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Lecture Notes: Spring 2000

Joseph E Flaherty Amos Eaton Professor Department of Computer Science Department of Mathematical Sciences Rensselaer Polytechnic Institute

Troy, New York 12180

Spring 2000

Outline

1 Introduction

1.1 Historical Perspective

1.2 Weighted Residual Methods

1.3 A Simple Finite Element Problem

2 One-Dimensional Finite Element Methods

2.1 Introduction

2.2 Galerkin's Method and Extremal Principles

2.3 Essential and Natural Boundary Conditions

2.4 Piecewise Lagrange Approximation

2.5 Hierarchical Bases

2.6 Interpolation Errors

3 Multi-Dimensional Variational Principles

3.1 Galerkin's Method and Extremal Principles

3.2 Function Spaces and Approximation

3.3 Overview of the Finite Element Method

4 Finite Element Approximation

4.1 Introduction

4.2 Lagrange Bases on Triangles

4.3 Lagrange Bases on Rectangles

6.2 One-Dimensional Gaussian Quadrature

6.3 Multi-Dimensional Gaussian Quadrature

11 Linear Systems Solution

11.1 Introduction

11.2 Banded Gaussian Elimination and Prole Techniques11.3 Nested Dissection and Domain Decomposition

11.4 Conjugate Gradient Methods

11.5 Nonlinear Problems and Newton's Method

v

vi

1] A.K Aziz, editor The Mathematical Foundations of the Finite Element Method withApplications to Partial Di
erential Equations, New York, 1972 Academic Press.

2] I Babuska, J Chandra, and J.E Flaherty, editors Adaptive Computational Methodsfor Partial Di
erential Equations, Philadelphia, 1983 SIAM

3] I Babuska, O.C Zienkiewicz, J Gago, and E.R de A Oliveira, editors AccuracyEstimates and Adaptive Renements in Finite Element Computations John Wileyand Sons, Chichester, 1986

4] K.-J Bathe Finite Element Procedures Prentice Hall, Englewood Clis, 1995

5] E.B Becker, G.F Carey, and J.T Oden Finite Elements: An Introduction, ume I Prentice Hall, Englewood Clis, 1981

vol-6] M.W Bern, J.E Flaherty, and M Luskin, editors Grid Generation and AdaptiveAlgorithms, volume 113 of The IMA Volumes in Mathematics and its Applications,New York, 1999 Springer

7] C.A Brebia The Boundary Element Method for Engineers Pentech Press, London,second edition, 1978

8] S.C Brenner and L.R Scott The Mathematical Theory of Finite Element Methods.Springer-Verlag, New York, 1994

9] G.F Carey Computational Grids: Generation, Adaptation, and Solution Strategies.Series in Computational and Physical Processes in Mechanics and Thermal science.Taylor and Francis, New York, 1997

10] G.F Carey and J.T Oden Finite Elements: A Second Course, volume II PrenticeHall, Englewood Clis, 1983

11] G.F Carey and J.T Oden Finite Elements: Computational Aspects, volume III.Prentice Hall, Englewood Clis, 1984

vii

12] P.G Ciarlet The Finite Element Method for Elliptic Problems North-Holland,Amsterdam, 1978.

13] K Clark, J.E Flaherty, and M.S Shephard, editors Applied Numerical ics, volume 14, 1994 Special Issue on Adaptive Methods for Partial DierentialEquations

Mathemat-14] R.D Cook, D.S Malkus, and M.E Plesha Concepts and Applications of FiniteElement Analysis John Wiley and Sons, New York, third edition, 1989

15] K Eriksson, D Estep, P Hansbo, and C Johnson Computational Di
erentialEquation Cambridge, Cambridge, 1996

16] G Fairweather Finite Element Methods for Di
erential Equations Marcel Dekker,Basel, 1981

17] B Finlayson The Method of Weighted Residuals and Variational Principles demic Press, New York, 1972

Aca-18] J.E Flaherty, P.J Paslow, M.S Shephard, and J.D Vasilakis, editors Adaptivemethods for Partial Di
erential Equations, Philadelphia, 1989 SIAM

19] R.H Gallagher, J.T Oden, C Taylor, and O.C Zienkiewicz, editors Finite ements in Fluids: Mathematical Foundations, Aerodynamics and Lubrication, vol-ume 2, London, 1975 John Wiley and Sons

El-20] R.H Gallagher, J.T Oden, C Taylor, and O.C Zienkiewicz, editors Finite ments in Fluids: Viscous Flow and Hydrodynamics, volume 1, London, 1975 JohnWiley and Sons

Ele-21] R.H Gallagher, O.C Zienkiewicz, J.T Oden, M Morandi Cecchi, and C Taylor,editors Finite Elements in Fluids, volume 3, London, 1978 John Wiley and Sons

22] V Girault and P.A Raviart Finite Element Approximations of the Navier-StokesEquations Number 749 in Lecture Notes in Mathematics Springer-Verlag, Berlin,1979

23] T.J.R Hughes, editor Finite Element Methods for Convection Dominated Flows,volume 34 of AMD, New York, 1979 ASME

24] T.J.R Hughes The Finite Element Method: Linear Static and Dynamic FiniteElement Analysis Prentice Hall, Englewood Clis, 1987

viii

26] C Johnson Numerical Solution of Partial Di
erential Equations by the Finite ment method Cambridge, Cambridge, 1987

Ele-27] N Kikuchi Finite Element Methods in Mechanics Cambridge, Cambridge, 1986

28] Y.W Kwon and H Bang The Finite Element Method Using Matlab CRC ical Engineering Series CRC, Boca Raton, 1996

Mechan-29] L Lapidus and G.F Pinder Numerical Solution of Partial Di
erential Equations

in Science and Engineering Wiley-Interscience, New York, 1982

30] D.L Logan A First Course in the Finite Element Method using ALGOR PWS,Boston, 1997

31] J.T Oden Finite Elements of Nonlinear Continua Mc Graw-Hill, New York, 1971

32] J.T Oden and G.F Carey Finite Elements: Mathematical Aspects, volume IV.Prentice Hall, Englewood Clis, 1983

33] D.R.J Owen and E Hinton Finite Elements in Plasticity-Theory and Practice.Pineridge, Swansea, 1980

34] D.D Reddy and B.D Reddy Introductory Functional Analysis: With Applications

to Boundary Value Problems and Finite Elements Number 27 in Texts in AppliedMathematics Springer-Verlag, Berlin, 1997

35] J.N Reddy The Finite Element Method in Heat Transfer and Fluid Dynamics.CRC, Boca Raton, 1994

36] C Schwab P- And Hp- Finite Element Methods: Theory and Applications in Solidand Fluid Mechanics Numerical Mathematics and Scientic Computation Claren-don, London, 1999

37] G Strang and G Fix Analysis of the Finite Element Method Prentice-Hall, glewood Clis, 1973

En-38] B Szabo and I Babuska Finite Element Analysis John Wiley and Sons, New York,1991

39] F Thomasset Implementation of Finite Element Methods for Navier-Stokes tions Springer Series in Computational Physics Springer-Verlag, New York, 1981

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40] V Thomee Galerkin Finite Element Methods for Parabolic Problems Number 1054

in Lecture Notes in Mathematics Springer-Verlag, Berlin, 1984

41] R Verfurth A Review of Posteriori Error Estimation and Adaptive Renement Techniques Teubner-Wiley, Stuttgart, 1996

Mesh-42] R Vichevetsky Computer Methods for Partial Di
erential Equations: Elliptic tions and the Finite-Element Method, volume 1 Prentice-Hall, Englewood Clis,1981

Equa-43] R Wait and A.R Mitchell The Finite Element Analysis and Applications JohnWiley and Sons, Chichester, 1985

44] R.E White An Introduction to the Finite Element Method with Applications toNonlinear Problems John Wiley and Sons, New York, 1985

45] J.R Whiteman, editor The Mathematics of Finite Elements and Applications V,MAFELAP 1984, London, 1985 Academic Press

46] J.R Whiteman, editor The Mathematics of Finite Elements and Applications VI,MAFELAP 1987, London, 1988 Academic Press

47] O.C Zienkiewicz The Finite Element Method Mc Graw-Hill, New York, thirdedition, 1977

48] O.C Zienkiewicz and R.L Taylor Finite Element Method: Solid and Fluid ics Dynamics and Non-Linearity Mc Graw-Hill, New York, 1991

Mechan-x

1.1 Historical Perspective

The
nite element method is a computational technique for obtaining approximate tions to the partial dierential equations that arise in scienti
c and engineering applica-tions Rather than approximating the partial dierential equation directly as with, e.g.,

solu-
nite dierence methods, the solu-
nite element method utilizes a variational problem thatinvolves an integral of the dierential equation over the problem domain This domain

is divided into a number of subdomains called nite elements and the solution of thepartial dierential equation is approximated by a simpler polynomial function on eachelement These polynomials have to be pieced together so that the approximate solutionhas an appropriate degree of smoothness over the entire domain Once this has beendone, the variational integral is evaluated as a sum of contributions from each
nite el-ement The result is an algebraic system for the approximate solution having a
nitesize rather than the original in
nite-dimensional partial dierential equation Thus, like

nite dierence methods, the
nite element process has discretized the partial tial equation but, unlike
nite dierence methods, the approximate solution is knownthroughout the domain as a pieceise polynomial function and not just at a set of points.Logan 10] attributes the discovery of the
nite element method to Hrennikof 8] andMcHenry 11] who decomposed a two-dimensional problem domain into an assembly ofone-dimensional bars and beams In a paper that was not recognized for several years,Courant 6] used a variational formulation to describe a partial dierential equation with

dieren-a piecewise linedieren-ar polynomidieren-al dieren-approximdieren-ation of the solution reldieren-ative to dieren-a decomposition ofthe problem domain into triangular elements to solve equilibrium and vibration problems.This is essentially the modern
nite element method and represents the
rst applicationwhere the elements were pieces of a continuum rather than structural members

Turner et al 13] wrote a seminal paper on the subject that is widely regarded

1

2 Introduction

as the beginning of the
nite element era They showed how to solve one- and dimensional problems using actual structural elements and triangular- and rectangular-element decompositions of a continuum Their timing was better than Courant's 6],since success of the
nite element method is dependent on digital computation whichwas emerging in the late 1950s The concept was extended to more complex problemssuch as plate and shell deformation (cf the historical discussion in Logan 10], Chapter1) and it has now become one of the most important numerical techniques for solvingpartial dierential equations It has a number of advantages relative to other methods,including

two-
the treatment of problems on complex irregular regions,

the use of nonuniform meshes to reect solution gradations,

the treatment of boundary conditions involving uxes, and

the construction of high-order approximations

Originally used for steady (elliptic) problems, the
nite element method is now used

to solve transient parabolic and hyperbolic problems Estimates of discretization errorsmay be obtained for reasonable costs These are being used to verify the accuracy of thecomputation, and also to control an adaptive process whereby meshes are automaticallyre
ned and coarsened and/or the degrees of polynomial approximations are varied so as

to compute solutions to desired accuracies in an optimal fashion 1, 2, 3, 4, 5, 7, 14]

1.2 Weighted Residual Methods

Our goal, in this introductory chapter, is to introduce the basic principles and tools ofthe
nite element method using a linear two-point boundary value problem of the form

Lu] := ;

d dx

(p(x)du dx

) +q(x)u=f(x)
0< x <1
(1.2.1a)

u(0) =u(1) = 0: (1.2.1b)The
nite element method is primarily used to address partial dierential equations and ishardly used for two-point boundary value problems By focusing on this problem, we hope

to introduce the fundamental concepts without the geometric complexities encountered

in two and three dimensions

Problems like (1.2.1) arise in many situations including the longitudinal deformation

of an elastic rod, steady heat conduction, and the transverse deection of a supported

cable In the latter case, for example, u(x) represents the lateral deection at position

x of a cable having (scaled) unit length that is subjected to a tensile force p, loaded by

a transverse force per unit length f(x), and supported by a series of springs with elasticmodulus q(Figure 1.2.1) The situation resembles the cable of a suspension bridge Thetensile forcep is independent ofxfor the assumed small deformations of this model, butthe applied loading and spring moduli could vary with position

00

00 11

11 00

func-us multiply (1.2.1a) by a test or weight function v and integrate over (0
1) to obtain

to represent the integral of a product of two functions

The solution of (1.2.1) is also a solution of (1.2.2a) for all functions v for which theinner product exists We'll express this requirement by writingv 2 L

4 IntroductionEquation (1.2.2c) is referred to as a variational form of problem (1.2.1) The reason forthis terminology will become clearer as we develop the topic.

Using the method of weighted residuals, we construct approximate solutions by placing u and v by simpler functions U and V and solving (1.2.2c) relative to thesechoices Speci
cally, we'll consider approximations of the form

re-u(x) U(x) = N

X j=1 c j



j(x)
(1.2.3a)

v(x) V(x) = N

X j=1 d j



j(x): (1.2.3b)The functions 

2 will suce for the present discussion.)Thus, it's natural to expect U to also be an element of C

2(0
1) Mathematically, we gardU as belonging to a
nite-dimensional function space that is a subspace ofC

N the trial space and regard thepreselected functions 

j(x), j = 1
2
:
N, as forming a basis for S

N.Likewise, since v 2 L

2, we'll regard V as belonging to another
nite-dimensionalfunction space ^S

N called the test space Thus, V 2 S^

N

 L

2 and 

j(x), j = 1
2
:
N,provide a basis for ^S

N.Now, replacing v and u in (1.2.2c) by their approximationsV and U, we have

(V
LU]; f) = 0
8V 2 S^

N : (1.2.4a)The residual

r(x) := LU]; f(x) (1.2.4b)

is apparent and clari
es the name \method of weighted residuals." The vanishing of theinner product (1.2.4a) implies that the residual is orthogonal in L

2 to all functionsV inthe test space ^S

N.Substituting (1.2.3) into (1.2.4a) and interchanging the sum and integral yields

N X j=1 d

j( j

LU]; f) = 0
8d

j

j = 1
2
:
N: (1.2.5)Having selected the basis

j,j = 1
2
:
N, the requirement that (1.2.4a) be satis
ed forallV 2 S^

N implies that (1.2.5) be satis
ed for all possible choices of d

k,k = 1
2
:
N.This, in turn, implies that

( j

LU]; f) = 0
j = 1
2
:
N: (1.2.6)Shortly, by example, we shall see that (1.2.6) represents a linear algebraic system for theunknown coecients c

k, k = 1
2
:
N.One obvious choice is to select the test space ^S

N to be the same as the trial spaceand use the same basis for each thus, 

k(x) =

k(x), k = 1
2
:
N This choice leads

to Galerkin's method

( j

Lu]; f) = 0
j = 1
2
:
N
(1.2.7)which, in a slightly dierent form, will be our \work horse." With 

N This is typicallythe case As we shall see, it will be dicult to construct continuously dierentiableapproximations of
nite element type in two and three dimensions We can constructthe symmetric variational form that we need by integrating the second derivative terms

in (1.2.2a) by parts thus, using (1.2.1a)

(v 0 pu

0 +v u ; vf)dx ; vpu

0 j 1

0 = 0 (1.2.8)where ( )0 = d( )=dx The treatment of the last (boundary) term will need greaterattention For the moment, let satisfy the same trivial boundary conditions (1.2.1b) as

6 Introduction

u In this case, the boundary term vanishes and (1.2.8) becomes

A(v
u);(v
f) = 0 (1.2.9a)where

A(v
u) =

Z 1 0

(v 0 pu

0+v u)dx: (1.2.9b)The integration by parts has eliminated second derivative terms from the formulation.Thus, solutions of (1.2.9) might have less continuity than those satisfying either (1.2.1) or(1.2.2) For this reason, they are called weak solutions in contrast to the strong solutions

of (1.2.1) or (1.2.2) Weak solutions may lack the continuity to be strong solutions, butstrong solutions are always weak solutions In situations where weak and strong solutionsdier, the weak solution is often the one of physical interest

Since we've added a derivative to v by the integration by parts,v must be restricted

to a space where functions have more continuity than those in L

2 Having symmetry inmind, we will select functions u and v that produce bounded values of

A(u
u) =

Z 1 0

(u

0)2+u

2]dx: (1.2.10)Functions where (1.2.10) exists are said to be elements of the Sobolev space H

1 We'vealso required that u and v satisfy the boundary conditions (1.2.1b) We identify thosefunctions in H

1 that also satisfy (1.2.1b) as being elements of H

1

0 Thus, in summary,the variational problem consists of determiningu 2 H

1

0 such that

A(v
u) = (v
f)
8v 2 H

1 0

The substitution of (1.2.3b) with 

j replaced by 

j in (1.2.12a) again reveals the moreexplicit form

A( j

U) = (

j

f)
j = 1
2
:
N: (1.2.12b)Finally, to make (1.2.12b) totally explicit, we eliminateU using (1.2.3a) and interchange

a sum and integral to obtain

N X k=1 c k

A( j



k) = (

j

f)
j = 1
2
:
N: (1.2.12c)Thus, the coecientsc

k,k = 1
2
:
N, of the approximate solution (1.2.3a) are mined as the solution of the linear algebraic equation (1.2.12c) Dierent choices of thebasis 

deter-j, j = 1
2
:
N, will make the integrals involved in the strain energy (1.2.9b)and L

2 inner product (1.2.2b) easy or dicult to evaluate They also aect the accuracy

of the approximate solution An example using a
nite element basis is presented in thenext section

;1

(x)dx= 1

and

0< x 1

< x 2

0
otherwise :

where x

j;1=2 = (x

j +x j;1)=2 Using (1.2.6), show that the approximate solution

U(x) satis
es the dierential equation (1.2.1a) on the average on each subinterval( ), = 1 2

1.3 A Simple Finite Element Problem

Finite element methods are weighted residuals methods that use bases of piecewise nomials having small support Thus, the functions (x) and (x) of (1.2.3, 1.2.4) arenonzero only on a small portion of problem domain Since continuity may be dicult toimpose, bases will typically use the minimum continuity necessary to ensure the existence

poly-of integrals and solution accuracy The use poly-of piecewise polynomial functions simplifythe evaluation of integrals involved in the L

2 inner product and strain energy (1.2.2b,1.2.9b) and help automate the solution process Choosing bases with small support leads

to a sparse, well-conditioned linear algebraic system (1.2.12c)) for the solution

Let us illustrate the
nite element method by solving the two-point boundary valueproblem (1.2.1) with constant coecients, i.e.,

;pu

00+qu=f(x)
0< x <1
u(0) =u(1) = 0
(1.3.1)where p >0 and q 0 As described in Section 1.2, we construct a variational form of(1.2.1) using Galerkin's method (1.2.11) For this constant-coecient problem, we seek

where

(v
u) =

Z 1 0

A(v
u) =

Z 1

(v 0 pu

0+v u)dx: (1.3.2c)

With u and v belonging to H

nding the simplest continuous piecewise polynomial approximations of u and v Thiswould be a piecewise linear polynomial with respect to a mesh

0 = x 0

< x 1

< : : < x

N = 1 (1.3.3)introduced on 0
1] Each subinterval (x

;x j;1

if x j;1

x < x

j x

j+1

;x xj+1;xj ifx

j-1

j (x)

N x

0
otherwise : (1.3.4b)Using this basis with (1.2.3), we consider approximations of the form

U(x) =N;1

X j=1 c j



j(x): (1.3.5)Let's examine this result more closely

10 Introduction

x j

j+1 c

1

U(x)

Figure 1.3.2: Piecewise linear
nite element solutionU(x)

1 Since each 

j(x) is a continuous piecewise linear function of x, their summation

U is also continuous and piecewise linear Evaluating U at a node x

k of the meshusing (1.3.4b) yields

U(x

k) =N ;1 X j=1 c j



j(x

k) =c k :

Thus, the coecients c

k, k = 1
2
:
N ;1, are the values of U at the interiornodes of the mesh (Figure 1.3.2)

2 By selecting the lower and upper summation indices as 1 andN ;1 we have ensuredthat (1.3.5) satis
es the prescribed boundary conditions

U(0) = U(1) = 0:

As an alternative, we could have added basis elements 

0(x) and 

N(x) to theapproximation and written the
nite element solution as

U(x) = N

X j=0 c j



j(x): (1.3.6)Since, using (1.3.4b), U(x

N ;1 X k=1 c k

A( j



k) = (

j

f)
j = 1
2
:
N ;1: (1.3.8)Equation (1.3.8) can be evaluated in a straightforward manner by substituting replacing



k and

j using (1.3.4) and evaluating the strain energy andL

2 inner product according

to (1.3.2b,c) This development is illustrated in several texts (e.g., 9], Section 1.2).We'll take a slightly more complex path to the solution in order to focus on the computerimplementation of the
nite element method Thus, write (1.2.12a) as the summation ofcontributions from each element

N X j=1

A

j(V
U);(V
f)j] = 0
8V 2 S

N 0

(1.3.9a)where

j(V
U) =

Z x j xj;1 pV 0 U 0 dx
(1.3.9c)

A M

j (V
U) =

Z x j

x j;1

qV Udx
(1.3.9d)(V
f)j =

Z xj xj;1

j arising from inertial eects or sources of energy

Matrices are simple data structures to manipulate on a computer, so let us write therestriction of U(x) to x



j(x)



=  j;1(x)


j(x)]

 c j;1 c j



j(x)



=  j;1(x)


j(x)]

 d j;1 d j

12 IntroductionOur task is to substitute (1.3.10) into (1.3.9c-e) and evaluate the integrals Let us begin

by dierentiating (1.3.10a) while using (1.3.4a) to obtain

1=h j

h

j =x j

; x j;1

j = 1
2
:
N: (1.3.11b)Thus, U

pd j;1

d

j]



;1=h j

1=h j

 dx

x j;1 p



1=h 2 j

;1=h 2 j

;1=h 2

j 1=h

2 j

 dx

!

 c j;1 c j

 :

The integrand is constant and can be evaluated to yield



Kj = p

h j



1 ;1

;1 1

 : (1.3.12)The 2 2 matrix Kj is called the element stiness matrix It depends on j through h

j,but would also have such dependence if p varied with x The key observation is that

Kj can be evaluated without knowing c

j;1, c

j,d j;1, or d

j and this greatly simpli
es theautomation of the
nite element method

j (V
U) =

Z x j

x j;1

qd j;1

d

j]



 j;1

 j



 j;1



j]

 c j;1 c j

 dx:

Withq a constant, the integrand is a quadratic polynomial inx that may be integratedexactly (cf Problem 1 at the end of this section) to yield

The
nal integral (1.3.9e) cannot be evaluated exactly for arbitrary functions f(x).Without examining this matter carefully, let us approximate it by its linear interpolant

f(x) f

j;1

 j;1(x) +f

d j;1

d

j]



 j;1

 j



 j;1



j]

 f j;1 f j

lj = h j

6



2f j;1+f

j f

j = N X j=1

d j;1

d

j]p h



1 ;1

;1 1

  c j;1 c j

 :

The
rst and last contributions have to be modi
ed because of the boundary conditionswhich, as noted, prescribe c

j = d

1]p h

1]c

1] + d 1

d

2]p h



1 ;1

;1 1

  c 1 c 2



+

+d N;2

d N;1]p h



1 ;1

;1 1

  c

N ;2 c

N ;1



+ d

N ;1]p h

1]c

N ;1]:

Although this form of the summation can be readily evaluated, it obscures the need for thematrices and complicates implementation issues Thus, at the risk of further complexity,we'll expand each matrix and vector to dimensionN ;1 and write the summation as

N X k=1 A S

j = d 1

d 2

d

N ;1]p h

2

6 6 6 6 4

7 7 7 7 5

2 6 6 6 4

c 1 c 2

c

3 7 7 7 5

14 Introduction

+d 1

d 2

d

N;1]p h

2 6 6 6 6 4

1 ;1

;1 1

3 7 7 7 7 5

2

6 6 6 4

c 1 c 2

+ + d

1

d 2

d

N ;1]p h

2 6 6 6 6

4 1 ;1

;1 1

3 7 7 7 7 5

2

6 6 6 4

c 1 c 2

+d 1

d 2

d

N ;1]p h

2 6 6 6 6 4

1

3 7 7 7 7 5

2

6 6 6 4

c 1 c 2

c N;1

3

7 7 7 5

Zero elements of the matrices have not been shown for clarity With all matrices andvectors having the same dimension, the summation is

N X j=1 A S

j =dTKc
(1.3.16a)where

K= p h

2 6 6 6 6 6 6 6 4

2 ;1

;1 2 ;1

;1 2 ;1

;1 2 ;1

;1 2

3 7 7 7 7 7 7 7 5

(1.3.16b)

c= c 1

c 2

c

N ;1]T

d= d 1

d 2

in Figure 1.3.3, the elemental indices determine the proper location to add a local matrixinto the global matrix Thus, the 2 2 element stiness matrix K is added to rows

2 = d 1

d

2] p h



A S

3 = d

2

d

3]p h



K= p h

2

6 6 6 6 6 6 6 6 6 6 4

Figure 1.3.3: Assembly of the
rst three element stiness matrices into the global stinessmatrix

j ;1 and j and columns j ;1 and j Some modi
cations are needed for the
rst andlast elements to account for the boundary conditions

j =dTMc
(1.3.17a)

N X j=0

(V
f)j =dTl (1.3.17b)where

M= qh

6

2 6 6 6 6 6 4

4 1

1 4 1

1 4 1

1 4

3 7 7 7 7 7 5

f

0+ 4f

1+f 2 f

1+ 4f

2+f 3

f N;2+ 4f

N ;1+f

N

3

7 7 7 5 : (1.3.17d)

16 IntroductionThe matrixM and the vector lare called the global mass matrix and global load vector,respectively.

Substituting (1.3.16a) and (1.3.17a,b) into (1.3.9a,b) gives

k, k = 1
2
:
N ;1, of the
nite element solution are mined by solving a linear algebraic system With c known, the piecewise linear
niteelement U can be evaluated for any x using (1.2.3a) The matrix K+M is symmetric,positive de
nite, and tridiagonal Such systems may be solved by the tridiagonal algo-rithm (cf Problem 2 at the end of this section) in O(N) operations, where an operation

deter-is a scalar multiply followed by an addition

The discrete system (1.3.19) is similar to the one that would be obtained from acentered
nite dierence approximation of (1.3.1), which is 12]

(K+D)^c= ^l
(1.3.20a)where

D =qh

2

6 6 6 4

11

1

3

7 7 7 5

^l=h

2

6 6 6 4

f 1 f 2

f N;1

3

7 7 7 5

c^=

2

6 6 6 4

^

c 1

^

c 2

^

c N;1

3

7 7 7 5 : (1.3.20b)

Thus, the qu and f terms in (1.3.1) are approximated by diagonal matrices with the

nite dierence method In the
nite element method, they are \smoothed" by couplingdiagonal terms with their nearest neighbors using Simpson's rule weights The diagonalmatrix D is sometimes called a \lumped" approximation of the consistent mass matrix

M Both
nite dierence and
nite element solutions behave similarly for the presentproblem and have the same order of accuracy at the nodes of a uniform mesh

Example 1.3.1 Consider the
nite element solution of

Relative to the more general problem (1.3.1), this example has p=q= 1 and f(x) =x.

We solve it using the piecewise-linear
nite element method developed in this section onuniform meshes with spacing h = 1=N for N = 4
8
:
128 Before presenting results,

it is worthwhile mentioning that the load vector (1.3.15) is exact for this example Eventhough we replaced f(x) by its piecewise linear interpolant according to (1.3.14), thisintroduced no error since f(x) is a linear function ofx

Letting

e(x) =u(x); U(x) (1.3.21)denote the discretization error, in Table 1.3.1 we display the maximum error of the
niteelement solution and of its
rst derivative at the nodes of a mesh, i.e.,

1:= max

1<j<N je 0

(x

;

j )j: (1.3.22)

We have seen that U

0(x) is a piecewise constant function with jumps at nodes Data inTable 1.3.1 were obtained by using derivatives from the left, i.e.,x

;

j = lim!0

x j

; Withthis interpretation, the results of second and fourth columns of Table 1.3.1 indicate that

=hare (essentially) constants hence, we may conclude thatjej

1 =O(h

2)and je

=h 2 je 0 j 1 je 0 j 1

The
nite element and exact solutions of this problem are displayed in Figure 1.3.4 for

a uniform mesh with eight elements It appears that the pointwise discretization errorsare much smaller at nodes than they are globally We'll see that this phenomena, calledsuperconvergence, applies more generally than this single example would imply

Since
nite element solutions are de
ned as continuous functions (of x), we can alsoappraise their behavior in some global norms in addition to the discrete error norms used

in Table 1.3.1 Many norms could provide useful information One that we will use quite

18 Introduction

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

p(e

0)2+q

2]dx

 1=2 : (1.3.23a)This expression may easily be evaluated as a summation over the elements in the spirit

of (1.3.9a) With p=q= 1 for this example,

kek 2

A=

Z 1 0

(e 0

)2

+e 2

]dx

 1=2 : (1.3.23b)Other global error measures will be important to our analyses however, the only one

that we will introduce at the moment is the L

2 norm

kek

0 :=

Z 1 0 e

2(x)dx

 1=2

Results for theL

2 and strain energy errors, presented in Table 1.3.2 for this example,indicate that kek

0 = O(h

2) and kek

A = O(h) The error in the H

1 norm would beidentical to that in strain energy Later, we will prove that these a priori error estimatesare correct for this and similar problems Errors in strain energy converge slower thanthose inL

2 because solution derivatives are involved and their nodal convergence isO(h)(Table 1.3.1)

N kek

0 kek 0

=h 2 kek A kek A

2 and strain energy for the piecewise-linear
nite element solution

of Example 1.3.1 (Numbers in parenthesis indicate a power of 10.)

Problems

1 The integral involved in obtaining the mass matrix according to (1.3.13) may, ofcourse, be done symbolically It may also be evaluated numerically by Simpson'srule which is exact in this case since the integrand is a quadratic polynomial Recall,that Simpson's rule is

Z h 0

x j;1



 j;1

 j



 j;1



j]dx:

Using (1.3.4), determine Mj by Simpson's rule to verify the result (1.3.13) Theuse of Simpson's rule may be simpler than symbolic integration for this examplesince the trial functions are zero or unity at the ends of an element and one half atits center

2 Consider the solution of the linear system

AX=F
(1.3.24a)

20 Introductionwhere F and X are N-dimensional vectors and A is an N N tridiagonal matrixhaving the form

2 6 6 6 6 6 4

a 1 c 1 b 2 a 2 c 2

b N;1 a N;1 c N;1 b

N a N

3 7 7 7 7 7 5 : (1.3.24b)

Assume that pivoting is not necessary and factorA as

A =LU
(1.3.25a)where L and U are lower and upper bidiagonal matrices having the form

L =

2 6 6 6 6 6 4

l

N 1

3 7 7 7 7 7 5

(1.3.25b)

2 6 6 6 6 6 4

u 1 v 1 u 2 v 2

u N;1 v N;1 u N

3 7 7 7 7 7 5 : (1.3.25c)

Once the coecientsl ,j = 2
3
:
N,u

UX=Y
(1.3.26b)then,

LY=F: (1.3.26c)2.1 Using (1.3.24) and (1.3.25), show

u

1 =a 1

l =b j

=u j;1

u

j =a j

; l c j;1

j = 2
3
:
N

= = 2 3

2.2 Show that Y and X are computed as

Y

1 =F 1

Y

j =F j

; l Y j;1

j = 2
3
:
N

X

N =y N

=u N

X

j = (Y j

; v j X j+1)=u j

where pand q are positive constants and f(x) is a smooth function

3.1 Show that the Galerkin form of this boundary-value problem consists of
nding

(v 0 pu

0+v u)dx ;

Z 1 0

vf dx= 0
8v 2 H

1 0 :

For this problem, functions u(x)2 H

1

0 are required to be elements of H

1 andsatisfy the Dirichlet boundary condition u(0) = 0 The Neumann boundarycondition at x= 1 need not be satis
ed by either u orv

3.2 Introduce N equally spaced elements on 0 x 1 with nodes x

j = jh,

j = 0
1
:
N (h= 1=N) Approximateu by U having the form

U(x) = N

X j=1 c k

0(1) = 0 nor does it have to We will discussthis issue in Chapter 2.)

22 Introduction3.3 Write a program to solve this problem using the
nite element method devel-oped in Part 3.2b and the tridiagonal algorithm of Problem 2 Execute yourprogram with p = 1, q = 1, and f(x) = x and f(x) = x

2 In each case, use

N = 4, 8, 16, and 32 Lete(x) =u(x); U(x) and, for each value ofN, pute jej

N 0

 H 1

0 satis
es (1.2.12).Letting e(x) =u(x); U(x), show

A(e
e) =A(u
u); A(U
U)where the strain energyA(v
u) is given by (1.3.2c) We have, thus, shown that thestrain energy of the error is the error of the strain energy

1] I Babuska, J Chandra, and J.E Flaherty, editors Adaptive Computational Methodsfor Partial Dierential Equations, Philadelphia, 1983 SIAM.

2] I Babuska, O.C Zienkiewicz, J Gago, and E.R de A Oliveira, editors AccuracyEstimates and Adaptive Renements in Finite Element Computations John Wileyand Sons, Chichester, 1986

3] M.W Bern, J.E Flaherty, and M Luskin, editors Grid Generation and AdaptiveAlgorithms, volume 113 of The IMA Volumes in Mathematics and its Applications,New York, 1999 Springer

4] G.F Carey Computational Grids: Generation, Adaptation, and Solution Strategies.Series in Computational and Physical Processes in Mechanics and Thermal science.Taylor and Francis, New York, 1997

5] K Clark, J.E Flaherty, and M.S Shephard, editors Applied Numerical ics, volume 14, 1994 Special Issue on Adaptive Methods for Partial DierentialEquations

Mathemat-6] R Courant Variational methods for the solution of problems of equilibrium andvibrations Bulletin of the American Mathematics Society, 49:1{23, 1943

7] J.E Flaherty, P.J Paslow, M.S Shephard, and J.D Vasilakis, editors Adaptivemethods for Partial Dierential Equations, Philadelphia, 1989 SIAM

8] A Hrenniko Solutions of problems in elasticity by the frame work method Journal

14] R Verf!urth A Review of Posteriori Error Estimation and Adaptive Renement Techniques Teubner-Wiley, Stuttgart, 1996.

1 Is the Galerkin method the best way to construct a variational principal for a partialdierential system?

2 How do we construct variational principals for more complex problems? Specically,how do we treat boundary conditions other than Dirichlet?

3 The nite element method appeared to converge asO(h) in strain energy andO(h

2)

inL

2 for the example of Section 1.3 Is this true more generally?

4 Can the nite element solution be improved by using higher-degree polynomial approximations? What are the costs and benets of doing this?

piecewise-We'll tackle the Galerkin formulations in the next two sections, examine higher-degreepiecewise polynomials in Sections 2.4 and 2.5, and conclude with a discussion of approx-imation errors in Section 2.6

2.2 Galerkin's Method and Extremal Principles

\For since the fabric of the universe is most perfect and the work of a mostwise creator, nothing at all takes place in the universe in which some rule ofmaximum or minimum does not appear."

1

2 One-Dimensional Finite Element Methods

- Leonhard EulerAlthough the construction of variational principles from dierential equations is animportant aspect of the nite element method it will not be our main objective We'llexplore some properties of variational principles with a goal of developing a more thoroughunderstanding of Galerkin's method and of answering the questions raised in Section 2.1

In particular, we'll focus on boundary conditions, approximating spaces, and extremalproperties of Galerkin's method Once again, we'll use the model two-point Dirichletproblem

Lu] :=;p(x)u

0]0+q(x)u=f(x)
0< x <1
(2.2.1a)

u(0) =u(1) = 0
(2.2.1b)with p(x)>0, q(x)0, andf(x) being smooth functions on 0 x 1

As described in Chapter 1, the Galerkin form of (2.2.1) is obtained by multiplying(2.2.1a) by a test function v 2 H

where

(v
f) =

Z 1 0

(v 0 pu

0+v u)dx
(2.2.2c)and functions v belonging to the Sobolev space H

1 have bounded values of

Z 1 0

0 if it also satises the trivial boundary conditions

v(0) =v(1) = 0 As we shall discover in Section 2.3, the denition of H

1

0 will depend onthe type of boundary conditions being applied to the dierential equation

There is a connection between self-adjoint dierential problems such as (2.2.1) andthe minimum problem: nd w 2 H

1

0 that minimizes

Iw] =A(w
w);2(w
f) =

Z 1

p(w

0)2+qw

2

;2wf]dx: (2.2.3)

Maximum and minimum variational principles occur throughout mathematics and physicsand a discipline called the Calculus of Variations arose in order to study them The initialgoal of this eld was to extend the elementary theory of the calculus of the maxima andminima of functions to problems of nding the extrema of functionals such as Iw] (Afunctional is an operator that maps functions onto real numbers.)

The construction of the Galerkin form (2.2.2) of a problem from the dierential form(2.2.1) is straight forward however, the construction of the extremal problem (2.2.3)

is not We do not pursue this matter here Instead, we refer readers to a text on thecalculus of variations such as Courant and Hilbert 4] Accepting (2.2.3), we establishthat the solution u of Galerkin's method (2.2.2) is optimal in the sense of minimizing(2.2.3)

Theorem 2.2.1. The function u 2 H

1

0 that minimizes (2.2.3) is the one that satises(2.2.2a) and conversely

Proof Suppose rst that u(x) is the solution of (2.2.2a) We choose a real parameter 

and any function v(x)2 H

1

0 and dene the comparison function

w(x) = u(x) +v(x): (2.2.4)For each functionv(x) we have a one parameter family of comparison functionsw(x)2 H

1 0

with the solution u(x) of (2.2.2a) obtained when  = 0 By a suitable choice of  and

v(x) we can use (2.2.4) to represent any function in H

1

0 A comparison function w(x)and its variationv(x) are shown in Figure 2.2.1

ε v(x) w(x)

4 One-Dimensional Finite Element MethodsExpanding the strain energy and L

2 inner products using (2.2.2b,c)

Withp >0 and q 0, we have A(v
v)0 thus, u minimizes (2.2.3)

In order to prove the converse, assume that u(x) minimizes (2.2.3) and use (2.2.4) toobtain

0(0) = 2A(v
u);(v
f)] = 0:

Thus, u is a solution of (2.2.2a)

The following corollary veries that the minimizing function u is also unique

Corollary 2.2.1. The solution u of (2.2.2a) (or (2.2.3)) is unique

Proof Suppose there are two functions u

1

u 2

2 H 1

0 satisfying (2.2.2a), i.e.,

A(v
u

1) = (v
f)
A(v
u

2) = (v
f)
8v 2 H

1 0 :

Subtracting

A(v
u 1

; u

2) = 0
8v 2 H

1 0 :

Since this relation is valid for all v 2 H

; u 2

u 1

u 1

; u

2) is positive unless u

1 = u

2 Thus, itsuces to consider cases when either (i) q(x)  0, x 2 0
1], or (ii) q(x) vanishes atisolated points or subintervals of (0
1) For simplicity, let us consider the former case.The analysis of the latter case is similar

When q(x)  0, x 2 0
1], A(u

1

; u 2

u 1

; u

2) can vanish when u

0 1

; u 0

Corollary 2.2.2. If u
w are smooth enough to permit integrating A(u
v) by parts thenthe minimizer of (2.2.3), the solution of the Galerkin problem (2.2.2a), and the solution

of the two-point boundary value problem (2.2.1) are all equivalent

Proof Integrate the dierentiated term in (2.2.3) by parts to obtain

Iw] =

Z 1 0

The last term vanishes since w 2 H

1

0 thus, using (2.2.1a) and (2.2.2b) we have

Iw] = (w
Lw]);2(w
f): (2.2.5)Now, follow the steps used in Theorem 2.2.1 to show

A(v
u);(v
f) = (v
Lu]; f) = 0
8v 2 H

1 0

and, hence, establish the result

The minimization problems (2.2.3) and (2.2.5) are equivalent when w has sucientsmoothness However, minimizers of (2.2.3) may lack the smoothness to satisfy (2.2.5).When this occurs, the solutions with less smoothness are often the ones of physicalinterest

F(x
w
w

0)dx (2.2.6a)when w satises the \essential" (Dirichlet) boundary conditions

w(0) =
w(1) =: (2.2.6b)Letw 2 H

Smooth stationary values of (2.2.6) would be minima in this case and correspond

to solutions of the dierential equation (2.2.1a) and boundary conditions (2.2.1b)