -norm, we use the Aubin Nitsche duality technique [10]. Setting e = u u h and letting γ H 2 (Ω) be the solution of the dual problem on the convex polygonal domain Ω corresponding to (4.5.4b), we obtain Since Π h γ Z h Z, (4.5.4) implies ν( e, γ) = ν(e, γ) = (e, e). (4.5.10) ν( e, Π h γ) = 0. (4.5.11) Therefore, we have ν( e, γ) = ν( e, (γ Π h γ)). (4.5.12) By (4.4.2) and (4.5.9) (4.5.12), we have e 2 L 2 (Ω) e H 1 (Ω) γ Π h γ H 1 (Ω) c e H 1 (Ω)h γ H 2 (Ω). Since Ω is convex [9], the regularity estimate γ H 2 (Ω) c e L 2 (Ω) holds, and this yields e L 2 (Ω) ch e H 1 (Ω). Combining this result with (4.5.9), we obtain the L 2 error estimate u u h L 2 (Ω) ch s+1 u H s+1 (Ω) (4.5.13) In addition, by the inf sup condition and standard arguments [9], we have p h P W p L 2 (Ω) ν (u u h ) L 2 (Ω) Cνh s u H s+1 (Ω), s = 1, 2. (4.5.14) Then, the triangle inequality and the properties of the L 2 projection imply that the error in the pressure approximation satisfies p p h L 2 (Ω) p P W p L 2 (Ω) + p h P W p L 2 (Ω) ch( p H 1 (Ω) + νh s 1 u H s+1 (Ω)), s = 1, 2. 58

68 4.5.1 Convergence Analysis for the Post-Processed Pressure Solution Theorem 12 shows that the velocity and pressure approximations have different orders of convergence. In this section, we develop a locally computed, post processed pressure solution that has the same quadratic convergence as the velocity error by making use of the super convergence property (4.5.14). For h, we define p P 1() that satisfies p q = ν h u h q + f q, q P 1 (), (4.5.15a) p = p h, (4.5.15b) where the discrete laplacian operator h is defined piecewise with respect to, that is, h u h Ki = u h Ki for 1 i 4 and K i K r. Since p and q are both constant, (4.5.15a) yields p q = 1 (ν h u h + f) q, q P 1 (). (4.5.16) By letting q = x i, we derive a formula for the post-processed pressure solution p = c + x (ν h u h + f), (4.5.17) with the constant c chosen such that (4.5.15b) is satisfied. Theorem 13. Suppose that the solution of the weak problem (u, p) H 3 (Ω) H 2 (Ω). Let p P 1 ( h ) satisfy (4.5.15) on each h, i.e., p = p, where p (4.5.17). Then, there holds Proof. Since p, P W p L 2 0(Ω), we have p p L 2 (Ω) ch 2 (ν u H 3 (Ω) + p H 2 (Ω)). Ω (p P W p) = 0. is defined by Fix 1 h and choose h with = 1. Let q 1 = 1, q = 1, and q = 0 otherwise. Thus, q W h L 2 0(Ω), and by the definition of the L 2 projection, (p P W p) = (p P W p). 1 1 This identity holds for all h, and it follows that 0 = (p P W p) = (p P W p) Ω h 59

69 = h (p P W p) = Ω (p P W p) Thus, we have (p P W p) = 0 for all h. Then, applying Poincaré Friedrich s inequality on h, we obtain p p L 2 () c( p p L 2 () + h (p p ) L 2 ()), (4.5.18) where (p p ) denotes the mean of (p p ) over. The definition of P W p and (4.5.15b) imply Therefore, p p = p p = P W p p h = P W p p h. p p L 2 () = P W p p h L 2 () P W p p h L 2 (). (4.5.19) By the triangle inequality, (4.5.15a) and the Cauchy-Schwarz inequality, we obtain (p p ) 2 L 2 () = (p p ) (p q) + (p p ) (q p ) (4.5.20) = (p p ) (p q) + ν h (u u h ) (q p ) (p p ) L 2 () (p q) L 2 () + ν h (u u h ) L 2 () (q p ) L 2 (), for all q P 1 (). Again, by the triangle inequality, we have (q p ) L 2 () (q p) L 2 () + (p p ) L 2 () (4.5.21) Note that P W p P 0 () P 1 () and letting q = P W p in (4.5.21) yields: Using (4.5.22) in (4.5.20), we obtain (q p ) L 2 () (p p ) L 2 (). (4.5.22) (p p ) L 2 () c( (p q) L 2 () + ν h (u u h ) L 2 ()). (4.5.23) Combining (4.5.18), (4.5.19) and (4.5.23) results in p p L 2 () c( p h P W p L 2 () + νh h (u u h ) L 2 () + inf q P 1 () h (p q) L 2 ()). (4.5.24) In addition, by the Bramble-Hilbert lemma and the regularity of p, we have inf h (p q) L 2 () ch 2 p H 2 (). (4.5.25) q P 1 () Furthermore, by the inverse inequality and the triangle inequality, for any v P 2 (K r ) 60

70 H 1 (), there holds h h (u u h ) L 2 () h h (u v) L 2 () + c u h v H 1 () c(h h (u v) L 2 () + u v H 1 () + u u h H 1 ()). Let v be the nodal interpolant of u. Then, (4.5.26) yields (4.5.26) h (u u h ) L 2 () c(h 2 u H 3 (w()) + u u h H 1 ()), (4.5.27) where w() denotes the set of quadrilaterals that intersect with. Applying (4.5.27) and (4.5.25) to (4.5.24), we obtain p p L 2 () c( p h P W p L 2 () + ν u u h H 1 () + h 2 (ν u H 3 () + p H 2 ())). Summing over h and using (4.5.9), p p L 2 (Ω) c(ν u u h H 1 (Ω) + h 2 (ν u H 3 (Ω) + p H 2 (Ω))) ch 2 (ν u H 3 (Ω) + p H 2 (Ω)). 4.6 IMPLEMENTATION In this section, we describe how we build the velocity space V h () in detail. More precisely, we present the computation and the implementation of a basis of V h (). We note that, unlike the serendipity Taylor-Hood pair, our local velocity space is defined on a physical element of the mesh, and it is not invariant under bilinear maps. These restrictions may suggest that the basis must be solved locally on each quadrilateral h, and this leads to solving a (16 16) linear system for each element. Here, we discuss an alternative construction which is more efficient and may possibly be extended to isoparametric elements Construction of a Canonical Basis Our goal is to construct an affine, bijective map between a physical element and a reference element. In Chapter 2, we have used the unit square as the reference element. However, since there does not exist an affine map between a quadrilateral and the unit square, here we can not use the unit square as the reference element, therefore, we define a skewed version 61

71 of the unit square as the reference element and denote it by ˆÂ [23]. In the construction of ˆÂ, we employ three vertices of to build an affine map and then we use this affine map to identify  that characterizes ˆÂ. For a quadrilateral h, denote by T 1, T 2 the two triangles obtained by splitting from opposite vertices. Let A = (A 1, A 2 ) be the unique vertex of T 2 that is not a vertex of T 1 and denote by ˆT the reference triangle with the vertices (0, 0), (1, 0), (0, 1). Then, define an affine bijection F : ˆT T 1, and set ˆT 2 = F 1 (T 2 ) and  = (Â1, Â2) := F 1 (A). Moreover, set ˆT 1 := ˆT. Then, the union ˆT 1 ˆT 2 gives a convex quadrilateral with the vertices (0, 0), (1, 0), (0, 1) and (Â1, Â2) (See Figure 4.3). We define the skewed reference element as ˆÂ := ˆT 1 ˆT 2. We further note that F : ˆÂ is an affine, bijective transformation. Let V( ˆÂ) denote the velocity space defined on the reference quadrilateral ˆÂ. the set of vertices and edge midpoints of ˆÂ by {â j } 8 j=1 as shown in Figure 4.4. define {ˆv (k) i Denote Then, } V( ˆÂ) that satisfy ˆv (k) i (â j ) = δ i,j e k, where e 1 = (1, 0) t and e 2 = (0, 1) t, for 1 i, j 8 and 1 k 2. Thus, {ˆv (k) i } forms a canonical basis of V( ˆÂ). By using a modified Piola transform, we define the following velocity functions on : where Clearly, v (k) i β(k) 1 β (k) 2 v (k) i = B(β (k) 1 ˆv (1) i + β (k) 2 ˆv (2) i ), := B 1 e k, x = F(ˆx), ˆx ˆ. P 2 (K r ) H 1 (). Moreover, the divergence of v (k) i div(v (k) i ) = β (k) 1 divˆv (1) i + β (k) 2 divˆv (2) i. Furthermore, at a j = F(â j ) with â j ˆ, we have satisfies v (k) i (a j ) = B(β (k) 1 ˆv (1) i (â j ) + β (k) 2 ˆv (2) i (â j )) = B(β (k) 1 δ i,j e 1 + β (k) 2 δ i,j e 2 ) = δ i,j B = δ i,j e k. Thus, {v (k) i } is a canonical basis for V h (). β(k) 1 β (k) 2 62

72 A T 2 F (0, 1) F T 1 (0, 1) ˆT 2  ˆT (0, 0) (1, 0) ˆT 1 (0, 0) (1, 0) Figure 4.3: Construction of the skewed reference element. Affine transformation F mapping the reference element ˆÂ (right) to the physical element (middle) Derivation of the Reference Basis Functions In this section, we describe an efficient way of computing the basis on the skewed reference element ˆÂ. In the two diagonal split of ˆÂ, we let {ĉ j } 4 j=1 denote the midpoints of the interior edges and ĉ 5 be the point of intersection of the two diagonals (See Figure 4.4). Moreover, we let {ŵ i } 5 i=1 P 2 ( ˆK ˆ r ) H 1 ( ˆÂ) satisfy ŵ i (ĉ j ) = δ i,j and ŵ i (â j ) = 0, and {ˆv i } 8 i=1 P 2 ( ˆK ˆ r ) H 1 ( ˆÂ) satisfy ˆv i (ĉ j ) = 0 and ˆv i (â j ) = δ i,j. Thus, {ŵ i, ˆv i } form the canonical basis of P 2 ( ˆK ˆ r ) H 1 ( ˆ). Note that ŵ i ˆ = 0, therefore, ŵ i are quadratic bubble functions. Furthermore, {ŵ i e k, ˆv i e k } forms the canonical basis for P 2 ( ˆK ˆ r ) H 1 ( ˆ). 63

73 â 3 =  â 4 ĉ 2 â 5 ĉ 3 â 2 â 6 ĉ 5 ĉ 4 ĉ 1 â 7 â 1 â 8 Figure 4.4: Labeling of the skewed reference element. The basis of V( ˆÂ) is regarded as the Lagrange subbasis {ˆv i e k } corrected by the functions {ŵ i } to enforce the constant divergence constraint. In particular, the degrees of freedom and the definition of V( ˆÂ) yield the following result. Lemma 32. Let ˆv (k) i V( ˆÂ) denote the canonical basis of V( ˆÂ). Then, for each i and k, there exist unique vectors {b (k) i,j }5 j=1 R 2 such that ˆv (k) i = e kˆv i + 5 b (k) i,j ŵj. In particular, j=1 the vectors are uniquely determined by the following constraint, which represents a (10 10) system. div(ˆv (k) i ) = e k ˆ ˆv i + 5 j=1 b (k) i,j ˆ ŵ j P 0 ( ˆÂ). (4.6.1) Remark 7. The Lagrange basis {ˆv i, ŵ i } are easily computable, even on the skewed quadrilateral ˆÂ. The (10 10) system given in (4.6.1) can be solved symbolically in terms of the point Â. We assume that  is opposite to the origin as shown in Figure 4.3. Remark 8. The condition for each ˆT i, therefore, yields 8 constraints in total. (k) div(ˆv i ) ˆTi P 0 ( ˆT i ) imposes two linearly independent constraints div(ˆv (k) i ) P 0 ( ˆÂ) only adds 2 constraints due to the singular vertex. The additional continuity condition 64

74 4.7 NUMERICAL EXPERIMENTS In this section, we present some numerical results supporting the theoretical results and compare the convergence rates of the divergence free macro elements with the convergence rates of the serendipity elements for some test cases on both tensor product and general convex quadrilateral meshes (See Figure 4.5). We note that the degrees of freedom on quadrilateral meshes are less than the degrees of freedom on tensor product meshes (See Tables 4.2, 4.3) Experiment 1: Stokes Problem We compute the finite element method (4.5.2) on a sequence of refined meshes obtained from tensor product and quadrilateral grids. In this experiment, we consider the domain Ω = (0, 1) 2 and take the viscosity constant as ν = 1. We choose the data such that the velocity solution u and the pressure solution p are as follows: u(x, y) = 2x2 y(2y 1)(y 1)(x 1) 2, 2xy 2 (y 1) 2 (2x 1)(x 1) p(x, y) = x 2 y 2. The error estimates with the rates of convergence on the tensor product and quadrilateral meshes are listed in Tables 4.4, 4.5 and Tables 4.6, 4.7, respectively. Moreover, for comparison, we list the errors obtained by using the serendipity elements on the same sequence of meshes in Tables 4.8, Discussion on the Experimental Results for the Stokes Problem Experiment 1 shows that the velocity solution obtained from the divergence free macro element method have optimal order convergence on both the tensor-product and quadrilateral meshes (See Tables 4.4, 4.6). In addition, second order convergent pressure solutions are derived via post-processing (See Tables 4.5, 4.7). These results support the theoretical results stated in Section 4.5. On the other hand, we observe that the velocity solutions obtained by using serendipity elements on the quadrilateral meshes are suboptimal and the pressure solutions are only first order convergent (See Table 4.9). 65

75 (a) Tensor-product mesh. (b) uadrilateral mesh. Figure 4.5: Mesh with h = Table 4.1: Mesh statistics for h = Mesh Type Tensor-Product uadrilateral Elements Vertices Boundary Vertices Edges Boundary Edges