ACCURACY AND CONVERGENCE OF FINITE-ELEMENT/

GALERKIN APPROXIMATIONS OF TIME-DEPENDENT

PROBLEf-,ISWITH EMPHASIS ON DIFFUSION AND CONVECTION

by

L. C. Wellford, Jr. and J. T. Oden

Department of Aerospace Engineering andEngineering Mechanics

Division of Engineering Mechanics

Texas Institute for Computational Mechanics

The University of Texas at Austin

AUSTIN, TEXAS

December, 1973

ACCURACY AND CONVERGENCE OF FINITE-ELEMENT/

GALERKIN APPROXIMATIONS OF TIME-DEPENDENT

PROBLEMS WITtI EMPHASIS ON DIFFUSION AND CONVECTION

L. C. Wellford Jr. and J. T. Oden

1. Introduction

With the exception of a number of important papers in the

engineering literature of the early 1960's, the mathematical

theory of finite element approximations reached a degree of mat-

urity only in 1969 and 1970. It was around this time that the

strength and elegance of the method and its relationship to con-

temporary research areas in interpolation theory began to be

appreciated by the mathematical world. There has since been a

flood of literature on mathematical features of the method, and a

number of books and expository articles have already appeared

[1-5]. The latest chapter in the theory, time-dependent problems,

is naturally a very new and incomplete addition. However, portions

of this chapter stand to be developed relatively quickly, for,

with minor modifications, they can be drawn from the existing and

more extensively developed theory of finite-differences, much of

which has bearing on certain finite-element approximations. In

fact, it is now widely accepted that the natural way to approach

many time-dependent problems is to approximate the spatial varia-

2

tion of the dependent variables using finite-element methods and

to approximate the temporal behavior by finite-differences. This

practice should make it possible to optimally exploit inherent ad-

vantages in both methods.

In very recent times, several distinct methods have emerged

for studying the central mathematical questions surrounding finite

element schemes for time-dependent problems: accuracy, convergence,

and stability. The ultimate utility of the method in applications

to evolution problems hinges on the answers to these questions. In

the present paper, we attempt to assess the degree to which these

questions have been answered thus far, as well as to describe sev-

eral techniques that can be used to arrive at them.

More specifically, this paper is largely an exposition aimed

at analyzing a number of techniques for arriving at error estimates

of finite-element/finite-difference approximations of certain time-

dependent problems. We shall emphasize linear diffusion equations

for two reasons: first, to keep the scope of the paper within

reasonable limits; second, because the underlying theory is more

completely developed for this class of problems. However, we shall

also consider certain finite-element approximations of the convec-

tion problem which, to date, has not been treated deeply in the

literature. Indeed, much of what we consider can be applied to

finite-element approximations of all types of linear boundary/

initial-value problems.

In the exposition to follow, we describe a number of basic

techniques for determining rates-of-convergence of approximations

of a class of linear time-dependent problems. We choose to cate-

gorize these techniques as follows:

(i)

(ii)

(iii)

(i v)

Semigroup Theoretic Estimates

Energy Methods

L2 Methods

Other Methods

3

(i) It is a widely known result in the theory of partial-

differential equations that for a broad class of evolution equations

the fundamental solution operator is a member of a semigroup of .

operators (e.g. [6,7]). By using certain basic properties of semi-

groups, error estimates can often be easily obtained, particularly

in those cases in which it is possible to transform the problem into

a one-parameter family of elliptic problems. This makes the problem

of estimating the spatial rate-of-convergence relatively straight-

forward. The rate-of-convergence of the temporal approximations can

be established by exploiting certain properties of fundamental solu-

tions. For example, if the fundamental solution is a member of a

semigroup, then it has a matrix exponential form which can be approx-

imated using the Pade~ matrix-approximation theory. The use of semi-

groups in difference approximations of time-dependent problems has

been discussed by several authors (e.g. Peetre and Thomee [8], Widlund

[9]). Some features of the methods we discuss were used by Babuska

andAzizin collaboration with Fix [10].

(ii) In most mathematical models of physical problems a norm

can be developed which is equivalent to the total energy in the

system. Normally, the study of convergence of various Galerkin

approximations in an appropriate energy norm is a very natural under-

taking. This is due to the fact that the weak forms of most boundary-

value problems of mathematical physics can be interpreted physically

in terms of changes in energy. The use of energy methods can be found

4

in the fini te- difference 1itera ture (e.g. [11]), and variants have

been used by Fujii [12] and Oden and Fost [13] for finite-element

analyses.

(iii) A fairly extensive literature has accumulated in recent

years on Galerkin approximations of the diffusion equation in which

L2-estimates in spatial variations and L2- or Loo-estimates are ob-

tained in the temporal variations. This "L2-theory" has been largely

developed by Douglas and Dupont [14, 15, 16], Varga [17], Wheeler [18],

and others. Interestingly enough, the methods do not involve energy-

error estimates and enable one to go directly to stronger L2-estimates.

(iv) There are, of course, several other techniques in use for

studying finite element approximations of time-dependent problems.

The projection methods of Thomee [19], for example, make use of the

fact that the finite-element technique produces a system of differ-

ence equations in Rn. By using standard difference concepts and

projections from Rn back into the space V in which the original prob-

lem is posed, error-estimates for certain finite-element approxima-

tions can be obtained. Alternately, certain parabolic problems can

be shmvn to have coercive properties under an appropriate choice of

norm (see, e.g., Lions and Magenes [20]). Thus, "elliptic -type"

error estimates can be obtained, as shown by Cella and Cecchi [21].

It is only a matter of interpretation as to whether or not these

"projection" and "coercive-operator" techniques do not actually be-

long to the semigroup and the energy methods described previously.

As in elliptic theory, the study of convergence of finite

element approximations rests firmly on certain results from inter-

polation theory. For this reason we discuss in the section follow-

ing this introduction certain features of interpolation theory which

(Z.l)

...5

are essential for our investigation. Next, in Section 3, we des-

cribe finite-element/Galerkin models of a general class of diffusion

problems, and in Sections 4,5, and 6 we obtain error estimates for

these models using the semigroup theory, energy methods, and LZ-

methods, respectively. In section 7 we describe finite-element

approximations of certain linear convection problems, and in Sections

8 and 9 we apply semigroup and LZ techniques to obtain error-estimates

associated with these approximations.

2. Finite Element Approximation and Interpolation

Consider a Hilbert space H whose elements are functions u(~)

of points x = (x ,x ,... ,xn) in some bounded domain Q of Rn. In~ 1 2

subsequent sections, the context shall make clear the specific

properties of H, but for the moment we need only assume that it

is endowed with an inner product, (u ,u ). A finite-element model1 2

~of Q (and H) is another region Q which is partitioned into a fin-

ite number E of disjoint open sets Qe called finite elements:

/\ EQ = LJ"ITe

e=l

Here "ITeis the closure of Qe'

a set of local basis functions

that

Within each element we identifya

~~(e)(~) which have the property

~~(e) (x) = 0N ~

x :f Qe

M,N = 1,2, ••• ,Ne;

e,f = 1,Z, ... ,E ; lal :. k (Z.Z)

Here ~~ is a nodal point labelled M in element Qf, o~~, o~, o~,are Kronecker deltas, N is the number of nodes in element Q ,e e

6

and we have used multi-index notation; i.e., ~ and ~ are ordered

n-tuples of non-negative integers, a =

following conventions are used:

(a , a , ... , a ), and the1 2 n

+ C/. + ••• + a2 n

(2. 3)

The local representation of a function in terms of the basis

functions ~~(e)(~) is of the form

Na~(e)

(2.5)

is of the formG

E E A~Xg(x)lal:'k ~=l ~ ~ -

and the global representationE

Vex) = LJ V (x ) =- 1 e -ee=

X~ (x)fi -

Here X~(x) are global basis functions given byfi -

E N (e)= U re Q N~~ (e) (x )

e=l N=l fi N -e(2.6)

(e)Nwhere Q fi defines a Boolean transformation of the disconnected

(e)Nsystem of elements into the connected model Q (i.e., Q ~ = 1 if

(e)node N of Q coincides with node x~ of Q and Q ~ = 0 if other-e _ u

wise).~ a G

Suppose Q = Q. Then the set of functions {X-ex)} -l;lal<kfi - ~- -

defines a finite-dimensional subspace of H which we shall denote

Sh(Q). Here h is the mesh parameter of the finite element mesh

(i.e., if he = dia(Q)e' h = max(h1,h

2, ••• ,hE)). We define Sh(Q)

more precisely subsequently. For economy in notation, we shall

re-label the global basis functions X~(x) as ~N(x) N=1,2, ... ,NOfi - -

7

(. . d h' ~al.e., we lntro uce an automorp lsm aN

-

global representation is of the form

Vex) (2.7)

Returning now to the space H, consider a typical element

u(=u(x)). The pair {(. ,.) '{~N}NO } define an orthogonal projec-- N=l

tion Qh : H+Sh(Q) such that

_ No NQhu = W = E (u,~ )~N(x)

N=l

where ~M(~) ::;/~N'~M)-l~M(~)' The function

(2.8)

(2.9)

is referred to as the (pointwise) interpolation error of the finite

element approximation W(x) = Qhu(~). Its properties depend ex-

plicitly on the properties of the subspace Sh(Q).

To appreciate the importance of the interpolation error ln

finite-element/Galerkin theory, consider the abstract boundary-

value problem, find uEH such that

(P(u),v) = (f,v) 'f v E M (2.10)

where P:H+M is a linear operator. The finite-element/Galerkin

approximation of the solution u of (2.10) is the function UESh such

that

The function

(P(U),V) = (f,V) (2.11)

e(~) :: u(~) - U(~) (2.12)

is the (pointwise) approximation error of U(~). Since U(~) £ Sh(Q),

there is a mapping ITh : H+Sh(Q) such that IThU = U; but ITh is not ~n

8

orthogonal projection into Sh(Q) relative to {(. ,.) '{~N}}' Indeed,

by setting v = V in (2.10) and subtracting (2.11) from the result,

we see that (Pe,V) = 0; thus, Pe is orthogonal to She

The function

£(~) = TIhu(~) - Qhu(~) = U(~) - Qhu(~) (2.13)

is referred to as the projection error. In most of the develop-

ments to follow, we show that it is possible to bound certain

norms of the projection error by the corresponding norm of the"

"interpolation error; i.e., we derive relationships of the type

(2.14)

Since e = E-£, use of the triangle inequality glves

(2.15)

Thus, (i) if the coefficient (1 + C(h)) remains bounded as h+O, and

(ii) if I lEI I +0 as h+O, we have proved convergence of the methodH

in the I 1·1 IH norm. Criteria (i) is a question of stabil~ of the

approximation, while (ii) is a question of consistency. The latter

question depends explicitly on H and the properties of Sh(Q), so

that the convergence of the method is connected to the interpolation

error in a fundamental way.

In many instances, the space H is a Sobolev space Hm(Q), the

elements of which are functions whose partial derivatives of order

< m are square integrable on Q. The inner-product in Hm(Q) lS

then

and the norm is

f E D~uD~vdQQ I al<m~ -

E f (D~u)2dQ1~I~m Q

(2.16)

(2.17)

where dQ = dx dx ...dx .1 2 n In such cases we construct the finite-

9

element subspaces S (Q) so as to have the following properties:h

(i) For every u£Hm(Q), there is a constant C such

that IIQhullm:' Cllullm for all It > O.

(ii) If p(~) is a polynomial of degree < k,

QhP(~) = p(~)

(iii) Let h+O uniformly (i.e., for each refinement of

the mesh, let the radius Pe of the largest sphere that can be In-

scribed in Qe be proportional to he)' Then there is a constant K

independent of h such that

IIElln:' Khk+l-nlulk+l

for n < m, where lulk+l is the semi-norm

lul2:: ~ f (D~u)2dQm I a l2,m Q

(2.18)

(2.19)

Interpolation results such as (2.18) were derived by Strang [22],

Ciarlet and Raviart [23], and others. We shall henceforth assume

that the spatial interpolation spaces Sh(Q) have properties (i)-

(iii) .

3. Finite-Element Approximation of the

Diffusion Equation

We now consider a class of time-dependent problems charac-

terized by equations of the form

au(~,t) + A(~)u(~,t) = f(~,t)

~ £ Q; t £ (O,T]

D~u(x,t) = 0 , X £ an; lal < m - 1~ - -

= u (x)o -

X £ Q (3.1)

where A is the 2m-th order differential operator

A(x) = E (-1) I~ID~A (x)D~- I a/ , I B 1< m g~- ~-

10

(3.2)

and the coefficients A (x) are such that A is m-elliptic (stronglya B --,-

elliptic); i.e., there exists a sesqui1inear form

a(u,v) :: (Au,v) = J E A B(X)D~UD~VdQQlal,IBI<m ~ ~-~ ~-

such that there are positive constants µ and µ for whicho 1

a(u,u) > µ IIul12- 0 m

and

(3. 3)

(3.4)

a(u,v)< µ Ilullllvll (3.5)- 1 m m

We then replace (3.1) by the equivalent (weaker) variational prob-

lem,

(dU(t) ,v) + a(u(t) ,v) = (f(t) ,v)at 0 0

(u(',O),v) = (u ,v) , V V E Hm (Q)o 0 0 0

V t E (O,T] (3.6)

~here Hm(Q) is the Sobolev space of Hm(Q) functions with compacto

support in Q.

Now consider a Galerkin approximation of (3.5) which involves

seeking a function U(~,t) E Sh(Q)XC1(0,T) such that

( au ( t) V) + a (U ( t) ,V) = ( f (t) ,V)at ' 0 0

t E (O,T]

(3. 7)(U ( • ,0) ,V) = (u , V)o 0 0

Since a continuous dependence on t is still assumed, U(~,t) lS

referred to as a semidiscrete Galerkin approximation. Now, the

coefficients AN in (2.7) are functions of time t. Thus, intro-

ducing (2.7) into (3.7) we obtain a system of first-order differ-

ential equations for the specific coefficients AN(t) corresponding

to the finite element approximation:

11

~oc AM(t) + ~oK Ar>'[(t) = f (t)~1=1 NM M=l NM N

~oc AM(O) = 9- (3 8)NM N .M=l

Here

fN = 9-(~N) ;

9-N = (u , ~N)I 0 1

and AM(t) = dAfv1(t)/dt.

In practical calculations, we introduce the partition P of

[0,T] comp 0sed 0 f the set {t ,t ,..., tR} \II her e 0 = t < t <o 1 0 1

< tR = T with t 1 - t = ~t; and we introduce the sequence {Un}Rn+ n n=o

to denote the value of the function U(t) E Sh(Q)XCl(O,T) at the

time points of partition P. Thus {Un}R = {Vet )}R . Then wen=o n n=o

construct a family of finite difference-Galerkin approximations

associated with parameter B (0 ~ e ~ 1) which represent solutions

to the following equation:

(OtUn,V) ° + (l-e)a(Un+l,V):+ Ba(Un,V) = (f (t) ,V)

°t E CO,T]

(3.9)v V E S h (Q)

l.e., 0tUn =

Evaluation of (3.10) with the finite element approxi-

(UO,V) = (tio,V)° °

denotes the forward difference operator;where 0t

Un+l _ Un~t

mation (2.7) leads to a system of algebraic equations for the

coefficients An = AN(t ).N n .

N ° n+lM~l [CNM + ~t(l- e) °NQKQM]A~l

N .EOC AO = 9-

M=l N~r-M N(3.10)

12

4. Error Estimates for the Diffusion Equation

Using Semigroup Theoretic Results

The calculation of error estimates for equation (3.6) can be

embedded in the theory of the approximation of elliptic partial

differential equations by using semigroup tools. In particular,

the traditional characterization of the semigroup through the resol-

vent operator can be used to transform the parabolic diffusion prob-

lem into a one-parameter family of elliptic problems. The error ln

the spatial discretization for the parabolic problem can then be

related to errors encountered in modeling elliptic problems. On

the other hand, errors due to the discretization in time can be

determined through the exponential function representation for the

semigroup and its relationship to the Pade~ approximations.

Initially, we must define the components of the error of the

approximation scheme. u(~,t) is the exact solution to (3.6);

U(~,t) is the solution to the semidiscrete Galerkin approximation

(3.7); and Un(x) is the solution to the finite difference-Galerkin-approximation (equation (3.~)) at time t = n~t. We introduce the

definitions,

e(x,n~t)-o(~,n~t)

= u(x,nfit) - Un(~) = approximation error (4.1)

= u(~,nfit) - U(~,nfit) = semidiscrete approxi-

mation error (4.2)

Un(x) = temporal approximation

error (4.3)

Then, for any choice of norm on ~,

Ile(t)11 = Ilu(·,n~t) - Un(o) + U(',nfit) - U(',nfit) I 1

= 110 +TII ~ 11011 + IITII (4.4)

g(O) = So

13

The evolution problem (3.1) is reminiscent of the linear

dynamical system

g - ~g = f

the solution of which is

q = e~tq + fte~(t-s)f(s)ds- -0 0 -

For simplicity, let us assume that ~(s) = O. Then the matrix

~(t) given by

is called the fundamental solution operator and

q(t) = E(t)q- "'-0

The operator ~(t) is a member of a multiplicative semigroup, G;

i.e., if E (t) and E (t) are in G, then the semigroup properties,-1 -2

(i) E'E E G-1 ~2

(closure) (4.5)

(ii) E •(E •E ) = (E 1 • E ).E (assoc iat ivi ty)-1 -2 ~3 - -2 ~3

are satisfied. In fact, we also have the important properties,

§(t)§(s) = ~(t + s)

limt+O E (t) = I- - (4.6)

How does one construct the fundamental solution operator ~(t)

from a given matrix A? Let L(g(t))

form of get). Then

g denote the Laplace trans-

Thus, if s is not an eigenvalue of ~,

- )-19 = (s!-~ goE(t) = L-l(sI-A)-l- - - (4.7)

NOlV a similar situation is encountered -in the use of the

weak formulation of partial differential equations of the type

(3.6). In this case we define a fundamental solution operator

E(x,t) such that- -

14

u(x,t) = E(x,t)u (x)- - - 0-

Then it is clear from (4.7) and the definition of the resolvent-1operator R(x,s) = (sI-A) that

....., ....., - -

Hence, if u = L[u]

R(x,s) = f~e-stE(x,t)dt- - 0 --

(4.8)

and, of course,

R(x,s)u (x)- .....,0-

(4.9)

(4.11)

where r is a contour in the complex plane. Now we take the Laplace

transform of the weak parabolic partial differential equation (3.6)

and set f(t) = 0, for convenience. Then

L[(()1(t) ,v) ] + L[a(u(t) ,v)] = 0at 0

Nmv

and

L[( a~(t) ,v)].at

= f ft€stdu(x,t)v(x)dtdQQ 0 dt -

= sfQu(~;s)v(~)dQ - fQuo(~)v(~)dQ

= s(u(s) ,v) - (u ,v)000

(4.12)

L [a(u(t) ,v)] = a(u(s),v) (4.13)

Hence introducing (4.12) and (4.13) into (4.11), we obtain the

boundary-value problem

s(u(s),v) + a(u(s),v) = (u ,v)000

(4.14)

Similarly, if U is the Galerkin approximation to the exact

15

solution, then we define the approximate fundamental solution

operator ~h(~,t) so that

(4.15)

If we introduce the approximation ~h to matrix 6, an approx-

imate resolvent operator ~h(~'s) is defined such that

~h(~ ' s) = (s I-~h) -1

Then the approximate resolvent is related to the approximate fun-

damental solution operator through

~h(~,t) = L-l[(S!-~h)-l] = L-l[~h(~'s)]

Thus,

(4.16)

Hence, 'ifU = L[U], then the transformed version of (4.15) glves

iJ(~,s) = ~h (~,s)uo (~)

and, of course,

U(~,t) = 2;' fr~(~,s)estdsuo(~)1

(4.17)

Now if we take the Laplace transform of the semidiscrete

Galerkin equation (3.7) and set f(t) = 0 for convenience, we ob-

tain an approximate boundary-value problem for (4.14)

s (U(s) ,V) + a(U(s) ,V) = (u ,V) V V£Slt(Q) (4.18)a a a

Subtracting (4.18) from (4.14) and defining the transformed approx-

L[a(t)] = L[u(t)-U(t)], we get

V V £ Sh(Q) (4.19)

a(s) by u(s) - U(s) =S(0(5) ,V) + a(o(s) ,V) = a

a

Thus it is clear that if we can estimate the approximation error

imatiori error

a(s) for the boundary-value problem (4.14), then we can deduce the

approximation error for problem (3.6) by using the inverse

16

Laplace transform, (4.10), and (4.17) ~

a (t) = L -1 [0 (s)] = L -1 [u(s) - IT(s) ]

To insure that the inverse Laplace transform exists, we assume

that the operator ~ is the infinitesimal generator of a strongly

continuous semigroup. The Hille-Yosida-Phillips theorem (Friedman

[24]) guarantees that if the operator ~ is the infinitesimal gen-

erator of a strongly continuous semigroup, there exist real numbers

M and w such that for every R s > w, s is in the resolvent set ofeA (i.e., (s !-6)-1 exists) and

IIR(x,s)nll < M- - - (s-w)lI n = 1,2,3, ... (4.21)

Thus if we select the contour r in the complex plane so that if

S E r, S is-i~ the resolvent set, Res> B, and Res = B as s+oo

where B is a small negative constant, then the integrand in (4.20)

is bounded in accordance with. (a:2l) as s increases. Thus the

Laplace transform (4.20) exists. We choose B so that lsi < µ .u

The operator A satisfies the m-elliptic condition (3.4).

Thus the operator ~ + Re sl occurring in (4.14) also satisfies

an m-elliptic condition since

= a(v, v) + Res (v,v) 0

= a(v,v) + Res Ilvll~

> a(v,v) + Bllvl12

m> µ II v II

2 m (4.22)

Thus the boundary value problem (4.14) is strongly elliptic for

17

each s € r. Equation (4.22) leads to the following error estimate.

Theorem 4.1. If TI(s) is the Galerkin approximation in Sh to ~(s)

the solution of problem (4.14), then

(4.23)

Proof: From the m-elliptic property of operator A + Re s~ (4.22)

we have that if TI*(s) is an arbitrary element of Sh

µ211~(s) - U(s) II~:. Res(~(s) - U(s) ,u(s) - U(s))

+ a(~(s) - TI(s), ~(s) - TI(s))

= R s(~(s) - TI(s), ~(s) - U*(s) +eU* (s) - U ( s)) + a (u(s) - IT(s),

u(s) - U*(s) + U*(s) - U(s))

= Res (~(s) - TI(s), ~(s) - U* (s))

+ a(~(s) - U(s), ~(s) - U*(s))

(4.24)

To obtain the last result, we have used the identity

Res(~(s) - TI(s), U*(s) - U(s)) + a(u(s) - U(s), U*(s) - TI(s))

= 0

obtained by taking the real part of (4.19) and setting V = U*(s)

TI(s). Now using the Schwarz inequality (3.5) in (4.24)

µ.llu(s) - U(s)hI2 < ResM Ilu(s) - U(s)llo Ilu*(s) - U(s)llo2 m - 1

+ µ 11~(s) - U(s) II IIU*(s) - U(s) II1 m m

< ( R sM. M + µ ) II ~ (s) - U(s) I te 1 21m

IIU*(s) - TI(s)II (4.25)mThus, simp1if~ing (4.25), we obtain

11~(s) - TI(s)II < (ResM1M2 + µl) IIIT*(s) - TI*(s)IIm - m

µ 2

(4.26)

Now let U*(s) be the arbitrary element of Sh which interpolates

li(s). Then we see from the interpolation result (2.18) that

18

Ilu(s) - U(s)11 <m -

This completes the proof.

Thus the error estimate for the semidiscrete approximation (3.7)

to (3.6) can be determined.

Theorem4.2. If u(x,n~t) is the solution to (3.6) at time t = nfit-and U(x,nfit) is the solution to (3.7) at time t = n~t, then

Ilcr(x,nfit)II = Ilu(x,nfit) - U(x,nfit) II- m - - m

:. C2hk+1-mlu(x,nfit) I

k+

l(4.27)

Proof: 'Using the trans form (4.20) -be tween the error involved in

the approximation of equation (3.6) and (4.14) the result follows

Ilcr(~,nfit)11m = Ilu(~tnfit) - U(~,n~t) 11m

= II 1 J (u(s) - U(s)) estds II2lTi r m

< C hk+ 1- mJ Iu(s) I est ds- 2lTi r k+l

= C hk+l-mlu(x t) I2 _, k+l

Now let E} (X;t,T) be the fundamental solution operator assoc-I -

iated with the semidiscrete Galerkin approximation (3.7) and

E}~t,e(X;(V+l)fit,vfit) be the fundamental solution operator (oftenI _

called the amplification matrix) associated with the finite dif-

ference-Galerkin approximation (3.9). Then

(4.28)and

Un+l(x) = Eh~t,e(~;(V+l)fit,v~t)un(~) (4.29)

The operators Eh and Eh6t,e have the semigroup property (4.6).

Thus

(4.31)

19

(4.32)

Now the semidiscrete problem (3.7) is assumed to be well-posed and

the finite difference-Ga1erkin problem (3.9) is assumed to be

stable (depends continuously on the initial data). Thus the fun-

damental solution operator and the amplification matrix are bounded

IIEh(~;(V-l)fit,O)u(~)llm:' C31IU(~)"m v = O,l, ... ,N

V U £ Sh(Q) (4.33)

IIEh~t,e(~;n~t,vfit)V(~) 11m:' C41IV(~) 11m 0:. v:.n:.N

V V £ S h (Q)

(4.34)

Equation (4.34) implies that

IIEhfit,e(~;n6t,vfit) II =

o < v < n ::. N (4.35)

Consider the semidiscrete finite element-Galerkin equation

(3.8). Solving (3.8) for the value of the solution vector AM(v~t)

based on the "initial value" of AM((v-l)fit) and setting fn=O

AM(V6t) = EM8A8((V-l)~t) (4.36)

where E~,18= e-fitBM~ and B~lB = C~,ln-lKn 8. The matrix EM8 is a dis-

crete approximation for the fundamental solution operator Eh. Sim-

ilarly, solving the finite difference-Galerkin equation (3.10) for

the value of A~ in terms of A~-l and setting fn =0, lve obtain

( 4. 37)

where

20

The operator F~S in (4.37) is a discrete analogue for Ehfit,e.

A useful approximation for the term EMS in (4.36) can be ob-

tained through the Pade~ approximations. The Pade~ approximations

Rp,q (~tBMS) are a rational matrix approximation for e-fitBMS de-

fined by

where

Rp,q(~tB~ ..IB) = nptq(~tB~IS) [dp,q(~tBMS)]-l

= e-fitB~I13+ O(~tBrl13)r (4.38)

r =

q 1- kE (p+q-k)! [(p+q) !k! (q-k)! ]-l.-LitBM13)

k=OpE (p+q-k)! [(p+q) !k! (p-k)! ]-l(~tBt'IS)k

k=O

p+q+l

From (4.36) ,'(4.37), and (4.38) we see that the following re-

lationships hold

RO,l = F~IS = EMS + 0(~t)2

R = F~13 = EMS + 0(~t)2 (4.39)1 , 0

1R 1 , 1 = Ft113= EMS + 0 (~t) 3

Here-the' choice of e corresponds to the forward difference, back-.ward difference, and Crank-Nicholson schemes, respectively. The

notation indicates that each term of the matrix FM~ can be expressed

as the sum of the corresponding term in the matrix EMS plus a term

of order of magnitude ~tr. Equation (4.39) implies that

No\V

EMS

Ei'-IB

EMS

F~B = 0(~t)2

F~lS = 0(~t)2

FAs = O(fit)3

(4.40)

(4.41)

21

Using (4.40), and (4.41), we find that

IIEh(~;v~t,(v-l)fit)

IIEh(~;vfit,(v-l)~t)

IIEh (~;vfit,(v-I) fit)

- E~t,l(~;V~t,(V-l)~t) II <

- E~t'O(~;v~t,(V-l)~t) II <1

- E~t , 'Z (~ ; v fi t , (v - 1) ~t) II <

(4.42)

Based on the previous results, the temporal error estimate theorem

can be introduced. We will prove the temporal error estimate the-

orem only for the forward difference approximation (8=1). Error

estimates for other temporal operators can be derived in similar

fashion.

Theorem 4.3. Let U(x,n~t) be the solution to the semidiscrete-

Galerkin equation (3.7) at time t=nfit and Un(x) be the solution to

the forward differenced Galerkin approximation, (3.9) with 6=1, at

time t=n~t. In addition let (4.33), (4.35), and (4.42) hold. Then

the temporal approximation error is

IIT(x,nfit)11 = IIU(x,nfit) - Un(x)11 < C ~t IIUO(x)11- m - - m- 8 - m

(4.43)

Proof: Using the semigroup property (4.31) and (4.32)

II T(x,nfit)II = II U(x,n~t) - Un(x) II- m - - m

II(~h (~;nfit,O) - E~t, 1 (~;nfit,O))UO(~) I'm

nI I z: Eh~t,l(x;n~t,v~t) [Eh~t,l(x;v~t,(V-l)~t)

1 ~ ~v=

n< E IIE}~t,l(x;n~t,vfit)11 IIEhfit,l(x;v~t,(V-l)~t)

1 1 - ~v=

22

And using (4.33), (4.35), and (4.42)

IIT(x,nfit)11 < nC C ~t2C IIUo(x)11~ m- 45 3 - m

The total error estimate of the approximation is given in the

following theorem.

Theorem 4.4. If the components of the error of the approximation

are given by (4.1), (4.2), and (4.3), then

Ile(~,n~t) 11m :. C2hk+l-mlu(~,n~t) Ik+l + C/~t IIUu(~) 11m

(4.44)

Proof: From (4.4)

Ile(~,n~t) 11m:' Ilcr(~,n~t)11m + IIT(~,n~t) 11m

But introducing (4.27) and (4.43)

Ile(~,nfit)11m:' C2hk+l-mlu(~,n~t) Ik+l + Ca~t

S. Error Estimates for the Diffusion

Equation Using the Energy Method

We often find it mathematically convenient and physically

appealing to seek an error estimate that indicates the error in

energy of an approximation. This estimate of the error in energy

of the approximation requires the definition of an appropriate

energy norm. For the diffusion problem the most natural energy

norm can be constructed using the bilinear form a(u,v) connected

with the operator ~ (3.3). The bilinear form a(u,v) satisfies

the m-ellipticity condition (3.4). Thus effectively, an inner

product space HA is introduced in association with the operator

~ and the inner-product a(u,v). The norm associated with this

23

inner-product is called the energy ~,

IluliA = la(u,u) (5.1)

the properties of which depend intrinsically on the operator A.

The energy space HA is topologically equivalent to the Sobolev

space Hm(Q); i.e., positive constants y and y exist such thato 1

y 0 I Iu I 1m:" I I u I I A :. y 11 I u I Im (5.2)

In the subsequent developments we find it convenient to replace

the energy norm II' II A by the Sobolev norm II' II m us ing (5.2).

Now suppose ul . is the exact solution to (3.6) evaluated atn

time point t=n~t and Un is the solution to the finite difference-

Galerkin equation (3.9) at time point t=n~t. We seek an estimate

in energy for the difference between uln and Un. In this section

we consider only the case in which the temporal operator is the

forward difference operator in time. Thus we set 8=1 in (3.9).

Now let Wn be an arbitrary function in the subspace Sh(Q). Then

we define the error components

en ::: U 1'1 - Unn

En = Wn _ Un

En = u In - Wn

n ,', IE =-~ul - 0tU it,. at n

(5.3)

We can now cite a theorem which specifies the behavior of the

error.

Theorem 5.1. Let en, En, En, and En be the error components de-

fined in (5.3). Then

a(En En) :::-(0 en En) - a(En En) - (En En), t' 0 ' , 0(5.4)

24

Proof: Subtracting (3.9) with 8=1 from (3.6) evaluated at t=nfit

- 0tul + 0tul. - 0tUn,V) + a(ul: - wn + wn - Un V) = 0n non '

V E Sh(Q)

(5.5)

(aulat n

(~In - 0tUn,V)o + a(uln - Un,V) = 0

where~ln denotes }f evaluated at time point

in the form

t=nfit.

(5.5)

Rewriting

and introducing (5.3), we get

(En + 0 en V) + a (En + En,V) = 0 V E Sh (Q) (5.6)t ' a

Now setting V=En and using the bilinearity of (.,.) and a(','),0

we obtain (5.4).

The error component En can be estimated using the following

theorem.

Theorem 502. If the error components en, En, En, and En are re-

related by (5.4), then there exist positive constants µ , y , µ , andall

M such that2

µ IIEnl1 < y~IIEnll + 2µ IIEnl1 + 2~.IIIEnll (5.7)o m- m 1 TIl 2 0

Proof: The last term on the right hand side of (5.4) can b~ es-

timated using the Schwarz inequality for the L2 inner-product and

the embedding result IIEn 110 :.. c II En II m' There exist positive con-

stants M and M such that1 2

(En, En) a :. M 1 II En II a II En II 0 :. ~\ II En II a II En II m

Then introducing (3.5) and (5.8) into (5.4)

a(En,En) < _(Oten,En) + µ IIEnl1 liEn/1m- 0 1 m+ H II En II II En II m

2 a

Now to estimate (Oten,En)o set V=En in (5.6)

_(Oten,En)o = a(En,En) + a(En,En) + (En,En)o

(5.8)

(5.9)

(5.10)

2S

Introducing (5.1), (5.2), and (5.8) into (5.10)

_(Oten,En) < y211Enili + µ IIEnllm IIEnllm + t-I IIEnl1

o - 1 1 2 0

Now introducing (5.11) into (5.9) and using the m-elliptic property

of a(' ,.) (equation (3.4)), we obtain the energy estimate.

µ IIEnl12 < a(En En) < y211Enl12 +o m- , - 1 m

+ 2M II En II2 0

2 µ II En II IIEn II1 m m

II En II m

The resul t (5. 7) follows by dividing by II En IIm'

Then the estimate of the approximation error en is contained

ln the following theorem.

Theorem 5.3. If the error components en, En, En, and En are defined

by (5.3) and En satisfies (5.7), then there exist positive constants

M , M , M , and M such that4 5 6 7

and

II en II <!\I IIEn II + M IIe;nIIm- 5 m 4 0(5.12)

Ilenll < M hk+l-mlulk+l + M fit II lull I (5.13)m - 6 7 2

where IIIull12 = f(d2U)2dQ.

2 Q dt2

Proof: The first result follows from (5.3) and (5.7). Equation

(5.7) implies that

IIEn IIm :. M 3 IIEn IIm + M4 IIe;nII 0

But from (5.3)

Ilenllm= I IEn+Enl 1m

< II En II m + II En I 1m< (1 +M ) II En II + M IIe;nII

3 m 4 0

= M IIEn II +;"1 IIe;nII5 m 4 0

(5.14 )

26

The second result can be obtained by expanding u in a Taylor's

series about t=n6t, introducing the result into (5.3) , and taking4

the L2 norm.

II £n II < cfit III u III (5.15)0- 2

Then introducing (Z.18) and (5.15) into (5.14), we obtain (5.13).

6. Error Estimates for the Diffusion Equation

Using L2 Methods

The derivation of LZ error estimates for finite element-Galerkin

models for the diffusion equation follows very naturally from the

interpolation theory developed in section 2. The key point in the

derivation of the L2 error estimate is establishing a bound on the

approximation error for all time points based on the pointwise error

estimate in time. Traditionally this has been carried out through

a discrete version of Gronwall's Inequality (Lee [26]). The method

used here to derive error estimates for the finite element-Galerkin

solution circumvents the complex arguments involved in using the

discrete Gronwall's inequality. For simplicity, the finite element-

Galerkin solution will be defined as piecewise linear between dis-

cretization points in time. Then we identify the nodal values in

time, effectively, through a collocation procedure at the temporal

node points. Thus, we embed the approximate solution in the temporal,

as well as the spatial, variables in a subspace of the original sol-

ution space. This procedure leads to the use of the previously

developed interpolation results in time, as well as space.

We consider the solution of the weak parabolic problem (3.6)

with f(t) = O. The exact solution to this problem has been charac-

terized by Goldstein [27]

27

(6.1)

However if the coefficients AaSin equation (3.2) are such that~-A~~ £ COO(Q) and Dr(A~~) £ Loo(Q) for all ~, ~, and y, then

u £ COO[O,T;Hm(Q)] (6.2)

We assume in this section that the coefficients AaS are constants~

so that (6.2) holds. However, this choice is not basic to the

method of derivation of error estimates to be introduced here. In

fact, the methods used here are valid as long as the regularity

of the exact solution u is such that

Let us define the space ~1 [O,T;Sh(Q)]. If PC1[0,T;Sh(Q)]

is the space of functions which are continuous with piecewise con-

tinuous derivatives between the points of the partition P of [O,T]

introduced in Section 3, then pel [O,T;Sh(Q)] is the subspace of

PC1[0,T;Sh(Q)] of functions which are piecewise linear; Sh(Q)

is the finite dimensional subspace of Hm(Q) described in Section 2.

We seek an approximate Galerkin solution Un to (3.6) using the

finite difference-Galerkin approximation (3.9) with 8=1 (this cor-

responds to the forward difference in time case). The solution of

this problem is fully equivalent to the solution of the following

problem. Find a U £ ~l[O,T;Sh(n)] such that

(!¥(!l,V)~=n6t + a(U(t) ,V)t=n~t = 0 V V £ PC1[0,T;Sh(Q)]

n = O, ... ,R

(U ( • ,0) , V) = (u ,V)o 0 0

(6.3)

where the notation indicates that a collocation is performed at

time points t=n~t, n=O, ... ,R. Our subsequent developments pertain

to (6. 3) .

28

Now let Wet) be that element of 'P"C1 [O,T;Sh(Q)] which inter-

polates u in the sense of (2.8), and let Un = U(n~t). We can

define the error components of the approximation by

e = u(t) - U(t)

E 1 = Wet) - U(t)

E = Wet) - Un(6.4)

2

E = u(t) - Wet)

Then the following theorem describes the behavior of the error.

Theorem 6.1. If u(t) is the exact solution to (3.6), U(t) is the

approximate solution to (3.6) defined by (6.3), Wet) is the inter-

polant of u(t) defined by (2.8), and the error components are

defined by (604)" then.

rtfit:.t :. (n+l)fit

Proof: Subtracting (6.3) from (3.6)

(6.5)

(duCt) Vet)) _(dUet) V(t))t=n~t + a(u(t) ,Vet))at.' 0 dt ' 0

- a(U(t),V(t))t=n~t = 0

Tf V E F'C'l[o,T; h(Q)] (6.6)

But (dU(t) V(t)t=n~t = (dU(t) Vet)) for nfit< t < n+l(~t), sinceat' 0 '\... ' - '-

U(tl is piecewise linear in t. Thus.1.1.-

( d u.~~)_d Uit),V (t)) + a (u(t)-Un ,V (t)) = 0 Tf V E ~ 1 [0,T ;Sh (Q) ]o

(6. 7)

29

Rewriting (6.7)

(au(t)_aW(t)+aW(t)_aU(t) Vet)) + a(u(t)-W(t)+W(t)-Un,V(t))=Oat at at at' 0

Then introducing (6.4) into (6.8) and setting V = El

(U1,E) + aCE ,E ) = _(aE,E "\ - a(E,E )()L 1 2 1 at 1-'0 1

(6.9)

The result C6~5) follows directly from (6.9) by splitting up the

second term on the left:.hand side.

The following theorem describes the variation of E with time:1

Theorem 6.2. Let the hypotheses of Theorem 6.1 hold, then2

~ LIIE 112 :.~11~112 + nilE 112 +!:l. IlaU(t)(t_n~t)112dt 1 0 4nat 0 1 0 4µ at mo

n~t < t < (n+l) fit (6.10)

Proof: E -E = U (t) -un = a ¥ ( t) (t-:n~t) for nfit < t < (n+1) 6t .2 1 t-

Thus using the Cauchy-Schwarz inequality (3.5)

a (E -E E) = a (au (t) (t-nfit) E )2 l' 1 at ' 1

< µ II au (t) (t-nfit) II II E II nfit < t < (n+1) fit1 ~r m 1 m

Now using the elementary inequality ab < ~ [Ea2+(.!.)b2] for E > a- £

(with a = µ IlaU(t)(t_n~t) II b = liE II and E = ~)1 at m' 1 m' 2µ o

aCE -E ,E ) < ~II aU(t) (t-nfit)112 + µ II E 112 n~t < t < (n+l) fit2 1 1 - 4µ at mOl m

o(6.11)

Now by the definition of the interpolation operation (2.8) used

to define Wet) £ 'i'5t'l[O,T;Sh(Q)]and the best approximation property

of Hilbert spaces (O'de'n1[28]), we have that

(E, V) = 0o

(6.12)

30

Now for the moment we suppress the dependence on time. If V E

IZl[O,T;Sh(Q)], then for any particular time point, V E Sh(Q). Now

Sh(Q) contains all polynomials of degree ~k, and A is the

operator of order 2m introduced in (3.2). In this development we

restrict the subspace Sh to the class of subspaces which allow inter-

polation in the sense of (2.8) of derivatives through the m-lth order

of the solution U. And assume that A maps Sh(Q) into Sh(Q) .. This implies

-. -1if V £ Pc1 [O,T;'Sh(Q)], then AV £ PC [O,T;Sh(Q)J. Thus

a(E,E ) = (E,AE )1 1 0

= 0

Introducing (6.11) and (6.13) into (6.5) gives

(6.13)

+ ~II au (t) (t-nfit)112

4µ Clt mo

+ µ liE 112

o 1 mn~t < t < (n+l)fit (6.14 )

Now using the definition of the LZ norm and the m-elliptic property

of (3.4), we get

l.-dIIE 112 + µ liE 1122 dt 1 0 0 1 m

+ µ 'IE 112

o 1 mnfit < t < (n+l)fit (6.15)

2Thus cancelling µo IIE 111m on each side of (6.15) and

inequality (a,b) < !..../laI12+ nllbl12 to estimate the

- 4non the right hand side in (6.15), we obtain (6.10).

using the

first term

31

The following theorem gives an estimate for E at the dis-1

cretization points in time.

Theorem 6.3. Let the hypothesis of Theorem 6.1 hold. Then

II E "L ( ) = sup I I E (M~t) I I ~ c [I I E (0) I I + I IE I IH 1 (L )1 00 L2 O<M<R 1 0 It 1 0 2

+ fit II duet) II ],\, LZ(HITI)

(6.16)

where, for example, the notation L2(Hm) indicates the function lS

in LZ in the time variable and Hm in the spatial variable.

Proof: Integrating (6.10) from t=nfit to t=(n+l)~t gives

liE (n+l)~tI12- liE (nfit)I 12 < ~ f(n+l)fifldEI12dt

1 0 1 0 - Zn nfit dt 0

(n+l)~t 2+ Zn fnfit IIElllodt

2 (+ 1)+ ~ f n tJtII dUet) (t-n~t) 112dt

2u 0 n~ t d t m(6.17)

Now simplifying the last term on the right hand side

(n+1) tJtII dU (t) (t-n~t) II J~dtf atnH "(n+1) 6t au (t) 112 (t-nH) 2dt

= f II dt mnfit

I" ,t !

r ' t I

d ~t2 f(n+l)~tlldU(t)112dt3 - n4! ',_ d t m

)-... -- --- J.......... s-

t I • '-::1 (6.18)

Introducing (6.17) into (6.18)

liE (M~t) II 2 - lIE ( 0) I I 21 0 1 0

and summing from n=O to n=M, we obtain

< L ll£lt I I~ II 2 d t + Zn lIt:. l I I E II 2 d t- 2n 0 dt 0 0 1 0

+ U ~ fit2 /'tJ t II dU (t) II 2dt (6 .19)6u 0 dt mo

•

32

But II aU(t) II < c II au(t) II for all t £ [O,M t]. C is of theat m - 2 at m 2

form l+f(h,~t). We do not introduce the explicit expression for

f into our equations because it can only result in higher order

terms in the error estimate. This result then implies that

T+ 2n f IIE II2dto 1 0

(6.20)

2 1 II aE II 2II E 1 (M~ t) II ~ :. [II E 1 (0) II 0 + 2n" at L 2 (L 2)222

11 C2 2 lau(t)11 ]+ --L-~t I at L2(Hm)6110

But the classical Gronwall's inequality (Bellman [29]) implies

that if Ix(t) 1< a + ft1x(s) Ie ds where a and C are positive con-- 0

stants, then Ix(t) I~ a ect. Thus

I I E ( M~t) I I 2 < [I I E ( 0) I I 2 + Lz I I ~E I IL (L )1 0 - 1 0 n at 2 2

+ µ~C 22~ t 21 I aU ( t) I I 2 ] e Z n T

6110 at L2(Hm)

or uSlng the embedding result II~EIIL (L ) ~ C /IEIIH

1(at 2 2 3 LZ)

IIE1(M~t)"0:' C4[IIE1(0)llo + IIEIIl-p(L2) + fit·/Ia~~t)IIL2(Hm)]

(6.21)

The result (6.16) follows by taking the supremum of (6.21) for all

integers M ,,,ith0 < M < R. Note that PCl [0,T;S (Q)] is a space ofh

piecewise linear functions. Thus since El £ PLl[O,T;Sh(Q)], El

attains;its maximum at one of the discretization points in time.

Thus the supremum of the El at the discretization points in time

is exactly the Loo norm.

33

The approximation error can then be established through the

following theorem.

Theorem 6.4. Let the hypotheses of theorem 6.1 hold. Then

IleIILz(Lz) :. cs[lle(O) 110 + IIEIIH1(L2

) + 6t '1Ia~~t)IILz(Hm)]

(6.22)

Proof: Using (6.4),

IleIIL2(L2) = IIEIIL2{L2) + IIE11IL2(L2)k

< IIEll L (L ) + Til E 1IIL (L )Z 2 00 2

(6.23)

Then introducing (6.16) into (6.23), we have~ ~

I Ie IIL (L ) < TC IIE (0) II +·1 I E II . + T C I IE II 12 2 - ~ 1 0 _L2(L2) 4 H (Lz)

k+ TC fit:llau(t)1I

4 at L2 (HID)

< C [lle(O)11 + IIEII 1 + ~t:llau(t)IJ ]soH (LZ) at L2 (Hm)

We can now use the interpolation results of section Z to

derive the final error estimate.

Theorem 6.5. If the space pel[O,T;Sh(Q)] satisfies conditions

(i), (ii), and (iii) in section 2, then

IIell L (L ) :. c II e (0) II + C hk+lll u II k 12 2 5 e, 8 H ~(H + )

+ C ~t II au II5 at L2 (Hm)

(6.24)

Proof: Th~ interpolation error E is caused by an orthogonal pro-

jection Qx in space and Qt in time. The projection operator Qh

34

(equation (2.8)) is the composition

Qh = QxQt

Then using the triangle inequality and the fact that IIQ I I < 1x

But from the interpolation result (2.18)

k+lII u - Qxu II I-[l (L ) :... C 6h II u I IH1 (IIk + 1)

" 2

Ilu-QtuIIH~(L2):' C7f.t IluIIH2(L2)

(6.25)

(6.26)

Then introducing (6.25) and (6.26) into (6.22), we obtain the

estimate (6.24).

7. Finite Element Approximations

for the Convection Equation

We now consider a class of time-dependent problems character-

ized by equations of the form

au(~,t) + L(x)u(x,t) = f(x,t)at X E Q t £ [O,T]

X £ aQ- (7.1)

X £ Q

where L is the first order differential operator

L(x) = E Aa(x)D~1~1=1

(7.2)

35

Now we replace (7.1) by the weaker but equivalent problem,

(dU(t) v) + (Lu(t) ,v) = (f(t) ,v)dt ' 0 0 0

(u(',O),v) =(u ,v)o 0 0

'if t £ (O,T] (7 .3)

Now we construct a semidiscrete Galerkin approximation of

(7.2) which involves seeking a function U(~,t) E Cl [O,T;Sh(Q)]

( d ~ (t) ,V) + (L U(t) ,V) = (f (t) ,V)tOO 0

( U ( • , 0) , V) = ( u ,V)o 0 0

'rJ t E (O,T] (7.4)

Now introducing the finite element approximations (2.7) into

(7.3) we obtain a system of first order differential equations

for the coefficients AN(t)

(7.5)

where

<PN

temporal operator is replaced by a finite difference operator in

time through a procedure similar to that introduced in Section 3 ..In this manner we construct a family of finite difference - Galerkin

approximations associated with parameter 8 (0 :. 8 ~ 1) which rep-

resent solutions to the following equation:

+ (1-8) (LUn+l V), 0 + 8(LUn,V) = (f(t),V)o 0

(U 0 , V) 0 = (u0 ' V) 0

t £ (O,T) V V E Sh(Q)

(7.6)

..36

where t denotes the forward difference operator in time; i.e.,

8tUn = Un+l_Un. Evaluation of (7.5) with the finite elementfit

approximation (2.7) leads to a system of algebraic equations for

the coefficients A~ = AN(tn)

N Na n+l 0 nE [CNM + fit(1-8)0NQDQM]Ai,f = E [CNM - ~t80NQDQM]AN + fitfN

~I=l ~I=l(7.7)

8. Error Estimates for the Convection Equation

Using Semigroup Theoretic Methods

We seek an estimate for the error encountered in the approx-

imation of (7:3) by the finite difference-Ga1erkin approximation

(7.6) with 8=1 (the forward difference in time case). Let u(x,n~t)

be the solution to (7.3) at t=nfit, U(x,nfit) be the solution to' the

semidiscrete Galerkin approximation (7.4) at time t=nfit, and Un(x)-be the solution to the finite difference-Galerkin approximation at

t=n~t. Then we can define the error components of the approxima-

tion

e(~,nfit) = u(~,nfit)

o(~,nfit) = u(~,nfit)

Un(~) = approximation error

U(x,nfit) = semidiscrete approx-

imation error

(8.1)

(8.2)

temporal approximation

error (8.3)

Now to establish the semidiscrete approximation error we use

a technique which is essentially the same as adopted in section 4.

37

We take the Laplace transform of the weak hyperbolic equation

(7.3) and denote the transformed solution by u(~,s) = L[u(~,t)]

s(u(s),v)o + (Lu(s),v)o = (uo,v)o (8.4)

Note that in developing this equation we have assumed that f(t) = 0

for convenience. Similarly taking the Laplace transform of the

semidiscrete Galerkin approximation (7.4) and denoting the trans-

formed solution by U(~,s) = L[U(~,t)]

s(U(s),V)o + (LU(s),V)o = (uo,V)o (8.5)

Now subtracting (8:5) from (8.4) and defining the transformed

approximation error by u(s) - U(s) = a(s) = L[aCt)] = L[u(t) - U(t)]

s(er(s),V) a + (La(s),V) = 0 (8.6)

We can now introduce a theorem establishing the magnitude of

erthe transformed semidiscrete approximation error.

Theorem 8.1. Let u(x,n~t) by the solution to (7.~, U(x,n~t) be- -the solution to (7.~ with e = 1, and a(~,n~t) be the semidiscrete

approximation error, then there exists a constant C1such that

(8. 7)

Proo~: Let U* be the element of Sh(Q) which interpolates U in the

sense of (2.18); and let ~ = u - U* and I = U* - U. Then a = ~+ E. - Thus (8.6) with V = E can be rewritten

s(~,E) + s(t,t) + (L~,t) + (LE,F) = 0

or

-- -- 1 -- 1--(E,E) = -(E,E) - s(LE,E) - s(LE,E) (8.8)

Note that (LI,t) can be expressed as an integral over the boundary.

38

Thus since the boundary conditions are homogeneous (LE,E) = o.Using this result and the Schwarz inequality, (8.8) becomes

II Ell 0 :. II If II 0 + } I ILE II 0

< IIE/lo + S.IIElilS

Now using the triangle inequality, we obtain (8.7).

The semidiscrete approximation error is established

through the following theorem.

Theorem 8.2. Let the hypotheses of Theorem 8.1 hold, then

/lcr(',nfit)llo:' 21IE(',nfit)llo + Clf~fitIIE(.,t)llldt (8.9)

Proof: This result follows from Theorem 8.1 by taking the inverse,

Laplace transform.

The semidiscrete approximation error can be evaluated

for finite element models by using the interpolation results of

section 2.

Theorem 8.3. Let the hypotheses of Theorem 8.1 hold. Then there

exist positive constants C and C such that2 3

II cr(.,nfit) II 0 < C hk+ 11u Ik+ 1 + C 3hk fn~ t Iu I dt- 2 0 k+l

(8.10)

Proof: This result can be obtained from (8.9) by using the inter-

polation result (2.18).

Note that the rate of convergence is determined by the

term involving hk. Thus the rate of convergence of the finite

element approximation for the convection equation 1S one power

10\ver than the rate of convergence for the finite element approx-

imation of,the diffusion equation.

The temporal approximation error is established by exactly

39

the same procedure as was used in section 4 and is given in

Theorem 4.3. We now cite the theorem establishing the total

error for the finite element-Galerkin approximation for the con-

vection equation.

Theorem 8.4. If the components of the error of the approximation

are given by (8.1-3), then there exist constants C , C , and C234

such that

II e ( • , nfi t) II < C h k + 11 u Io - 2 k+l + C hkfn~tlul dt

3 0 k+l

+ C fit II Uo II 04

(8. 11)

Proof: The result follows by using the triangle inequality 1n

conjunction with (8.10) and (4.43).

9. Error Estimates for the Convection Equation

Using L2 Methods

The L2 error estimates for the convection equation will be

developed using the technique introduced in section 6. Accordingly

we will use the space PC1[0,T;Sh(Q)] here. Reference should be

made to section 6 for its definition.

We consider the solution of the weak hyperbolic problem (7.3).

The exact solution to this problem has been characterized by Gold-

stein [27]. If the coefficients Aa in (7.2) are constants, then-

For convenience we set Aa equal to a constant for each a. We

note that the method of proof developed here is valid for a much

weaker specification of A~.

Now we seek an approximate Galerkin solution Un to (7.3) uS1ng

the finite difference-Galerkin approximation (7.6) with 8=1 (the

forward difference in time case). The solution of this problem is

40

fully equivalent to the solution of the following problem. Find

a U £ PC1[0,T;Sh(Q)] such that

(~~(t) ,V)t=nfit+ (LU(t),V)t=nfit = 0

(U ( • , 0) , V) 0 = (u 0 , V) 0

'if V £ PC1[O,T;Sh(Q)]

n = O, ... ,R

(9.1)

The notation indicates that a collocation is performed at t=n~t,

n=O, ... , R.

Now let Wet) be that element of PC1[0,T;Sh(Q)] which inter-

polates u in the sense of (Z.8). Then we define the error com-

ponents of the approximation by

e = u(t) - U(t)

El = Wet)

Ez = Wet)

U( t)

Un(9.2)

E = u(t) - Wet)

The following theorem relates the error components.

Theorem 9.1. If u(t) is the exact solution to (7.3), U(t) is the

function defined by (9.1), and Wet) is the interpolant of u(t)

defined by (2.8), and the error components are defined by (9.2),

then

nfit< t < (n+l)fit (9.3)

Proof: Subtracting (9.1) from (7.3)

(duCt) V) - (dUCt) V)t=n~t+ (Lu(t) V) - (LU(t) V)t=n~t = 0'. , 0 " ... ' 0 ' 0 ' 0

'if V £ PC1[O,T;Sh(Q)]

But

(a~(t) ,V)t=nfito CdU(t) V) for n~t < t < (n+l)fit

dt ' 0

41

Thus

(au (t) _ au (t) ,V) + (Lu (t) - Lun ,V) 0 = 0at at 0

~ V £ PC1[0,T;Sh(Q)]

Rewriting (9.4)

eel u (t) - aW (t) + aw ( t) - au ( t) V)at at at at' 0

+ (Lu(t) - LW(t) + LW(t) - LUn,V) = 0o

(9.4)

v V £ PC1[0,T;Sh(Q)] (9.5)

Equation (9.3) can be obtained by introducing (9.2) into (9.5).

The theorem governing the behavior of El can now be stated.

Theorem 9.2. Let the hypotheses of Theorem 9.1 hold, then there

exist positive constants 6, £, n, Cl' and Cz such that

1 .d.....IIE112 < ~llaEl12 + C211EI12 +2 dt 1 0 - 4n at 0 - 1

4£

+ £!..II au (t) (t _n~ t) II 248 at 1

n~t < t < (n+l)~t (9.6)

Proof: E - E = U(t) - Un = aU(t) (t-n6t) for nfit < t < (n+l) ~t.2 1 at

Now using the elementary inequality

(a,b):' ~Bllal12 + 811bl12

we find that

(L(E~-El),El) = (L(aU(t)(t-n~t),El), 0 at 0

< !.-IIL( aU(t] (t-nfit))112 + 811 E 112

- 46 at 0 1 0

< ~llaU(t)(t_n~t)112 + I3I1E 112

- 413 at 1 1 u

nfit< t < (n+l) fit

Similarly, using (9.7) we haveC

(LE,E) < 211EII2 + £IIE 1121 0 - 4£ 1 1 0

(9. 7)

(9 .8)

(9.9)

(9.10)

\I

42

and

-(~, E) < LII ~112 + nilE 112at 1 0 - 4n at 0 1 0

The term (LE ,E ) can be expressed in terms of an integral over1 1

the boundary. Thus since the boundary conditions are homogeneous,

(LE ,E ) = o. Introducing this resul t in conjunction wi th (9.3),1 1

(9.9), and (9.10) into (9.3), we obtain (9.6).

We are now ready to introduce the theorem which glves an

estimate for E at the discretization points in time.1

Theorem 9.3. Let the hypotheses of Theorem 9.1 hold, then there

exists a positive constant C such that6

sup liE (Mfit)II0< M< R 1 0

< c6[IIE (0)11 + IIEII + ~t Ilau(t)11 1]1 0 Hl(Hl) at L2(H)

(9.11)

Proof: Integrating (9.6) from t=n6t to t=(n+l)~t

liE ((n+l)~t)112 - liE (n~t)1121 0 1 0

< .L!(n+l)~tll~112dt + ~!(n+l)6tlIEI12dt- 2n n~t at 0 2£ n~t 1

(n+I)6t I 112+ 2(n+£+S) ! IE dtn6t 1 0

C (n+l)~t ()+ ~ ! II au t (t -nfi t) II 2 dt

26 n~t at 1

Now there exists a constant C such that I laU(t) I I <"t 1

for all t £ [0,M6t]. C is of the form 1 + f(h,fit).3

(9.12)

C IlauCt)113 at 1

Since the

function f(h,~t) causes only higher order terms in the error es-

timate, we do not include an explicit expression for it in our re-

sults. Use of this inequality in conjunction with the Schwarz

inequality gives the following result

43

j(n+l)~tll dUet) (t-n~t) 112dtn~t dt 1

< fit2 jCn+l)fitllaUCt)112dt- 3 n~t at 1

< C:fit2 j(n+l)~tllau(t)112dt (9.13)- n~t at 1

3Introducing (9.13) into (9.12), summing from n=O to n=M, and using

the embedding result II~II < C IIEIIH1( )' we find thatat LZ(LZ)- ~ L2

2

II E (Mfi t) II 2 < [II E ( 0) II 2 + C4 I lEI IH21 (L )

1 0 - 1 0 2n 2

+ SllEI12 + clC~~e Ildu(t) 112 ]2£ L2(H1) 6, Bat L2(H1)

+ 2(n+£+B) jMfitII E IIo2dt (9.14)o 1

Applying the Gronwall's inequality to (9.14) and using the embedding2

result 21lEI12l + SllEIIL

2(Lll)< CsIIEII1211(Lll)we obtain (9.11).2n H (L2) 2£ 2 r - 'I

The approximation error can be established through the following

theorem.

Theorem 9.4. Let the hypothesis of Theorem 9.1 hold, then

(9.15)

Proof: This proof is the same as the one given in Theorem 6.4.

We can now give the final error estimate which is based on the

interpolation results of Section 2.

Theorem 9.5. If the space ptl[O,T;Sh(Q)] satisfies conditions (i),

(ii), and (iii) in Section 2, then

IleIIL2(L2) ~ C71Ie(0) 110 + ClohklluIIH1(Hk+d

+ C fit IIau II7 at L 2 (Hl ) (9.15)

•44

Proof: The proof of this theorem is essentially the same as the

proof of Theorem 6.5. However, we should note that in this case

the spatial interpolation error is different. Using (2.18)

I IU-Qxul IH1(Hl) :. Cahk+l-llluIIHl(I-Ik+l)

CahkllullHl (Hk+l)

Thus for the convection problem the rate of convergence in the

spatial variable is one power lower than that obtained in the

diffusion problem.

Acknowledgement. Support of the U.S. Air Force Office of Scientific

Research under Contract F44-69-C-0124 to the University of Alabama

1n Huntsville is gratefully acknowledged. We also express thanks

to the Engineering Mechanics Division of the ASE-EM Department,

The University of Texas at Austin, for providing certain facilities

used during the course of the work reported here.

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