Supervised by Assist. Prof. Dr. Rabee Hadi Jari

Transcription

1 Republic of Iraq Ministry of Higher Education and Scientific Research Thi-Qar University College of Education for Pure Sciences Superconvergence of finite element approximations for elliptic problem with Dirichlet boundary condition A Thesis Submitted to The Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar as a Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics by Rasool Nadhim Jasim (B.Sc. 2010) Supervised by Assist. Prof. Dr. Rabee Hadi Jari 2015 A.D 1436 A.H

2 و أ وز ل انه ه ع ه ي ك ان ك ت اب و ان ح ك م ة و ع ه م ك م ا ن م ت ك ه ت ع ه م و ك ان ف ض م انه ه ع ه ي ك ع ظ يما صدق اهلل انعهي انعظيم سورة النساء اآلية 111

3 ا ؤلىداء ؤاميي ل يطيب انويل ؤال بشكرك ول يطيب اههنار ؤاىل بطاغتم... ول ثطيب انوحظات ؤال بذهرك.. ول ثطيب الآخرة ؤال بؼفوك.. ول ثطيب اجلنة ؤال برمحتم.. هللا جل جالهل. ؤاىل من بوؽ امرساةل وأأدى ا ألماهة وهصح ا ألمة ؤاىل هيب امرمحة وهور امؼاملني... سيدان محمد صىل هللا ػويو و اهل وسمل. اىل سادة اخلوق وامئة اميدى اذلين اذىب هللا غهنم امرجس وطيرمه ثطيريا... ال محمد ػوهيم امسالم. اىل من لكهل هللا ابمييبة واموكار ؤاىل من أأمحل أأمسو بلك افتخار... وادلي امؼزيز اىل من افتلده منذ امصغر اىل من سلتين من دهما وترغر غت يف حبر حهبا وحناهنا فاكهت امؼني واملوب اذلي يدغو يل... وادليت امؼزيزة من أ ج ل امساىرة اىل من اكهوا يل احلب يف احلياة و مالذ امروح ورفلاء ادلرب... زوجيت و اولدي الاغزاء اىل مشوع احلب و رايحني حيايت... أأخويت و أأخوايت الاغزاء اىل مجيع من ساػدين يف امتام ىذا امبحث أ ىدي هلم مجيؼا... مثرة هجدي مؼل فيو بؼض اموفاء رسول ناظم جاسم

4 ACKNOWLEDGEMENTS All the praise and thanks are to ALLAH, the most gracious and most merciful, for his grace that enables me to continue the requirements of my study. My sincerest gratitude is due to my supervisor, Asst. prof. Dr. Rabee Hadi Jari for his patience, help and prudent guidance throughout this work. His support and guidance allow me to complete one of my goals in my life. Without his help, completion of this goal would have been more difficult. I wish to express my sincere thanks to all my teachers and the staff members of the department of mathematics at the college of education for pure sciences for providing me with the opportunity to continue my higher studies. My thanks and priding are to my colleagues in the higher studies. Also, my thanks go to everyone who helps me during the fulfillment of my research. I cannot find words to express my indebtedness to my mother, wife, daughters, brothers, and sisters, for their love, support and inspiration at every step of my success and failure, without which none of this would have been possible. Rasool N. Jasim January, 2015

5 Supervision Certification I certify that this thesis entitled (Superconvergence of Finite Element Approximations for Elliptic Problem with Dirichlet Boundary Condition) written by was prepared under my supervision at Department of Mathematics/ College of Education for Pure Sciences/ University of Thi-Qar, as a partial fulfillment of the requirements for the degree of Master of Science in Mathematics. Signature: Supervisor: Asst. Prof. Dr. Rabee Hadi Jari Date: 16 / 4 /2015 Address: College of Education for Pure Science, Thi-Qar University Head recommendation of the mathematics department In view of the available recommendation; I forward this thesis for debate by the examining committee. Signature: Name: Dr. Fadhil Jasim Mohammed Head of the Department of Mathematics, College of Education for Pure Sciences, Thi-Qar University Date: 16 / 4 /2015

6 Committee Certificate We, the examining committee, certify that we have read this thesis entitled (Superconvergence for Finite Element Approximations for Elliptic Problem with Dirichlet Boundary Condition) and have examined the student written by (Rasool Nadhim Jasim) in its contents. We think in our opinion, it has met as a partial the requirements fulfillment for the degree of Master of Science in Mathematics, with ( ) grade. Chairman Signature: Name: Dr. Abdul-Sattar J. Ali Scientific Degree: Assist. Professor Member Signature: Name: Dr. Ali H. Shuaa Scientific Degree: Assist. Professor Date: 16 / 4 /2015 Date: 16 / 4 /2015 Member Signature: Name: M.Sc. Waleed K. Jaber Scientific Degree: Assist. Professor Member (Supervisor) Signature: Name: Dr. Rabee H. Jari Scientific Degree: Assist. Professor Date: 16 / 4 /2015 Date: 16 / 4 /2015 Approved for the University committee on Graduate studies. Signature: Name: Asst. Prof. Dr. Rabee Hadi Jari Dean of the College of Education for Pure Sciences Date: 16 / 4 /2015

7 Abstract The superconvergence in the finite element method (FEM) is a phenomenon in which the order of convergence of the numerical solution is higher than the order of convergence of the maximum of the finite element error. The main target in the superconvergence study is to improve the existing approximation accuracy by applying certain post- processing techniques which are easy to implement and little cost. Furthermore, superconvergence is also recognized as a useful tool in a posterior error estimations, mesh refinement and adaptivity. The second order elliptic problem with Dirichlet boundary condition will be considered in this thesis. The main aim of this thesis is to use -projection methods to achieve superconvergence of the finite element solution for the model problem with Dirichlet boundary condition. For the second order elliptic problem with Dirichlet boundary condition, superconvergence is investigated for conforming and nonconforming finite element methods by using -projection methods. Numerical examples are tested to verify and support the theoretical conclusion. Also we used MATLAB released in 2008 as the programming language and computer Laptop, system model LIFEBOOK AH530, processor: Int(R) core (TM) i5 CPU, M 2.40 GHz, to find the numerical results in this thesis.

8 List of Symbols and Abbreviations Exact solution. Approximation solution. Given function. Positive constant. Two-dimensional domain. Boundary of. Mesh size. Mesh size in a (coarse mesh). Regular triangulation with mesh size Regular triangulation with mesh size Finite dimensional sub pace of. (coarse mesh). Finite element space associated with the partition. Degree of polynomial. polynomials of degree Laplacian operator. Gradient. ( ) Hilbert space. O( ) Order. Union of all boundaries of all element k. Collection of all interior edges. Basis function. ( ) Hilbert space of order 1, in nonconforming finite element. FEM PDEs Finite element method. Partial differential equations.

9 Contents Abstract...VII List of Symbols and Abbreviations VIII List of Figures......XI List of Tables.... XIII Chapter One: Introduction 1.1 Overview of the FEM and - Projection Methods Organization of the Thesis Basic Steps of any FEM Intended to Solve PDFs Advantages of the FEM Triangulations Definitions Theorems....8 Chapter Two: Superconvergence for Conforming Finite Element Approximations for Second Order Elliptic Problem Introduction Conforming FEM for Second Order Elliptic Problem projection: A general Idea for Superconvergence Superconvergence of Conforming FEM by -Projection Methods Remarks on Programming of Finite Element Method in MATLAB Finite Element Data: Nodal Coordinate Matrix Finite Element Data: Element Connectivity Matrix Finite Element Data: Boundaries Matrix Constructing the Stiffness Matrix and Load Vector Assembling the Right-Hand Side Boundary Condition Finite Element Data: The Coarse Mesh Information...27

10 2.5.8 Derivation of a Linear System of Equations of Coarse Mesh Numerical Examples for Conforming FEM by - Projection Methods.31 Chapter Three: Superconvergence of Nonconforming Finite Element Approximations for Second Order Elliptic Problem Introduction Nonconforming FEM for Second Order Elliptic Problem Superconvergence of Nonconforming FEM by -Projection Methods Numerical Examples for Nonconforming FEM by -Projection Methods Conclusions References

11 List of Figures 1.1 (a) Situations not admitted in triangulations. (b) A finite element triangulation(conforming) Triangle incircle and diameter (a) Two adjacent elements and sharing midpoint (b) A finite element triangulation (nonconforming) Typical two dimensional discretization structures nodes for Numbered element Typical two dimensional discretization structures elements for Typical two dimensional discretization structures boundaries node for Typical two dimensional discretization structure coarse mesh Convergence rate of error in Example (a) -norm error, (b) -norm error Results for ( )( ( ) ( )) in Example (a) Surface plot of approximate solution. (b) Surface plot of Convergence rate of error in Example (a) -norm error, (b) -norm error Results for ( )( ) in Example (a) Surface plot of approximate solution. (b) Surface plot of Convergence rate of error in Example (a) -norm error. (b) - norm error Results for ( )( ( ) ) in Example (a) Surface plot of approximate solution. (b) Surface plot of Two adjacent elements and sharing edge e Convergence rate of error in Example norm error...45

12 3.3 Results for ( ) in Example (a) Surface plot of approximate solution. (b) Surface plot of Convergence rate of error in Example norm error Results for ( )( ) in Example (a) Surface plot of approximate solution. (b) Surface plot of Convergence rate of error in Example norm error Results for ( ) ( ) in Example (a) Surface plot of approximate solution. (b) Surface plot of Convergence rate of error in Example norm error Results for ( ) in Example (a) Surface plot of approximate solution. (b) Surface plot of...49

13 List of Tables Errors on uniform triangular meshes and in Example Errors on uniform triangular meshes and in Example Errors on uniform triangular meshes and in Example Errors on uniform triangular meshes and in Example Errors on uniform triangular meshes and in Example Errors on uniform triangular meshes and in Example Errors on uniform triangular meshes and in Example

14 CHAPTER ONE Introduction

15 1.1 Overview of the FEM and - Projection Methods The FEM is a computational technique for obtaining approximate solutions to the partial differential equations that arise in scientific and engineering applications. The FEM utilizes a variational problem that involves an integral of the differential equation over the problem domain. This domain is divided into a number of sub domains called finite elements and the solution of the partial differential equation is approximated by a simpler polynomial function on each element. These polynomials have to be pieced together so that the approximate solution has an appropriate degree of smoothness over the entire domain. The result is an algebraic system for the approximate solution having a finite size rather than the original infinite-dimensional partial differential equation [1]. Approximating functions in finite elements is determined in terms of nodal values of a physical field which is sought. A continuous physical problem is transformed into a discretized finite element problem with unknown nodal values. [2]. Today, the FEM is considered one of the well- established and convenient technique for the computer solution of complex problems in different fields of engineering: civil engineering, mechanical engineering, nuclear engineering, biomedical engineering, hydrodynamics, heat conduction and geomechanics. From the other side, FEM can be examined as a powerful tool for the approximate solution of differential equations describing different physical processes. The success of FEM is based largely on the basic finite element procedures used: the formulation of the problem in variational form, the finite element discretization of this formulation and the effective solution of the resulting finite element equations [3]. The term finite element was first coined by Clough in In the early1960s, engineers used the method for approximate solutions of problems in stress analysis, fluid flow, heat transfer, and other areas. Also the first book on the FEM by Zienkiewicz and Chung was published in In the late 1960s and early 1970s, the FEM was applied to a 1

16 wide variety of engineering problems [4]. Since Oganesyan and Rukhovetz first proved the superconvergence for linear element approximations on a uniform triangle mesh in (1969), superconvergence for the finite element solutions has been an active research area in numerical analysis for more than thirty years [5]. Wang proposed and analyzed the -projection method for the least- squares conforming finite element method on the second order elliptic problem [6]. The accuracy of the finite element approximation can be improved to obtain superconvergence by applying some post-process techniques to them, the - projection method is such technique, the idea of the method is to make an - projection of existing finite element solutions to another finite element space associated with a coarse mesh the advantage of this method it is easy to implement and can be applied to different elements with general meshes. In this thesis, we will apply the -projection method to obtain the superconvergence for the different finite element approximations of the second order elliptic problem with Dirichlet boundary condition. The finite element methods involved are the conforming finite element method and the nonconforming finite element method for the second order elliptic problem with Dirichlet boundary condition. 1.2 Organization of the Thesis This thesis is divided into three chapters Chapter One is an overview of the FEM and -projection methods, basic steps of any FEM intended to solve PDFs, advantages of the FEM, triangulations, definitions of some Hilbert spaces and their corresponding norms with some of their properties and we refer to some theorems. 2

17 Chapter Two describes the superconvergence of conforming finite element approximations for the second order elliptic problem by -projection methods. Chapter Three describes the superconvergence of nonconforming finite element approximations for the second order elliptic problem by methods. -projection Finally, conclusions on the theoretical methods and the numerical results obtained. 1.3 Basic Steps of any FEM Intended to Solve PDEs In all FEM variants there are always the same sequence of steps to be taken (1) Discretize the continuum: divide the solution region into smaller regions that we call elements. The elements contain inside a certain number of points we call nodes. (2) Select the type of trial function to use, we select what kind of functions we will take to describe the variation of the function inside each element (the trial function). This is equivalent to say, that we select the basis set of functions that will describe our solution. One of the usual choices is to take a polynomial like for instance ( ). (3) The formulation: given the PDE you want to solve, now you must find a system of algebraic equations for each element such that by solving it you got the values of at the position of nodes of the element (, -, - ), it's mean that you must find for each element the matrix, - and the vector, -. (4) Assembling the equations for different elements. (5) Solve the system of equations. (6) Compute secondary quantities: Once you know the values, you can compute other magnitudes using the values of, [7]. 3

18 1.4 Advantages of the FEM This method has a number of advantages that make it very popular [8]. They include the ability to (1) Model irregularly shaped bodies quite easily. (2) Handle general load conditions without difficulty. (3) Model bodies composed of several different materials because the element equations are evaluated individually. (4) Handle unlimited numbers and kinds of boundary conditions. (5) Vary the size of the elements to make it possible to use small elements where necessary. (6) Alter the finite element model relatively easily and cheaply. (7) Include dynamic effects. (8) Handle nonlinear behavior existing with large deformations and nonlinear materials. 1.5 Triangulations Let be a bounded two-dimensional domain with smooth or polygonal boundary. A triangulation, or mesh of is a set * + of triangles, such that, and such that the intersection of two triangles is either an edge, a corner, or empty. No triangle corner is allowed to be hanging, that is, lie on an edge of another triangle see Figure 1.1 (a) and (b). The corners of the triangles are called the nodes. The set of triangle edges is denoted by * +. We distinguish between edges lying with in the domain and edges lying on the boundary. The former belongs to the set of interior edges, and the latter to the set of boundary edges, respectively. For each K, we define: 4

19 the longest side of K, the diameter of the circle inscribed in K, the center of the inscribed circle is called the incenter and is located at the intersection of the three internal angle, (see Figure 1.2). We shall assume that there is a positive constant triangulation * + such that independent of the This condition means that the angles of the triangle K are not allowed to be arbitrary small. The constant is a measure of the smallest angle in any for any * +. We always assume the regular triangulation (see [9]-[10]). (a) (b) Figure 1.1: (a) Situations not admitted in triangulations. (b) A finite element triangulation (conforming). Figure 1.2: Triangle incircle and diameter. 5

20 Also, the mesh for nonconforming finite element, we start by including all the nodes in conforming finite element, but we create a new node at the midpoint of every element edge, and add all these nodes as well, (see Figure 1.3). (a) (b) Figure 1.3: Two adjacent elements sharing midpoint. (b) A finite element triangulation (nonconforming). 1.6 Definitions Definition [10]: For a bounded domain integrable functions as follows:, define the space of square ( ) { } the space ( ) is Hilbert space with norm ( -norm) ( ) ( ) For real valued functions ( ), we defined the -inner product by ( ) 6

21 Definition [10]: (Hilbert Spaces). Let and denoting a partial derivatives of order, the space ( ) of the function on defined by ( ) * ( ) ( ) +, where ( ) ( ) and ( ) is an -index, are non-negative integers and, or ( ) is called Hilbert space of order, with inner product ( ) ( ) ( ) ( ), ( ) ( ), with the norm and the semi-norm ( ) ( ( )), ( ) ( ( )). We usually write and instead of ( ) and ( ). For Define Hilbert space as follows: with inner product ( ) * ( ) ( ) +, 7

22 ( ) ( ), - with the norm (, - ), and the semi-norm ( ) finally, we define the special Hilbert space ( ) ( ) * ( ) ( ) + we shall use this space when considering a partial differential equation that is coupled with a homogeneous (Dirichlet) boundary condition: on We note here that ( ) is also a Hilbert space, with the same norm and inner product as ( ). For the Hilbert space ( ) ( ). 1.7 Theorems Theorem [11]: Let and ( ), be two functions, the following differential identity is well-known as Green's identity in two dimensions applied to the Laplace term in the second order boundary value problem: where, ( ), and are the direction cosines of the outward normal to the boundary and the element of arc length along. 8

23 Theorem [9]: (Poincare inequality). Let Then, there is constant,, such that for any ( ) be a bounded domain. ( ) ( ) Theorem [12]: (Discrete Poincare- Friedrichs inequality). There exists such that ( ). where Theorem [9]: (Inverse estimate).on a quasi-uniform mesh any satisfies the inverse estimate ( ) ( ) Corollary [13]: (The triangle inequality). Let and belong to ( ), then ( ), and ( ) ( ) ( ) Theorem [14]: (The Cauchy-Schwarz inequality). Let and belong to ( ), then ( ) ( ) ( ) 9

24 2.1 Introduction We will study the second order elliptic problem with a homogeneous Dirichlet boundary condition. This model problem seeks an unknown function satisfying in (2.1) on (2.2) where is a bounded domain in, with boundary, where + denotes the Laplacian operator, and is given function in ( ). The objective of this chapter is to investigate superconvergence of conforming finite element approximations for the second order elliptic problem by using -projection method. The rest of this chapter is organized as follows. In section 2.2, present the conforming FEM for elliptic problem. In section 2.3, describe the general idea of the -projection method for superconvergence. In section 2.4, - projection method is applied to the conforming finite element solution, superconvergence results are proved. In section 2.5 remarks on programming are given. In section 2.6, numerical examples are tested to confirm the theoretical results derived in section Conforming FEM for Second Order Elliptic Problem We seek a solution to the second order elliptic problem with homogeneous Dirichlet condition in V= ( ) * ( ) +. Let be the conforming finite element partition of the domain with characteristic mesh size and ( ) be the set of all the polynomials defined on with degree less or equal to. Define * ( ) ( ) + ( ) 11

25 To derive a weak formulation of the model problem (2.1)-(2.2), we multiply Equation (2.1), by a test function ( ), for both sides and integrating by parts, by using green s formula we have (2.3) (2.4) (2.5) since thus the weak form of (2.1)-(2.2) is to find ( ) such that ( ) ( ) ( ) (2.6) where ( ) ( ) ( ) (2.7) and ( ). The conforming finite element approximation problem for (2.1)-(2.2) is to find ( ) such that ( ) ( ) (2.8) where ( ) ( ) ( ) (2.9) Define a norm for V= ( ) as follows ( ) ( ) ( ) (2.10) 11

26 2.3 Projection: A general Idea for Superconvergence The -Projection technique was introduced by Wang [6]. It projects the approximate solution to another finite element dimensional space associated with a coarser mesh. The difference in size of the two meshes can be used to achieve a superconvergence. Now, we introduce another finite element partition (coarse mesh), with mesh size. Assume that is related to the original mesh size by:, (2.11) with ( ). It will be seen that the parameter plays an important role in the post-processing. For now, let us construct the finite element space ( ) for the exact solution consisting of piecewise polynomials of degree r associated with the partition. Let be the -projector onto the finite element space. can be considered a linear operator (projection) from ( ) onto the finite element space. The post-processing of the finite element approximation is simply given by the projection: post- processed. To derive the optimal error estimate, we need to prove the following lemma. Lemma 2.3.1: There exists a constant, C, independent of such that ( ) ( ) (2.12) Proof: Since the problem (2.1)-(2.2) satisfy the weak form ( ) ( ) ( ) where ( ) ( ) ( ) An application of Cauchy-Schwarz inequality reveals ( ) (2.13) 12

27 (2.14) By the Poincare inequality we have ( ) (2.15) ( ) (2.16) where Since where ( ), (, - ) We have ( ) The following theorem can be found in [15] Theorem 2.3.2: Let and be the solutions of (2.1)-(2.2) and (2.8), respectively. Then, there exists a constant, independent of such that (2.17) We will assume an regularity for the solution of (2.1)-(2.2): (2.18) 13

28 2.4 Superconvergence of Conforming FEM by - Projection Methods Let be the -projector onto the finite element space In the following, we will analyze the error of The following lemma provides an error estimate for Lemma 2.4.1: Suppose that (2.18) hold with and ( ) Then there is a constant, C, independent of and such that ( ) (2.19) where ( ) is defined in (2.11). Proof: Using the definition of and, we have ( ) ( ) From the definition of the -projection for, we have It follows directly that, ( ) ( ) ( ) ( ) (2.20) Consider the following problem: find ( ) such that in Ω (2.21) on (2.22) multiply Equation (2.21) by a test function ( ) for both sides and integrate by parts over Ω, by using Green s formula, we have ( ) ( ) ( ) (2.23) where ( ) ( ) ( ) 14

29 It follows from (2.18) (2.24) for some constant C. Using the inverse inequality on the right hand side of (2.24), we have ( ) (2.25) The difference of (2.6) and (2.8) and using the fact that ( ) gives ( ) (2.26) By replacing in (2.23) by and then using (2.26), we obtain ( ) ( ) ( ) (2.27) where is any finite element function belong to Using Lemma (2.3.1) and (2.17) we obtain from (2.27) that ( ) ( ) inf v V h (2.28) In addition, from (2.24), (2.25) and (2.11) we obtain ( ) ( ) ( ) The last estimate, together with (2.20) is obtained ( ) Now, we come to estimate. 15

30 Theorem 2.4.2: Assume that (2.18) hold with and ( ) If the exact solution ( ) ( ) ( ), then there exists a constant, C, such that ( ) ( ) (2.29) where ( ) is the finite element approximation of the solution. Proof: By the definition of and (2.11), we have ( ) (2.30) Using the triangle inequality and combining (2.30) with (2.19) gives ( ) ( ), which completes the estimate for in (2.29) Next, we estimate ( ) as follow: By the definition of and (2.11), we have ( ) (2.31) Multiplying both sides of (2.31) by, we get ( ) ( ). (2.32) Moreover, the inverse inequality gives ( ) (2.33) Multiplying both sides of (2.33) by implies, and using (2.11), the above inequality 16

31 ( ) (2.34) Consequently, from (2.19), we obtain ( ( ) ) (2.35) Using the triangle inequality and combing (2.32) with (2.35) gives ( ) ( ) ( ) ( ) ( ) which implies that ( ) ( ) ( ) This completes the proof of the theorem. We can bound the right hand of (2.29) as follows: ( ) ( ), (2.36) where ( ) The above error estimate is optimized if is selected to satisfy ( ) ( ) solving from above yields ( ) (2.37) The corresponding error estimate is given by ( ) ( ) ( ) (2.38) 17

32 Thus, we have the following theorem Theorem 2.4.3: Assume that (2.18) hold with, and ( ). Let be the finite element approximation of the solution, and the parameter is given by (2.37). Then we have the optimal error estimate (2.38) of the post- processed approximation. 2.5 Remarks on Programming of Finite Element Method in MATLAB Let us briefly discuss some of essential features of a typical computer program implementing a finite element method Finite Element Data: Nodal Coordinate Matrix To do so, we define a set of nodes, (numbered, 1 9) and a set of elements, (numbers in circles). The node coordinates are stored in the nodal coordinate matrix numbered from top to bottom and from left to right. The dimension of this matrix is where is number of nodes. Figure 2.1, shows a simple finite element discretization of, -, -. For this mesh the nodal coordinate matrix would be represent as follows: Figure 2.1: Typical two dimensional discretization structures nodes for. 18

33 The subroutine to compute the nodes matrix in two dimensional space is given by: node=zeros((n+1)^2,2); for i=1:n+1 for j=1:n+1 node((i-1)*(n+1)+j,2)=1-(i-1)*1/n; node((i-1)*(n+1)+j,1)=0+(j-1)*1/n; end end Finite Element Data: Element Connectivity Matrix The element data are stored in the element connectivity matrix. This is a matrix of node numbers where each row of the matrix contains the connectivity of an element. Also, the elements connectivities are all ordered in a counterclockwise and start from the right angle for the element as shown in Figure 2.2. Figure 2.2: Numbered element. Also, see Figure 2.3, shows a simple finite element discretization of, -, -. For this mesh, the element connectivity matrix would be represented as follows: 19

34 Figure 2.3: Typical two dimensional discretization structures elements for. The subroutine to compute the elements matrix in two dimensional space is given by: elem=zeros(n^2*2,3); for i=1:n for j=1:n elem((i-1)*2*n+2*(j-1)+1,1)=(i-1)*(n+1)+j; elem((i-1)*2*n+2*(j-1)+1,2)=(i)*(n+1)+j; elem((i-1)*2*n+2*(j-1)+1,3)=(i-1)*(n+1)+j+1; elem((i-1)*2*n+2*(j-1)+2,1)=(i)*(n+1)+j+1; elem((i-1)*2*n+2*(j-1)+2,2)=(i-1)*(n+1)+j+1; elem((i-1)*2*n+2*(j-1)+2,3)=(i)*(n+1)+j; end end 21

35 2.5.3 Finite Element Data: Boundaries Matrix The boundaries data are stored in the boundaries node matrix. This is a matrix of node numbers where each row is the two node numbers which bound the corresponding edge on the boundary. The dimension of this matrix is where is a number of boundary edges. Figure 2.3, shows a simple finite element discretization of, -, -. For this mesh, the boundaries matrix would be represented as follows: Figure 2.4: Typical two dimensional discretization structures boundaries node for. The subroutine to compute the boundary matrix in two dimensional space is given by: BDNode=zeros(4*n,2); BDNode(1:n,:)=[2:n+1;1:n]'; % top BDNode(n+1:2*n,:)=[2*(n+1):n+1:(n+1)*(n+1);n+1:n+1:n*(n+1)]';% right BDNode(2*n+1:3*n,:) = [(n+1)*(n+1)-1:- 1:n*(n+1)+1;(n+1)*(n+1):-1:n*(n+1)+2]'; % bottom BDNode(3*n+1:4*n,:) = [ (n-1)*(n+1)+1:-n-1:1;n*(n+1)+1:-n- 1:n+2]'; % left 21

36 2.5.4 Constructing the Stiffness Matrix and Load Vector Let * + be the standard basis of hat functions for the space of continuous piecewise linears on a mesh of Ω with N nodes. The finite element approximation (2.8) of the weak form (2.6) is equivalent to ( ) ( ) (2.39) where ( ) ( ) ( ) and is the basis function or (hat function). Since, it can be written as the linear combination (2.40) where, is unknown coefficients, and substituting into (2.39), we finally obtain the following linear system of equations: ( ) ( ) (2.41) with N equations in N unknowns. The linear system of (2.41) can be written as, (2.42) where ( ) is the matrix with elements ( ) ( ),, where { } note that the subscript in is the local index, while in it is global index, where is the solution vector with ( ) and is the force vector with 22

37 ( ),, where. [ ], [ ], [ ]. The matrix is called stiffness matrix and the load vector. In order to obtain, we must compute the stiffness matrix and the load vector. The computation of will be decomposed into the computation of local stiffness matrix and the summation over all elements. The local stiffness matrix is determined by the coordinates of the vertices of the corresponding element. Let is triangle with vertices ( ) then the degrees of freedom for are the values at the vertices. A function ( ) is then uniquely determine by its values at the points (see[10],theorem 3.1). Let be the basis function for ( ) associated with the degree of freedom above ( ) where is 1 if and 0 otherwise. Any function v then has the representation: ( ) ( ) with ( ). Therefore ( ) (2.43) and have the system,,. (2.44) The solution of the system (2.44) is given by: 23

38 ( ), ( ), ( ), (2.45) where is area of the triangle, and (2.46) Substitution of Equation (2.45) into Equation (2.43) yields: where ( ) ( ) ( ) ( ), ( ) ( ) ( ), ( ), ( ) ( ) (2.47) Each of these three nodes corresponds to a non-zero hat function, given by ( ). (2.48) From the expression of ( ) we have Therefore we can get ( ) ( ) ( ) ( ) ( ) 24

39 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Also, the same feature for. Note that the gradient of is just the constant vector element ( ). Because there are only three non-zero hat functions on, we get a total of 9 integral contributions to the stiffness matrix from. These are the entries of the element stiffness matrix from. To compute the second term of the model problem using integration formula (see [9]), we have ( ) { Finally, we have { } ( ) ( ). To this end, we write the following subroutine to compute the local stiffness matrix in one triangle. N = size(node,1); NT = size(elem,1); u = zeros(n,1); % Assemble stiffness matrix aa = zeros(nt,3); cc=zeros(nt,3); bb = zeros(nt,3); n1 = elem(:,1); n2 = elem(:,2); n3 = elem(:,3); 25

40 x1 = node(n1,1); y1 = node(n1,2); x2 = node(n2,1); y2 = node(n2,2); x3 = node(n3,1); y3 = node(n3,2); aa(:,1) = y2-y3; bb(:,1) = x3-x2; aa(:,2) = y3-y1; bb(:,2) = x1-x3; aa(:,3) = y1-y2; bb(:,3) = x2-x1; cc(:,1)=x2.*y3-x3.*y2; cc(:,2)=x3.*y1-x1.*y3; cc(:,3)=x1.*y2-x2.*y1; area = (x2.*y3+x1.*y2+x3.*y1-x1.*y3-x3.*y2-x2.*y1)./2; A = sparse(n,n); for i = 1:3 for j = 1:3 if i==j Aij =(area*1/6)+(aa(:,i).*aa(:,j)+bb(:,i).*bb(:,j))./(4*area); else Aij =(area*1/12)+(aa(:,i).*aa(:,j)+bb(:,i).*bb(:,j))./(4*area); end A = A + sparse(elem(:,i),elem(:,j), Aij, N,N); end end Assembling the Right-Hand Side In this subsection, we shall discuss how to form the load vector efficiently. To compute right hand side (The entries of the 3 1 element load vector ), we define the vector ( ) by we compute the term by 3-points quadrature rule in two-dimension. The subroutine to compute the load vector in two dimensional space is given by: see ([16]). 26

41 % Assembling right hand side by 3-point rule mid1 = (node(elem(:,2),:)+node(elem(:,3),:))/2; mid2 = (node(elem(:,3),:)+node(elem(:,1),:))/2; mid3 = (node(elem(:,1),:)+node(elem(:,2),:))/2; bt1 = area.*(rhs(mid2)+rhs(mid3))/6; bt2 = area.*(rhs(mid3)+rhs(mid1))/6; bt3 = area.*(rhs(mid1)+rhs(mid2))/6; b = accumarray(elem(:),[bt1;bt2;bt3],[n 1]); Boundary Condition The subroutine to compute the boundary condition in two dimensional space is given by: ---- Handle the boundary condition for E=1:size(BDNode,1) p=bdnode(e,1); A(p,:)=0; A(p,p)=1; b(p)=exactu(node(p,:)); end uh=a\b; Finite Element Data: The Coarse Mesh Information In this subsection, we shall discuss the structure the coarse mesh by - projection method. Figure 2.5, shows a simple finite element discretization of, -, -. For this mesh, the coarse mesh and mesh to fine mesh would be represented as follows: 27

42 Figure 2.5: Typical two dimensional discretization structure coarse mesh Derivation of a Linear System of Equations of Coarse Mesh The post-processing of the finite element approximation is simply given by the projection: Post- processed. The -projection satisfying ( ) ( ), (2.46) then we have ( ). (2.47) In the implementation, the quantity is -projection of to associated with We define as follows: * ( ) ( ) +, where ( ) is the space of quadratic functions on that is ( ). The stander basis of ( ) is * +. To derive a linear system of equations, in order to compute the, we first note that (2.46) is equivalent to 28

43 ( ) ( ), ( ) (2.48) where is the hat basis function spanning. Further, expanding, it can be written as the linear combination, (2.49) where, is unknown coefficients, and substituting into (2.48), we finally obtain the following linear system of equations: ( ) ( ), (2.50) the linear system of (2.50) can be written as, (2.51) where ( ) is the matrix with elements ( ),, where note that the subscript in is the local index, while in it is global index, where is the solution vector with ( ) and is the force vector with ( ) ( ), where is the local basis function of element, where ( ) ( ), The element matrix of force vector (the entries of the ) is given by: ( ) ( ), 29

44 then ( ) ( ) ( ) ( ) ( ) [( )] Also, the matrix of element (the entries of the ) given by: [ ] [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] Since the matrix, and the matrix of force vector, then the solution by -projection given by:, where is the inverse of matrix M. 31

45 2.6 Numerical Examples for Conforming FEM by - Projection Methods In this section, we present several numerical examples to verify Theorem The triangulation of is constructed by: (1) Dividing the domain into an rectangular mesh. (2) Connecting the diagonal line with the positive slope. Denote as the mesh size. By using Theorem 2.4.3, we expect to get a convergence rate of order O( ) for and order O. / for ( ). The following numerical results agree with our theory. Example 2.6.1: Let the domain, -, - Also, the exact solution is assumed to be ( )( ( ) ( )) Table shows that after the use of the post-processing method, all the errors are reduced. The exact solution in the -norm of has a convergence rate similar to.there is no improvement for u in the - norm. However, the error in the -norm has a higher convergence rate, which is shown as ( ) for ( ), see Figure 2.6 (a) and (b). Figure 2.7 (a) and (b) gives results for the finite element approximation of the problem given in (2.1)-(2.2), both before and after post-processing. All of these results confirm with our theoretical result e e e e e e e e e e e e e e e e e e e-5 ( ) Table 2.6.1: Errors on uniform triangular meshes and 31

46 Figure 2.6: Convergence rate of error in Example (a) -norm error, (b) - norm error. Figure 2.7: Results for ( )( ( ) ( )) in Example (a) Surface plot of approximate solution. (b) Surface plot of. Example 2.6.2: Let the domain, -, - Also, the exact solution is assumed to be ( )( ) e e e e e e e e e e e e e e e e e e e e-6 ( ) Table 2.6.2: Errors on uniform triangular meshes and 32

47 From the results shown in Table 2.6.2, it is clear that the exact solution, in the -norm has the superconvergence. However, there is no improvement for in the -norm, see Figure 2.8 (a) and (b). Figure 2.9 (a) and (b), shows that the approximate solutions and. This agrees well with the theory. Figure 2.8: Convergence rate of error in Example (a) -norm error, (b) - norm error. Figure 2.9: Results for ( )( ) in Example (a) Surface plot of approximate solution. (b) Surface plot of. Example 2.6.3: Let the domain, -, - Also, the exact solution is assumed to be ( )( ( ) ) 33

48 0.2786e e e e e e e e e e e e e e e e e e e e-5 ( ) Table 2.6.3: Errors on uniform triangular meshes and Table gives the error profile for example Notice that the gradient estimate is of order ( ), that is much better than the optimal order ( ) However, there is no improvement in the -norm, see Figure 2.10 (a) and (b). Figure 2.11 (a) and (b) shows the results of the finite element approximation of the solution to problem (2.1)-(2.2). In addition, the numerical results are highly consistent with the theoretical -projection. Figurer 2.10: Convergence rate of error in Example (a) -norm error. (b) -norm error. Figure 2.11: Results for ( )( ( ) ) in Example (a) Surface plot of approximate solution. (b) Surface plot of. 34

49 3.1 Introduction Let us consider the second order elliptic problem with a homogeneous Dirichlet boundary condition. This model problem seeks an unknown function satisfying in (3.1) on (3.2) Nonconforming finite element method will be used to discretize the second order elliptic problem (3.1)-(3.2), with Dirichlet boundary condition. The objective of this chapter is to investigate superconvergence of nonconforming finite element approximations for the second order elliptic problem by using - projection method. The rest of this chapter is organized as follows. In section 3.2, present the nonconforming finite element method for elliptic problem. In section 3.3, -projection method is applied to the nonconforming finite element solution, superconvergence results are proved. In section 3.4, numerical examples are tested to confirm the theoretical results derived in section Nonconforming FEM for Second Order Elliptic Problem We seek a solution to the second order elliptic problem with homogeneous Dirichlet condition. In the space of elementwise -functions with respect to the triangulation, (see [12]) ( ) * ( ) ( ) which is a Hilbert space with norm ( ) ( ), 35

50 and the semi-norm ( ). Let be a finite element partition of the domain for with characteristic mesh size. Assume that the partition is a regular triangulation. Let denote the union of all boundaries of all elements K of and the collection of all interior edges. Let be an interior edge shared by two elements and in and let and be unit normal vectors on pointing exterior to and, respectively see Figure 3.1. Figure 3.1: Two adjacent elements and sharing edge e. We define the jump of a function across an edge as, -. Let be the nonconforming finite element space associated with the partition defined as: * ( ) ( ) is continuous at midpoint of and zero at the midpoint of every boundary edge e on Ω +. To derive a weak formulation of the model problem (3.1)-(3.2), we multiply Equation (3.1), by test function and integrating it over the domain Ω obtain, (3.3) and apply integration by parts on each element give 36

51 , (3.4) by using Green s formula for (3.4), we find that (3.5), (3.6) The exact solution satisfies the following equation: ( ) (3.7) where ( ) ( ) ( ). The nonconforming finite element approximation problem for (3.1)-(3.2) is to find such that ( ), (3.8) where ( ) ( ) ( ). Define a norm for ( ) as follows: ( ) ( ) ( ). (3.9) 37

52 3.3 Superconvergence of Nonconforming FEM by -Projection Methods In this section, we will prove the superconvergence for nonconforming finite element method for the problem (3.1)-(3.2). In order to do this, we need to prove the following lemma. Lemma 3.3.1: There exists a constant, C independent of such that ( ) ( ) Proof: Since the problem (3.1)-(3.2) satisfy the bilinear form ( ) (3.10) An application of Cauchy s -Schwarz inequality reveals ( ) (3.11). (3.12) By discrete Poincare inequality, we have ( ) (3.13) ( ) (3.14) where Since where ( ), (, - ) 38

53 We have ( ) The following theorem can be found in [17]. Theorem 3.3.2: Let and be the solutions of (3.1)-(3.2) and (3.8), respectively. Then, there exists a constant, independent of such that (3.15) We assume an regularity: (3.16) Define the finite element space ( ) for the exact solution Here, consists of piecewise polynomials of degree associated with the partition Let be the -projector onto the finite element space in the following we will analyze the error of. The following lemma provides an error estimate for. Lemma 3.3.3: Suppose that (3.16) hold with and ( ) Then, there is a constant, C, independent of and such that ( ) (3.17) where ( ) is defined in (2.11). Proof: Using the definition of and, we have ( ) ( ) From the definition of the -projection for, we have It follows directly that, ( ) ( ) 39

54 ( ) ( ) (3.18) Consider the following problem: find ( ) such that in Ω, (3.19) on, (3.20) multiply equation (3.19) by a test function ( ) for both sides and integrate by parts over Ω, by using Green s formula, we have ( ) ( ), ( ) (3.21) where ( ) ( ) ( ). It follows from (3.16), we have, (3.22) for some constant C. Using the inverse inequality on the right hand side of (3.22) we have ( ) (3.23) The difference of (3.7) and (3.8) gives ( ) (3.24) By replacing in (3.21) by and then using (3.24), we obtain ( ) ( ) ( ) 41

55 ( ) ( ) ( ), ( ) ( ), (3.25) by replacing in (3.25) by where any finite element function belong to Using Lemma (3.3.1) and (3.15), we obtain from (3.25) that ( ) ( ) ( ) ( ) ( ), inf ( ) v V h ( ), Since similarly ( ) (see[18]). We have ( ) + (3.26) In addition, from (3.22), (3.23) and (2.11) we obtain ( ) ( ) ( ) 41

56 ( ) ( ) ( ) ( ) ( ) ( ) ( ). The last estimate, together with (3.18), we obtain ( ). Now, we come to estimate. Theorem 3.3.4: Assume that (3.16) hold with and ( ) If the exact solution ( ) ( ) ( ), then there exists a constant, C, such that ( ) ( ), (3.27) where ( ) is the finite element approximation of the solution. Proof: By the definition and (2.11), we have ( ) (3.28) Using the triangle inequality and combining (3.28) and (3.17) gives ( ) ( ), which completes the estimate for in (3.27). 42

57 Next, we estimate ( ) as follow: By the definition of and (2.11) we have ( ) (3.29) Multiplying both sides of (3.29) by we get ( ) ( ) (3.30) Moreover the inverse inequality gives ( ) (3.31) Multiplying both sides of (3.31) by implies, and using (2.11) the above inequality ( ) (3.32) Consequently, from (3.17) we obtain ( ( ) ) (3.33) Using the triangle inequality and combing (3.30) and (3.33) gives ( ) ( ) ( ) ( ) ( ) which implies that ( ) ( ) ( ) This completes the proof of the theorem The above error estimate is optimized if is selected to satisfy ( ) ( ) (3.34) Solving from above yields 43

58 ( ) (3.35) The corresponding error estimate is given by ( ( ) ) ( ). (3.36) Thus, we have the following theorem Theorem 3.3.5: Assume that (3.16) hold with and ( ). Let be the finite element approximation of the solution. And the parameter is given by (3.35).Then we have the optimal error estimate (3.36) of the post processed approximation. 3.4 Numerical Examples for Nonconforming FEM by - Projection Methods In this section, we present several numerical examples to verify Theorem derived in section (3.3). The triangulation of is constructed by: (1) dividing the domain into an rectangular mesh and (2) connecting the diagonal line with the positive slope. Denote as the mesh size. In the implementation, the quantity is -projection of to associate with We define as follows: * ( ) ( ) + By using Theorem 3.3.5, we expect to get a convergence rate of order O. / for ( ). The following numerical results agree with our theory. 44

59 Example 3.4.1: Let the domain, -, - Also, the exact solution is assumed to be ( ) Table shows that after the use of the post-processing method, all the errors are reduced. The error in the -norm has a higher convergence rate, which is shown as ( ) for ( ), (see Figure 3.2). Figure 3.3 (a) and (b) give results for the finite element approximation of the problem given in (3.1)- (3.2), both before and after post-processing. All of these results confirm with our theoretical result e e e e e e e e-3 ( ) Table 3.4.1: Errors on uniform triangular meshes and Figure 3.2: Convergence rate of error in Example norm error. 45

60 Figure 3.3: Results for ( ) in Example (a) Surface plot of approximate solution. (b) Surface plot of. Example 3.4.2: Let the domain, -, - Also, the exact solution is assumed to be ( )( ) e e e e e e e e-4 ( ) Table 3.4.2: Errors on uniform triangular meshes and. From the results shown in Table 3.4.2, it is clear that the exact solution u in the -norm has the superconvergence, (see Figure 3.4). Figure 3.5 (a) and (b), shows that the approximate solutions and This is in agreement with the previously stated theory. 46

61 Figure 3.4: Convergence rate of error in Example norm error. Figure 3.5: Results for ( )( ) in Example (a) Surface plot of approximate solution. (b) Surface plot of. Example 3.4.3: Let the domain, -, - Also, the exact solution is assumed to be ( ) ( ) e e e e e e e e-3 ( ) Table 3.4.3: Errors on uniform triangular meshes and 47

62 Table gives the error profile for example Notice that the gradient estimate is of order ( ) has the superconvergence, (see Figure 3.6). Figure 3.7 (a) and (b), shows that the approximate solutions and In addition, the numerical results are highly consistent with the theoretical -projection. Thus they validate our superconvergence theories. Figure 3.6: Convergence rate of error in Example norm error. Figure 3.7: Results for ( ) ( ) in Example (a) Surface plot of approximate solution. (b) Surface plot of. Example 3.4.4: Let the domain, -, - Also, the exact solution is assumed to be ( ) 48

63 0.3023e e e e e e e e-4 ( ) Table 3.4.4: Errors on uniform triangular meshes and. From the results shown in Table 3.4.4, it is clear that the exact solution u in the -norm has the superconvergence, (see Figure 3.8). Figure 3.9 (a) and (b), shows that the approximate solutions and This is in agreement with the previously stated theory. Figure 3.8: Convergence rate of error in Example norm error. Figure 3.9: Results for ( ) in Example (a) Surface plot of approximate solution. (b) Surface plot of. 49

64 Conclusions In this thesis, we had a study elliptic problem of second order with homogeneous Dirichlet boundary condition. This problem is represented by ( in ) and ( ). We discussed two cases. In the first case, we considered the elements mesh is conforming. In the second case we considered the elements mesh is non- conforming. For the first case we were able to improve the accuracy of the convergence (superconvergence) of the problem studied by applying certain post processing techniques ( - projection method) which are easy to implement and little cost and on the same terms of the problem, we found the following. The exact solution in the -norm of has a convergence rate similar to.there is no improvement for u in the -norm. However, the error in the -norm has a higher convergence rate, which is shown as ( ) for ( ). For the second case we were able to improve the accuracy of the convergence (superconvergence) of the problem studied by applying certain post processing techniques ( - projection method), we found the following. The exact solution in the -norm has a higher convergence rate, which is shown as ( ) for ( ), than. Add to that the practical side confirms the theoretical side in this thesis. 51