Solution Of Stochastic Partial Differential Equations (SPDEs)

Using Galerkin Method And Finite Element Techniques

Manas K. Deb, Ivo M. Babuska and J. Tinsley Oden

TICAM, University of Texas, Austin, Texas

(September 5, 2000)

Abstract

Stochastic equations arise when physical systems with uncertain data are modeled.

This paper focuses on elliptic stochastic partial differential equations (SPDEs) and

systematically develops theoretical and computational foundations for solving them. The

numerical problem is posed on D x Q, where D is the physical space domain and Q

denotes the space of all the admissible elementary events. Two types of SPDEs are

considered here: (a) Random-RHS type, i.e. only the source term contains randomness

and (b) Random-LHS type, i.e. only the PDE coefficient is random. It is assumed that the

mean and the covariance of the input stochastic functions are given and similar quantities

need to be obtained for the solution. This study identifies the necessary function spaces,

details the weak forms and discusses their properties, develops a priori error estimates for

the solution and its statistical moments and constructs finite-element-based solution

schemes for these SPDEs. In summary, this study establishes the necessary extensions to

the theory and solution schemes of conventional Galerkin approximation-based finite

element method to stochastic equations, thus, motivating application of vast amount of

existing knowledge in the deterministic finite element community to the solution of

SPDEs.

1. Introduction

To date, most of the literature on reliable computer simulation of physical events

pertain to deterministic models where the input descriptions are specified with complete

Sbl

,.

certainty. It is well known, however, that in actual physical systems, all data carry a

certain level of uncertainty. Material properties, loading scenarios, boundary conditions,

domain geometry, etc., in some instances, may contain significant variations that could be

treated as random processes. In such cases, the notion of deterministic solution is of

limited value - the solution more properly should be defined in terms of useful statistical

measures. The randomness in a physical process may be either (i) intrinsic or containing

random system parameters as in quantum mechanics or (ii) may be due to external stimuli

e.g., random wind loading on a structure. Also, there are the situations where,

theoretically speaking, it may be possible to exactly describe all the necessary physical

characteristics; however, this is prohibitively expensive, hence, not practical. Therefore, a

common approach is to use a random media idealization.

Stochastic partial differential equations (SPDEs), defined here as partial differential

equations with input data that are stochastic functions, can be used to describe physical

systems where such input uncertainty is expected to affect the output significantly. In the

area of numerical solution ofSPDEs, there are two main approaches:

I. Analytical, where the problem formulation includes the effects of randomness.

The resulting equations can be solved using various techniques including finite

difference or finite element or some suitable variation of them.

2. Simulations such as the Monte Carlo simulation (MCS), where a large number of

equiprobable realizaions of the system are obtained by some sampling technique,

a set of output is generated for these realizations using standard deterministic

solution procedures and finally, the output statistics are obtained from this set of

output. Of course, techniques like finite difference or finite element can be used to

generate the set of output needed for MCS.

It is also necessary to distinguish between the case where only uncorrelated (white) noise

exists and the case where input stochastic fields are correlated. The current work

addresses the correlated case. For mathematical theory and modeling of stochastic

differential equations involving white noise, see, for example, Holden et al. [8] or

oksendal [14].

2

Several classes of methods for obtaining approximate solution to SPDEs have been

proposed in the literature. The work of Ghanem [7] and Ghanem and Spanos [6] advocate

a hybrid finite element-spectral approach while the monograph of Kleiber and Hien [9]

utilizes a perturbation scheme. Elishakoff and Ren [5] examine finite element methods

for structures with large stochastic variations and point to limitations of some approaches.

A fuller description of work on computational methods for SPDEs used to model

stochastic behavior in problems of mechanics can be found in the survey of Schueller and

Pradlwarter [15], in the books of Klieber and Hien [9], Ghanem and Spanos [6], and in

Deb [4] and the references therein.

The main goal of this paper is to describe a general framework that facilitates a

theoretical analysis and development of robust solution schemes based on Galerkin

approximation and established finite element techniques. In this paper, this approach is

referred to as the Stochastic Oalerkin Finite Element (SGFE) method (see [3]). This paper

considers model problems characterized by elliptic SPDEs, and provides the

corresponding function setting. Related mathematical notions and finite element based

numerical schemes are introduced for the creation of the discrete equations and for the

computation of a priori and a posteriori error estimates. Numerical results obtained by

solving the model SPDE boundary-value problems using the our SGFE approach and are

compared with results obtained by other alternatives including Monte-Carlo simulation.

The paper concludes with a short discussion of the key benefits and challenges related to

the present approach and suggests some follow-on research topics.

2. Model problem, weak form, Galerkin and FE approximations

Stochastic partial differential equations of the following general form are considered:

V-(a(x)VU'(x))=](x) in D (2.1)

where xED c 9t d, d = 1,2,3, is open and bounded domain with a Lipschitz continuous

boundary 0D. The quantities, a and f are stochastic or random functions (often

3

(2.3)

(2.4)

termed as Random Space Functions or RSFs); these play the roles of the system

coefficient and the source term, respectively, just as in the case of standard deterministic

elliptic PDEs. A scalar coefficient will be considered here and it will have the usual

restriction:

00 > ama.~ ~ a(x) ~ amin > 0 almost surely (a.s.), and a.e. in D (2.2)

The SPDE in (2.1), in general, may be coupled with Dirichlet or Neumann boundary

conditions of the form, with g and J prescribed RSFs:

u(s) = g(s), on aDD' seaDDa(s)Vu(s)en(s) = 7(s), on aDN, seaDN

where n(s) is the outwardly normal to the boundary aDN, aDD ua DN = a D and

aDD rlaDN = ¢.

The randomness of various functions involved in the above statements is formalized

through a probability space, (Q,F,P), where, n,F,p are the set of random events, the

0" - algebra of the subsets of Q and the applicable probability measure, respectively. The

elements of this probability space appear as parameters in the input stochastic functions.

Ideally, one would want to have joint probability distributions as descriptions of the

random functions; however, in practice, these are often characterized by their means,

covariances, and perhaps a few higher statistical moments.

Before embarking on the mathematical formulation of the problem, following

definitions are noted [10]:

Definition 2.1 .

A random space function (RSF) a on D is a function mapping D x Q H 9\, which is

jointly measurable with respect to dxx dP , P being the applicable probability measure

and Q is the set of elementary events. Thus, a == a(x,m), xeD, OJ en. ##

4

Definition 2.2

If a: D x nH 9l is an integrable (w.r.t. to the probability measure dP) random

function, then its mean or expectation (or the first order moment) is the space function

J.La = E[a] = (a) = fadP = fa(x,m)dp(m). Also, fdp(m) = 1, by defInition. Similarly,n ",eO ",en

the covariance of a, c{j(j(X,Y), x, Y ED, is defined as ((a(x,m )a(y,m ))).

Definition 2.3 - Stochastic Sobolev Spaces

For a given k ~ 0, the Stochastic Sobolev Space of order k is defmed as follows:

jjJ'(Dxn)= {u(x,m)I u(.,m)EHk(D) as., (1Iu(.,m~l:t(D)) < oo}In this context, another useful defmition is:

The space fjk (D x n) is equipped with the inner product:

##

(2.5)

(2.6)

(2.7)

The Stochastic Sobolev spaces as defined above have suitable trace properties so that the

boundary conditions can be handled adequately (see Larsen [10]). Also, the

space fj~ (D x n) contains C; (D)x P (0) as a dense subspace, thus affording flexible

construction of approximating functions in that space.

Using the notation inDefinition 2.1, (2.1) can be rewritten as:

V'.( a(x,m )V'u(x,m)) = ](x,m), xED, mE Q

##

(2.8)

where n could be identified with 9lM , where M is a non-zero integer. The associated

boundary conditions can also be similarly rewritten.

Noting that the solution to the model problem i.e. (2.1) can be obtained by integrating

the differential equation with respect to both D and n (for the latter using the applicable

probability measure), the problem can be restated, without loss of generality, assuming a

5

zero boundary condition for the Dirichlet part of the boundary i.e. on aDD' using the

following weak form:

Find u(x,m)E ii~(Dxn) such that:

J(a(x,m )V'u(x,m). V'v(x,m ))dx = J(7(x,m )v(x,m ))dx +D D

f(7(x,m )v(s, (j)))ds 'tj v(x,CtJ)e jj~(D x 0.)aDN

(2.9)

As far as the representation of input stochastic functions are concerned, restricting the

discussion to second order processes, a Fourier-type expansion, popularly known as the

Karhunen-Loeve (KL) expansion, can be used to express a stochastic process a(x,(j)):M

a = a(x,m) = ao(x) + lim IAa,(x)Aj(m)M-+«> ,=\

(2.10)

where, ao(x) is the given mean of a(x, m), .JX: and aj(x) are the eigenvalues and

eigenfunctions associated with the prescribed covariance function of a(x,CtJ), and A,(m)

are mutually orthogonal random variables [11]. For actual calculations, a finte value of M

may be used, as long the approximation is reasonably accurate. In this context we note

the following:

1. The probability domain n could be identified with mM; in fact, here

o.c9tM•

2. Only the above fact is essential to the current formulation; the remaining

properties e.g., orthonormality of aj (x), etc. are only useful at an

implementation level, and have been utilized here appropriately.

It should also be mentioned here that the lower the correlation of a, the higher the M. In

fact, if the correlation becomes zero, a will be uncorrelated or a white noise process and

will not be amenable to K-L expansion.

In general, closed-form solutions of the (2.9) are not feasible and, hence, approximate

solutions via some suitable numerical procedure are usually constructed. In the current

6

x

.01 I I I I~

L' I I V//""hw

~ hx .~~~

Fig. 1. A typical computational domain and pertinent FE mesh parameters

work, a Galerkin type of approximation is sought using finite dimensional subspaces

VilA. c jj~ (D x n) .Thus, the Galerkin statement for (2.9) becomes:

Find uhA.,(x, m) E ~.h.. such that:

J(a(x,m )'VUhxh .. (x,m). 'VVhxh", (x,m ))dx = J(l(x,m )vhJl", (x,m ))dx +D D

S(7(s,m)vh,J1.,(s,m))ds rJ ~"I'" (x,m) E i!;'A.CDN

(2.11)

With reasonable restriction on input data, the problems in (2.9) and (2.11) have unique

solutions [4, 10].

It may be noted here that if u(x,m) is an element of iii(D x n), it can be expressed

in the form [10]:

'"u = u(x,m) = Lu,(x)U;(m)1=1

7

(2.12)

Let the domains D and n be discretized using piecewise polynomials of order p and

q;.;:I....,M • Let N .• and NO) be the degrees-of-freedom in D and n corresponding to the

discretization parameters h" and hllJ' respectively (refer to Fig. 1 for the case M=I).

Accordingly, the approximate solution, Uh,lrw

' can be represented as:

where,

N,. N..

ulrrh", (x,m) = 2:L>jk ¢j (x)\fIk (m)j=1 k=l

(2.13)

U jle = Deterministic unknown coefficients to be determined

¢/x) = Standard finite element shape functions defined on D, generally

polynomials of order p; ¢j(x)e H~(D)

\fIk (tV) = Stochastic (finite element like) shape functions in Q of order qi. j:l .....,M ;

Similarly, the test functions have the representation:Nr

vq(x,m) = v(x)'¥/tV) = :L¢p(x)\f'q(m)p=1

(2.14)

Substituting (2.10), (2.13) and (2.14) into (2.11), the linear system needed to compute ujk

can be generated using procedures similar to standard finite element technique.

A question can be raised as to the correctness of the formulation given by (2.9): see,

for example, [14] for a comparison of Ito and Stratonovich integral, or [12,13] where the

issues regarding multiplication of stochastic functions have been discussed and some

preliminary results using the notion of the Wick product has been presented; however, the

results are somewhat inconclusive. The present work used fine-quality Monte-Carlo

solutions (MCS) as one of the reference solutions for comparison with those obtained via

SGFE. The equivalence ofMeS and SGFE is not established.

As was mentioned earlier, it not necessary that all of the input that could be stochastic

need to be so for the problem to be termed stochastic. In fact, two distinct categories of

problems are of interest in the current study: Random-RHS type, where the PDE

8

coefficient is deterministic with stochastic source term, and Random-LRS type, where

only the PDE coefficient is stochastic. For convenience, we will assume that the

boundary conditions are always deterministic.

3. A simpler model problem: Algebraic stochastic equation

3.1 Formulation and analysis

(3.1)

na(m)f dP(m) < coWEO

where a = Ci(lV) is a given stochastic function with the properties

While the ultimate goal is to solve the stochastic partial differential equation (SPDE)

described in Section 2, first a relatively simple algebraic stochastic equation is

considered. Although simple, this equation will provide a clear and comprehensive

exposition on the mathematical foundations, relevant approximation-theoretic

considerations, and, finally, the key elements of the computational technique [3, 4]. The

model problem can be stated as:

Find U E 1:(0) S.t. Ci(m)U(lV) = 1

and co> ama." ~ Ci(m) ~ amin > 0, and U(lV) is the unknown solution to be found. In this

simple case the exact solution is formally given as:

(3.2)

The difficulty in the actual evaluation of the exact solution, its probability density and

statistical moments, depends on the nature of Ci(m).

The weak form for (3.1) can be stated as:

Find U E v(n) such that

B(u,v) = L(v) Vv EV(n) (3.3)

where the space of admissible trial and test functions v(n), bilinear form B(u,v) and

the linear form L(v) are given as:

9

v(n) == jjO(n) == 1:(n)

B(u, v) == (Ci(OJ)u(OJ )v(OJ))

L(v) == (v(w))

(3.4)

(3.5)

(3.6)

A Galerkin approximation of (3.3) can be constructed by considering finite-

dimensional subspaces, ~ c v(n) and seeking solutions un E ~ . In this setting, with~ w ~

the assumptions on the coefficient a(OJ), the existence of a unique solutions to (3.3) and

its finite dimensional counterpart can be established, in a straightforward manner, by the

classical Lax-Milgram theorem (see [4]).

Keeping in view the assumptions on the stochastic coefficient, the following finite-

dimensional (i.e. M -dimensional) approximation of Ci(OJ) is considered:

where,

M

Ci(OJ):::; ao+La; A;(OJi}, OJ == {OJ;} E Q C 9lM;=1

aO = (Ci(m )) = Mean value of a(OJ )

(3.7)

a; = Suitably bounded coefficients E 91

Ai = Uncorrelated random variables (uniformly distributed between - a and a

with a>O.)

and, the above are chosen such that the assumptions on a(OJ) remain valid.

This form parallels K-L expansion of correlated stationary random fields.

If Ii is sufficiently regular, and M-dirnensional cubes of size h(JJ and piecewise

continuous polynomials of degree q are used to discretize n, and if ~'" = U - un.. is the

finite element approximation error, then the a priori error estimate for the solution, can

be written as [4]:

lie II ~ Ch '1+1 Illill

" .. 0 (JJ 1/+1

10

(3.8)

..

-(f I I U, I aIe .1

M=~ n=[-a.a]c9\

Fig. 2. Joint probability density and discretization of n

In addition to Ileh.. llo' for practical use, one may be interested in the elTors related to

the computed statistical moments (e.g., mean, variance, standard deviation, etc.) of the

stochastic solution uh . Let these errors be defined as:..eµ == (u)-(uh..) (Error in Mean)

eq1 = ((uu)-(u)(u)) - ((Uh.,~.J-(UhJ(Uh..)) (Error in Variance)

eq = ((uu)-(u)(u) ~ - ((u/r., u/r., )-(u" ..)(uh.. ) ~ (Error in Std. Dev.)

(3.9)

(3.10)

(3.11)

It can be shown [4] that these errors are bounded by Ilelr.. llo and converge at rates faster

than lie;,.. 110 by O(h~+l). In particular:

Ieµ I, Ieall, leal :5 Clh:+111~JoIeµ I, Ieall, leal :5 C2hw 2('1+

1) Ilullp+

1

(3.12)

(3.13)

where the constants C1 and C2 will take different values depending which error is being

considered.

11

3.2 Finite element implementation and numerical experiments

In a manner similar to traditional finite element discretization, a partition of the

domain can be conceived where the elements are simply M - dimensional rectangles.

The basic layout of this discretization is shown in Fig. 2 for M = 1 and 2. In the spirit of

conventional finite element approximation, the unknown solution can be expanded as:

..N..

u" .. (cu) = I Ui'¥i(cu );=1

(3.14)

where, '¥; (cu) are the chosen basis functions and U; are the unknown coefficients to be

determined. The important point to be noted here is the fact that 'Pi are stochastic - they

are actually functions of the random variables Ak (aJk), k = 1,...,M used in the expansion

of the coefficient a(m) (see (3.7)). This form is consistent with the representation

prescribed in Section 2.

Of course, there is a lot of flexibility in making a particular choice for 'Pj' The

optimal choice primarily depends on the purpose behind the computation and on the level

of computational efficiency one would like to achieve. In the current work, piecewise

constant functions are used: the ith basis function will take value of unity on the ith

M - dimensional element n; . In other words:

= 0 everywhere else (3.15)

Although they provide low order approximation, piecewise constant functions, in this

context will help decouple systems of N (J) equations that have to be finally solved, thus

providing efficiency from a computational perspective. Besides, this choice also keeps

the amount of additional complexity introduced due to the stochastic components to a

reasonable size. With the particular choices that have been made so far, the equation for

Uj can be written explicitly as:

(3.16)

12

,

Here the fact that {\fI;\fIj) = 0 if i * j has been utilized. Also, note that the equations for

V; are decoupled: this has the obvious computational advantage in all settings. In this

particular case, it lets us compute the unknown coefficient in tIlis problem without having

to solve a system of equations. Benefits due to this type of decoupling in the case of

SPDEs will become apparent in later sections.

Once the Vi have been computed, the solution u/1 can be reconstructed using (3.14).'"

Using this expansion of the solution, the m - th moment M;' of the solution can beh..

computed as:

(3.17)

Using the above definition of M;' , the expressions for the mean µjj and the variance~ ~

0"£ can be expressed as:"..

2 2 (I J0"- = M- - M-U"OI Uhw u~

(3.18)

(3.19)

Finite element solutions have been computed for a amax / amin values of 1.5 and 199

and for different values of M. These solutions have been compared with exact solutions

(derived in [4]) and with MCS, and excellent agreement has been observed. Note that

higher the value of ama>; / amin, steeper is the variation of the solution and its probability

density function. Theoretical convergence rates for the solution and its moments been

also been verified; Fig. 3 shows a typical convergence pattern. Note that NDIV is the

nunlber of elements along any of the n directions.

13

s- o... 1

- - -~

ro>. -2

-

=L2_Error

-

I l~ -4

Slope=O.'9979

=- ~

Errorjn_ Variance

1~ -6

Slope=2.0055 '

III

I

E -8

i

....01:1_10

~ -12 JError_in_Average

... Slope=1.9859 '

....GlE -14

0 1 2 3In(NDIV)

4 5

Fig. 3. Convergence rates for M = 1 and amax / amin = 1.5

Using the residual-based a posteriori estimates, a set of h-adaptive computations have

also been perfoID1ed [4] that indicate significant benefit due to adaptivity for high values

3.3 Monte-Carlo simulation: Some remarks

The model problem, with the set of parameters as described earlier this section,

was also solved using Monte-Carlo simulation (MCS). In all cases, converged MCS

solutions compared very well with those obtained from finite element calculations. A

very favorable feature of MCS is that its convergence is independent of the number of

random variables in the system, i.e. M. On the other hand, MCS convergence is usually

very slow; the mean converges as ~Ns , where Ns is the number of MCS simulations.

Also, in the case of MCS, the error in the moments are always specified probabilistically,

i.e. a confidence interval must be associated with the predictions. As an example, in the

14

..

present case, for amax I amin =199 and M=I, based on calculations made during this

study, one would require approximately 2 x 106 simulations in order to achieve a one

percent error in the mean with 99% confidence. While no rigorous investigation related to

the MCS convergence was done, for all the cases considered here, coarse-mesh finite

element solutions appeared to be comparable with fine-quality MCS results .

4. Solution of Randoffi_RHS type SPDE

Let only the source term], in (2.1), be random. This leads to the example of a

Random_RHS type SPDE that will be considered here. Without loss of generality, a

homogeneous problem will be considered. It is assumed that f has zero-mean, a

I

( )--Ix, -XII

standard deviation of aj, a covariance given by CovJi XI,X2 = a}e b , and can be

approximated by a truncated K-L expansion as:

M

](x) = ](x,OJ) = LarA f (x)F; (OJ), mE Q c 9tM1=1

(4.1)

The steps required to generate the discrete equations follow the stochastic Galerkin

finite element (SGFE) procedure described in Section 2. Of course, in order to create the

system of equations, some concrete descriptions of the deterministic and stochastic shape

functions, i.e. of ¢j (x) and of \fIk (w), have to be picked. In the present work, piecewise

linears for ¢j (x) and piecewise constants for \fIk (OJ) have been used; the stochastic shape

functions are the same as those used in Section 3. With these choices of shape functions,

for the q-th test function vq, q = 1...NOJ' the LHS and the RHS can be written as:

15

..

and,

U2q

= [Std. FE Coeff. Matrix kXN, (\¥q \¥q )1.•

M

RHSq = O"f Lfi: fF;(w)\¥q(w)dP(liJ) f.t;(x)v(x)dxi=l /lien xeD

b\j

M b2i

= O"f Ifi:(F; \¥q )1.i=1 •

bN·"

(4.2)

(4.3)

Here, [u1q, ... ,UN.Jr is the Nx -long, q -th segment of the Nt x N(JJ -long vector of

unknown coefficients, [ulI,,,,,uN p""u1q"",uN q"",U1N "",uN N lr. Also, the vectorr ): AI r ., J

[b1P... ,bNJr is fOfmed exactly as in usual finite element method fOf a given source term,

.t;(x). These LHS and RHS components are assembled over q = 1...N/lI in order to obtain

the complete system. Note that .t;(x) is the i -th eigen function from the K-L expansion

of the stochastic source term f. Once the complete solution vector has been computed,

its statistical moment of any order, as a function of x, can be computed as:

M;:xh., (x) = (Uh:h., (x,cu)) = f~~".,(x, W )dP(CU)/liEn

(4.4)

Quantities such as the mean and the variance of the solution can then be derived from the

appropriate combinations of these moments (see Section 3).

As a result of the specific choices that have been made, in particular, the piecewise

constant stochastic shape functions, \¥JUJ), a uncoupled set of N/lI systems of linear

equations (of dimension Nx each) have been generated. Note that, within a scaling

constant, all of these N /lI systems of equations have the same LHS matrix. Thus, very

16

little additional computational effort is required to solve this Random-RHS problem as

compared to the usual deterministic finite element method provided we factor the LHS

once and reuse the factors.

For the chosen model problem, deterministic PDEs that directly yield mean and

covariance of the solution have been reported [2, 10] and fulther detailed in [4]. In case

of the solution mean, one simply needs to use the mean of the source term and the

boundary conditions. The equation for the covariance, with p(X) , x2) == COVjjjj (XI' x2). is

given by the fourth-order homogeneous PDE:

a:p 2 = (1(x\,(J))](X2,(J))) = CovJj(x\,x2) (4.5)Ox( Ox2

It should be pointed out here that (4.5) is valid for arbitrary M and admits the case where

Covll (XI' x2) is a Dirac delta i.e. corresponds to white noise.

In the present work, the above covariance equation has been used, along with MCS,

to validate SGFE solutions for the model Random-RHS problem and excellent agreement

has been noted even with fairly coarse FE mesh. In Fig. 4, a plot of the covariance of the

solution is presented; here the homogeneous problem is posed in D = (-0.5,0.5) with

source term standard deviation O"j and the correlation length b are taken as 0.1 and 1.0,

respectively. For this calculation, a 4x 4 finite element mesh employing piecewise linear

trial functions were used to discretize D; 32M with M=3 piecewise constant elements

were used to discretize n.

5. Solution of RandoID_LHS type SPDE

In this case we assume that only the coefficient i.e. a in (2.1) is stochastic. Let, ahave a mean ao (x), standard deviation 0'0 , and be amenable to an approximation by a

truncated K-L expansion of M-terms i.e.

17

L..

5E-05 ~o()

CoC/)

Fig. 4. Random-RHS SPDE: Covariance of the solution

M

a = a(x,co) = ao(x) + Lcyaft: a;(x)A;(m), co Ene mM

;=1

(5.1)

Using the SGFE formulation in Section 2 and the procedure outlined in Sections 3

and 4, with basis functions as described in Section 4, one can construct the LHS of the

discrete system as:.. Tfr~,,£,] 0 • • • 0

,=0 N,xN,

0 [fr".,£,] 01=0 N,xN,

• • • I (5.2)

• • •• • •0 0 • • • [fr~_N_l, ]

1=0 N~)(N:r

18

,

where,

rjq == JAj(m)'¥q(m)'¥q(m)dp(m), with Ao(m)==l4lIeO

fk == Ai} JaaA ak(x)V¢;(X)V¢j(x)dxxeD

fo == AI} f ao(x)V¢j(x)'V¢j(x)dx.,e f)

and, AI} is the standard finite element assembly operator. The corresponding vector of

unknown coefficients is:

The RHS vector can be assembled as:

{/3q=1 {b, ,b2 , .•• ,bN x}"," /3q=M {b"b2 , ... , bN x}} (5.3)

where the vector {bl,b2, ... ,bNx} is the standard finite element RHS vector; the quantity f3q

may be viewed as a scaling factor which is given by:

fl" == f'¥q(m)dp(m)wen

Again, the block-diagonal nature ofthe LHS is due to the use of piecewise constant basis

functions in the discretization of n. As described in Sections 3 and 4, the statistical

moments of the solution can be easily obtained once the above system has been solved.

The SGFE formulation presented here has been validated using a series of numerical

calculations and comparison of the SGFE results with MCS and analytical solutions;

excellent agreement. has been noted even when the FE mesh is coarse. In Figs. 5-6, a

graphical comparison of solution mean and variance computed using SGFE and MCS is

presented. In all calculations an 8x 8 mesh with piecewise linear basis functions has been

used for the discretization of one-quarter of the spatial domain D. In case of SGFE, an

8 x 8 x 8 mesh with piecewise constant basis functions has been used to discretize n;note that M := 3. In case of MCS, 65536 realizations' were used in order to compute the

mean and variance of the solution. The associated problem is described by:

'V ry .(a(x,y,m)V ry u(x,y,m)):= f(x,y), x,yeD, mEn

where,

19

0.05 <;::o ~

o ~

..

Fig. 5. Solution mean: SGFE (left) max value = 0.06323, MCS (right) max value =0.06324

M

a(x,y,m) = ao(x,y)+o-o LA aj(x,y )A;(m);=1

D = (- 0.5,0.5)x (- 0.5,0.5), Q = (- J3,Ji) , f(x,y) = 2( 0.5-x2 _ y2)with the boundary condition,

u(x,y,m) =0 on aD

For the present calculations, the values ao (x,y) = 1.0, 0-0 =0.1 and M = 3 are used.

..

A priori error estimates have also been computed for these M+d dimensional

problems. As an example, for the basis functions chosen here, provided the solution is

sufficiently smooth, the solution error in the norm decreases as hx and as h{J) while the

error in mean and variance decreases as h; and as h;.

Note that the current implementation uses h-meshes. Of course, one may use p- or hp-

meshes in one or both of x, (r). In fact, the so called Weiner Chaos polynomials used in

implementations such as [6, 7] can be viewed as employing special p-meshes in the m-

dimensions.

20

..

Fig. 6. Solution variance: SGFE (left) max value = 0.1876E-04, MCS (right) max value

= 0.1881E-04

6. Concluding remarks

This paper described a general theoretical and computational framework for elliptic

SPDEs using Galerkin approximation and finite element techniques (SGFE). A suite of

numerical results were computed and compared with exact solutions as well with MCS.

While we have not explicitly proved the equivalence of MCS and SGFE solution, the

former is well accepted in the computational community and hence used as one of the

reference techniques. The approach detailed in this work can be regarded as a logical

extension of deterministic finite element method. It is robust and does not suffer from the

'small variation' restriction of perturbation-based methods. It is clear that in this

framework, most of the established functional-analytic results from traditional finite

elements can be extended in order to examine well-posedness of the problem and to

explore a priori and a posteriori error estimates. Also, great flexibility exists in the

discretization of the M+d dimensional problem domain, both in the shape of the

pcu1itionsas well as in the choice of the basis functions. Thus, efficient solution schemes

can be devised for a wide range of physical problems using such a framework. In fact, by

appropriate choice of basis functions, one can recover the solution schemes based on

spectral techniques reported in [6, 7, 10].

2\

Referring to (2.8), it is noted that if (J) is fixed, the familiar deterministic elliptic PDE

is recovered - in the context of MCS, this coinciding with a particular realization. Also,

once a K-L type expansion is introduced, for a fixed x, one has to deal with uncorrelated

random variables as opposed to correlated random space functions, thus greatly reducing

the problem complexity.

It is clear that, in general, significantly higher amount of computation should be

expected when dealing with SPDEs as compared to deterministic PDEs. In case of MCS,

this is manifested in the vary large number of realizations typically needed for high

quality solution. In SGFE, it is the high dimensionality, in particular, the value of M, that

dictates the extra effort needed to build and solve the system of equations. This challenge

may be addressed using one or a combination of the strategies like (i) parallel

implementation (ii) solution-adaptive discretization using a posteriori error estimation

and (iii) application specific decomposition of the input stochastic functions that help

keep Mlow. Note that the choice of basis functions made in this study yields systems that

are block diagonal, thus, are very amenable to parallel computation. These issues provide

interesting areas for further research.

Acknowledgement: The support of this work by the Office of Naval Research under

contract N00014-95-1-0401 is gratefully acknowledged.

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22

&

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