A FREQUENCY{DOMAIN METHOD FOR FINITE ELEMENT

A FREQUENCY{DOMAIN METHOD FOR FINITE ELEMENT SOLUTIONS OF PARABOLIC PROBLEMS CHANG-OCK LEE, JONGWOO LEE, AND DONGWOO SHEEN Abstract. We introduce and analyze a frequency-domain method for par-

abolic partial dierential equations. The method is naturally parallelizable. After taking the Fourier transformation of given sources in the space-time domain into the space-frequency domain, we propose to solve an inde nite, complex elliptic problem for each frequency. Fourier inversion will then recover the solution in the space-time domain. Existence and uniqueness as well as error estimates are given. Fourier invertibility is also examined. Numerical experiments are presented.

1. Introduction Let be an open bounded Lipschitz domain in RN , N = 2; 3, J = [0; 1), and ? = @ . We are interested in a numerical method for the following parabolic problem: 1 u ? r (ru) = f; J; (1.1a) t u = 0; ? J; (1.1b) u(x; 0) = 0; x 2 ; (1.1c) where 2 L2( ) and 2 H 1( ) are positive functions of x de ned on , which satisfy , , jrj , where ; ; ; are positive constants. In this paper, instead of solving Problem (1.1) in the space-time domain, we analyze the Fourier-transformed problems of Problem (1.1), and propose a naturally parallelizable algorithm by solving the Fourier-transformed problems. Recall rst that the Fourier transform vb(; !) of a function v(; t) in time is de ned by

; !) =

v( b

Z 1

?1

v(; t)e?i!tdt

Date : September 1, 1997. 1991 Mathematics Subject Classi cation. Primary 65N30; Secondary 35K20. Key words and phrases. parabolic problems, nite element methods, parallel algorithm, Four-

ier transform. The research was supported in part by KOSEF 961-0106-039-2, GARC and BSRI-MOE-97. 1

2

CHANG-OCK LEE, JONGWOO LEE, AND DONGWOO SHEEN

and the Fourier inversion formula is given by Z 1 1 v(; !)ei!td!: b v(; t) = 2 ?1 We extend f and u by zero to t < 0 and transform the space-time formulation of the equations (1.1) to a space-frequency formulation by taking the Fourier transform of (1.1) in time. We then obtain a set of the following elliptic problems depending on !: i! 1 ub ? r (rub) = f;b x 2 ; (1.2a) ub = 0; x 2 ?: (1.2b) Note that if v(x; t) is a real function, its Fourier transform satis es the conjugate relation: vb(x; ?!) = bv(x; !); ! 2 R: Then, the Fourier inversion formula takes the form Z 1 1 (1.3) v(x; t) = Re vb(x; !)ei!t d!: 0 Here, we explain brie y the reason why we consider Problem (1.2) instead of Problem (1.1). Our primary interest lies in proposing and analyzing a naturally parallelizable algorithm with which one may use parallel machines to solve Problem (1.1) as eciently as possible. The most popular strategy to get numerical solutions of (1.1) is to solve the problem in the space-time domain by using a marching algorithm such as backwardEuler or Crank-Nicolson methods. Such methods have proven to solve eectively many practical problems. In order to advance to next time steps when one uses a marching algorithm, one needs to solve elliptic problems using informations on space grids at the current and/or previous time steps. There could be two approaches to parallelization in solving parabolic problems in a wide sense. One direction may be to use the idea of domain decomposition methods to decompose the space domain into subdomains in solving each elliptic problem corresponding to each xed time step. For such a direction, we refer [4, 5, 14, 15, 18, 19, 25] and recent publications in major numerical analysis journals. However, these methods require heavy communication cost among processors in order to pass informations between neighboring subdomains. The other direction may be attempts to devise and implement parallel algorithms in the marching axes, that is, the time axis. However, the nature of evolution makes it dicult to nd a naturally parallelizable algorithm which does not spend too much communication time among processors. In this sense, both approaches formulated in the space-time domain are not naturally parallelizable. In this paper, we propose an alternative method by taking the Fourier transformation of Problem (1.1) to obtain a set of elliptic problems (1.2) for discrete

FREQUENCY{DOMAIN METHOD FOR PARABOLIC PROBLEMS

3

number of frequency !'s of interest. This formulation is based on the observation that the set of elliptic problems (1.2) depending on the parameter ! is completely independent. We are thus able to solve the set of elliptic problems (1.2), by assigning each such problem to each processor. Independence of each problem guarantees no communication cost among processors. Then our numerical solutions in each time is recovered by a discrete inverse Fourier transform. The above procedure has been proven to be very ecient for solving wave propagations with absorbing boundary conditions in a parallel machine [9, 10]. Wave equations becomes Helmholtz-type equations in the space-frequency domain, which have eigensolutions with Dirichlet or Neumann boundary conditions. This is not the case with absorbing boundary conditions; with absorbing boundary conditions the Helmholtz-type equations are uniquely solvable, and thus a natural parallelization is possible by simultaneously solving Fourier-transformed problems with dierent !'s in dierent processors; for details, see [9, 10, 12, 20]. Indeed, it turns out that solving wave propagation is more subtle than solving parabolic problems. But it is worthy to stress that parabolic problems with Dirichlet and Neumann boundary conditions can be solved in a natural parallelizable way. See also [21] for an analysis of a linearized Navier-Stokes equations, where a similar treatment for the Dirichlet boundary condition has been done to (1.1) or (1.2). This paper is organized as follows. In x2, we show that the equation (1.2) has the unique solution ub(; !) for ! > 0, and regularity and stability results are proved for such solutions. In x3 we rst treat a nite element procedure for (1.2) and derive error estimates for the procedure for each !. We then derive a full error estimate for solving (1.1) via the inverse Fourier transformation. Finally, in x4, some results from numerical experiments are given. 2. Continuous Problems 2.1. Notations and Variational Formulation. All functions are assumed to have values in the complex eld C : But, they are considered in the real eld for the time-dependent problems. Standard notations for function spaces and their norms will be used in this paper. Let L2( ) and L2(?) be the spaces of square integrable functions on and on ?, respectively. Corresponding inner products and norms will be denoted by (; ); h; i? and kk; jj?, respectively. Let H m( ) and H m (?), for nonnegative integer m, denote the usual Sobolev spaces with norms k km and j jm;?, and H0m ( ) the completion of C01( ) in the norm of H m ( ); see [1, 8] for more details of function spaces and related norms. De ne the sesquilinear form a! (; ) : H01( ) H01( ) ! C by

a! (u; v) = i! 1 u; v + (ru; rv);

u; v 2 H01( ):

4


A variational formulation of problem (1.2) is then to nd, for given fb 2 H ?1( ), ub(; !) 2 H01( ) such that a! (ub; v) = hf;b vi; v 2 H01( ); (2.1) where h; i denotes the duality pairing between H ?1 ( ) and H01( ). 2.2. Uniqueness and Existence. In what follows, C will denote a generic positive constant which may dier from place to place. Theorem 2.1. Suppose that is a bounded Lipschitz domain in RN ; N = 2; 3; with the boundary ?. Assume that fb(; !) 2 H ?1( ). Then for each !, the equation (2.1) has a unique solution ub(; !) 2 H01( ): Proof. From the de nition and Poincare lemma, p ja!(ub; ub)j jRe a! (ub; ub)j = k rubk2 C kubk21; (2.2) where C only depends on and : Thus, a! (; ) is coercive. Also note that for ub; bv 2 H01( );

p p ja! (ub; bv)j !

p1 ub

p1 bv

+ k rubkk rbvk (2.3) C (1 + !)kubk1kbvk1; where C only depends on , and : Thus, a! (; ) is continuous. An application of the Lax-Milgram lemma [8, 23] gives uniqueness and existence. Remark 2.1. The estimate (2.2) implies that the coercivity is independent of !, and (2.3) implies jja! jj C (1 + !): 2.3. Stability and Regularity. We are now going to establish stability and regularity of the solution of the problem (1.2). Using coercivity of a! (; ), it is easy to see the following stability estimate for problem (2.1): kub(; !)k1 C kfb(; !)k?1: Indeed, improved estimates can be obtained. We begin with taking v = ub in (2.1): 1 i! ub; ub + (rub; rub) = (f;b ub); from which the real and imaginary parts give

! kubk2 !

p1 ub

2 = Im(f;b ub) kfbk kubk; (2.4a)

p (2.4b) krubk2 k rubk2 = Re(f;b ub) kfbk kubk: From (2.4a) it follows that kubk ! kfbk: (2.5)


5

Combining (2.4b) and (2.5) yields krubk2 1 kfbk kubk !1 kfbk2: Thus r krubk p1! kfbk: Summarizing the above estimates, one gets the following lemma. Lemma 2.1. If ub(; !) 2 H01 ( ) is a solution of problem (2.1), then 1 (2.6a) kub(; !)k C min 1; ! kfb(; !)k; 1 (2.6b) krub(; !)k C min 1; p! kfb(; !)k; 1 kub(; !)k1 C min 1; p! kfb(; !)k: (2.6c) Let us now turn to get an H 2( )-estimate for the solution ub of (1.2). Assume

is a convex polygon in R2 or a C 2{domain in R3; and recall the following [16]:

2 N X

@ 2u b

bk2; u b 2 H 2 ( ) \ H01 ( ): (2.7)

@x @x C ku i j i;j =1 We now have the following lemma. Lemma 2.2. Assume that ! is given and fb(; !) 2 L2 ( ): If ub(; !) 2 H 2 ( ) \ H01( ) be the solution of (2.1), then there exists a positive constant C such that kub(; !)k2 C kfb(; !)k: Proof. Using (2.7), (1.2), (2.6a) and (2.6b), the following estimate is obtained:

jubj2 C kubk C

i ! ub ? r rub ? fb

! b C kubk + kf k + krkkrubk n

o

C kubk1 + k k C kfbk: fb

This completes the proof. In particular, the estimate of Lemma 2.2 shows the existence of ub 2 H 2( ) \ H01( ) if fb 2 L2( ) by the method of Galerkin approximation [22]. We summarize the above results in the following theorem.

6


Theorem 2.2. Suppose is a convex polygon in R2 or a C 2{domain in R3: Then for any fb(; !) 2 L2( ), there exists a unique solution ub(; !) 2 H01( ) \ H 2( ) with kub(; !)k2 C kfb(; !)k: Remark 2.1. The estimate in Lemma 2.2 means that the elliptic regularity coecient C for Problem (1.1) corresponding to ! is not singular as ! tends to zero or 1: This comes from the nature of parabolicity of (1.1), which diers from that of hyperbolicity of wave equations resulting into Helmholtz equations [12]. As immediate results of Theorems 2.1 and 2.2, we have the followings. Corollary 2.1. If kfb(; !)k?1 is integrable over the frequency domain R with respect to !, then there exist Fourier inverses of ub(; !) and @ ub ; 1 i N:

@xi Corollary 2.2. If kfb(; !)k is integrable over the frequency domain R with respect @ ub and @ 2ub ; 1 i; j N: to !, then there exist Fourier inverses of ub(; !); @x @x @x i

3. Finite Element Approximation

i

j

3.1. Finite Element Method for a Single Frequency. Let h > 0 be a discretization parameter tending to zero and Vh H01( ) be a nite element space. Then the discrete problem corresponding to (2.1) reads: Find ub 2 Vh such that for a given fb 2 H ?1 ( ), a! (ub; v) = hf;b vi; v 2 Vh : (3.1) We shall assume that Vh satisfy the following property: There exist a positive constant C and an operator h : H 2( ) ! Vh , independent of h such that kv ? hvkk Ch2?k kvk2; v 2 H 2( ); k = 0; 1: (3.2) For such nite element spaces, we refer, for example, [2, 3, 6, 7, 13, 17]. Let ubh(; !) 2 Vh be the Galerkin approximation to ub(; !) of (2.1). Then ubh(; !) exists uniquely due to Theorem 2.1. Furthermore we have the following error estimates. Theorem 3.1. Suppose that is a convex polygon in R2 or a C 2{domain in R3 and fb(; !) 2 L2( ). Then the approximate solution ubh(; !) of (3.1) to the solution ub(; !) of (2.1) for each ! satis es that kub(; w) ? ubh(; w)k1 C (1 + !)hkfb(; !)k; (3.3a) 2 2 kub(; w) ? ubh(; w)k C (1 + !) h kfb(; !)k: (3.3b)


7

Proof. From (2.1) and (3.1), we have the error equation: a!(ub ? ubh; v) = 0; v 2 Vh : By coercivity and continuity of a! and the above error equation, for arbitrary 2 Vh , kub ? ubhk21 Ca!(ub ? ubh; ub ? ubh) = C fa! (ub ? ubh; ub ? ) + a! (ub ? ubh; ? ubh)g = Ca!(ub ? ubh; ub ? ) C (1 + !)kub ? ubh k1 kub ? k1: Then by using (3.2) and Theorem 2.2, an appropriate choice of yields (3.3a). For a proof of the second inequality the usual duality argument will be used. Let z 2 H 2( ) \ H01( ) be the solution of a! (z; v) = hub ? ubh; vi; v 2 H01( ): Then, by elliptic regularity, we have

kzk2 C kub ? ubhk:

Using the continuity of a! , (3.2), (3.3a), and the above estimate, we have kub ? ubhk2 = a! (z; ub ? ubh) = a! (z ? hz; ub ? ubh) C (1 + !)kub ? ubhk1 kz ? h zk1 C (1 + !)hkub ? ubhk1 kzk2 C (1 + !)2h2kfbk kub ? ubhk: This completes the proof. 3.2. Full Error Estimate. We are now going to give the full estimate of errors introduced by the truncation and discretization of a quadrature of the inverse Fourier transform, and caused by nite element approximations. First we need the following lemma. Lemma 3.1. Suppose that Z 1 0

2k

t

Z t 0

kf (; s)k2 ds dt < 1;

k = 0; 1; 2; : : : ; m; for some nonnegative integer m. Let u be the solution of Problem (1.1). Then we have the following estimates: for k = 0; 1; 2; : : : ; m

ktku(; t)k2L2((0;1);H 1( ))

C

Z 1 0

2k

t

Z t 0

kf (; s)k2 ds dt:

(3.4)

8


Proof. Multiply (1.1a) by ut(; t) to get, for " > 0, 1 ku (; t)k2 + 1 d k1=2ru(; t)k2 1 kf (; t)k2 + ku (; t)k2; t t 2 dt 4 from which we have with a choice of a suciently small 0 < < 1

kut(; t)k2 + dtd k1=2ru(; t)k2 C kf (; t)k2:

By integrating the last inequality with respect to t over [0; T ] for any positive T , we get

kru(; T )k C k ru(; T )k C 2

1=2

2

Z T 0

kf (; t)k2dt:

Multiplying by T both sides of the above inequality and then integrating over (0; 1) in T , we obtain the desired estimate (3.4). This completes the proof. We consider restricted sources such that jfb(; !)j are square integrable with respect to ! and thus negligible for large j!j: We then choose a suciently large ! > 0 so that both ub(; !) and fb(; !) are negligible for j!j > !: Also recall that the computation of ub(; !) for ! < 0 is not necessary. Let N! be a positive integer and de ne the discretization parameter ! of the frequency domain by the formula ! = !=N! ; and introduce the mesh points !j? 12 = (j ? 21 )!; j = 1; ; N! on the interval (0; !): Due to (1.3), the time-domain solution u of (1.1) will then be approximated by N! X uh!;! (x; t) = 1 ubh (x; !j?1=2)eit!j?1=2 !: j =1 2k

We now try to estimate the convergence of uh!;! (; t) to u(; t) for a xed time t. Setting Z ! 1 u! (x; t) = ub(x; !)ei!td! 0 and N! X 1 u! ;! (x; t) = ub(x; !j?1=2)eit!j?1=2 !; j =1 we have u(x; t) ? uh!;! (x; t) = (u(x; t) ? u! (x; t)) + (u! (x; t) ? u!;! (x; t)) ? + u!;! (x; t) ? uh!;! (x; t) E1(x; t) + E2(x; t) + E3(x; t): (3.5)


9

First, by Theorem 2.2 we get Z Z C kE1(; t)k1 krub(; !)kd! C kfb(; !)kd!: (3.6) !>! !>! Thus kE1(; t)k1 ! 0 as ! ! 1: We also have R R kE2(; t)k21 C2 0! rub(x; t)ei!td! 2 ? PNj=1! rub(x; !j?1=2)eit!j?1=2 ! dx

R

@ 2 rub(x;!)ei!t 2 4 C (!) @!2 L2 (0;!) dx R 4 2ru(x; ) C (!)

?t[

2 2 d (3.7) + 2ttru(x; ) ? t rub(x; )

L2 (0;!) dx n R 2ru(x; )k2 2 C (!)4 kt[ L (0;1) o 2 d + t ktru(x; )k2L2(0;1) + t4krub(x; )k2L2(0;1) dx n 4 C (!) kt2uk2L2((0;1);H 1( )) o + t2ktuk2L2((0;1);H 1( )) + t4kuk2L2((0;1);H 1( )) : Thus, using Lemma 3.1, we see that kE2(; t)k1 ! 0 as ! ! 0 : Finally, from Theorem 3.1 and Theorem 2.2, we have

? kE3(; t)k1 C

1 PNj=1! rubh(; !j?1=2) ? rub(; !j?1=2) eit!j?1=2 !

N! ru C ! P bh (; !j ?1=2) ? ru b(; !j ?1=2 ) j =1 P (3.8) C ! Nj=1! h(1 + !j ?1=2)kfb(; !j?1=2)k : Ch

(1 + !) fb(; !)

2 2 L! ((0;1);L ( ))

Thus, if we assume that

(1 + ! )fb(

; !)

we have that

L2! ((0;1);L2( ))

kE3(; t)k1 ! 0

Z 1 0

k(1 + !)fb(; !)k d! 2

1=2

< 1;

as h ! 0 : Combining the estimates (3.6), (3.7), and (3.8), and using Lemma 3.1, we have the full error estimate.

10


Theorem 3.2. Suppose that is as in Theorem 3.1. Assume that for k = 0; 1; 2 Z 1 0

and

t

2k

Z t 0

kf (; s)k2 dsdt < 1;

; !)

(1 + ! ) fb(

L2! ((0;1);L2( ))

< 1:

Then uh!;! (; t) converges to u(; t) for a xed time t; moreover, for t > 0;

ku(; t) ? uh!;! (; t)k1 C1 +C2(!

)2

Z

k k

fb( ; !) d! !>! Z 1 Z s 2 X 2?k

2k

t

k=0

+C3h (1 + !) fb(

; !

s

0

)

0

kf (; r)k dr ds

L2! ((0;1);L2( ))

2

1=2

;

with Cj ; j = 1; 2; 3; dependent only on the domain and the coecients and : Similarly, if we estimate kE1(; t)k; kE2(; t)k; and kE3(; t)k in (3.5), using, for the second error term kE2(; t)k; (3.3b) instead of (3.3a), we obtain the following theorem. Theorem 3.3. Suppose that is as in Theorem 3.1. Assume that for k = 0; 1; 2 Z 1 0

and

t

2k

Z t 0

kf (; s)k2 dsdt < 1;

2

(1 + ! ) fb(

; !)

L ((0;1);L ( )) < 1: 2

2

!

Then uh!;! (; t) converges to u(; t) for a xed time t; moreover, for t > 0;

ku(; t) ? uh!;! (; t)k C1 +C2(!

)2

Z

k k

fb( ; !) d! !>! Z 1 Z s 2 X 2?k

2k

t

k=0 +C3h2

(1 + !)2 fb(

0

s

; !)

0

kf (; r)k dr ds

L2! ((0;1);L2( ))

2

1=2

;

with Cj ; j = 1; 2; 3; dependent only on the domain and the coecients and :


11

140

120

100

80

60

40

20

0

-20 0

0.5

1

1.5

Time Shape of Source Function g(t): 's represent the original input data g(i=Nt); i = 0; 1; 2; ; Nt, Nt = 40; and the solid line represents the graph of the modi ed source extended periodically to [0; 2] by interpolation.

Figure 1.

4. Numerical Experiments Some numerical experiments were performed for the Problem (1.1) with = = 1. Let = (0; 1)2 in R2 and J = (0; T ). We x the simulation time T = 1. The source f (x; y; t) = (x; y)g(t) was chosen so that the analytic solution u to Problem (1.1) is u(x; y; t) = sin(2x) sin(3y)h(t); (x; y) 2 ; t 2 J; where h(t) = t 1+10t2 . For the discrete input data for g , assume that g (iT=Nt ); i = 0; 1; Nt, with Nt = 40; are given. Notice that, in computing the Fourier transform bg(!) of g(t), if more informations of g(t) in [0; 1) are given, then more precise bg(!) may be obtained. However, in practice only a limited nite number of data are given. Therefore it is natural to expand the domain of data to a large interval. In our experiments, we assumed that g(2T + iT=Nt) = g(iT=Nt); i = 0; 1; ; Nt; regarding g as a periodic function with period 2T , thereby supplying 2Nt + 1 data points of time source

2

12


N! 20 30 40 50 60 70 80 160

(!) (5/4) (5/6) (5/8) (5/10) (5/12) (5/14) (5/16) (5/32)

Nx = Ny = 50 Nx = Ny = 100 Nx = Ny = 200 4.046E-2 (9.844E-3) 4.042E-2 (8.243E-3) 4.042E-2 (7.864E-3) 3.928E-2 (1.058E-2) 3.924E-2 (8.992E-3) 3.924E-2 (8.615E-3) 3.889E-2 (1.083E-2) 3.885E-2 (9.250E-3) 3.885E-2 (8.873E-3) 3.872E-2 (1.095E-2) 3.867E-2 (9.368E-3) 3.867E-2 (8.992E-3) 3.862E-2 (1.101E-2) 3.858E-2 (9.433E-3) 3.858E-2 (9.056E-3) 3.856E-2 (1.105E-2) 3.852E-2 (9.471E-3) 3.852E-2 (9.095E-3) 3.853E-2 (1.107E-2) 3.848E-2 (9.496E-3) 3.848E-2 (9.120E-3) 3.844E-3 (1.114E-2) 3.839E-2 (9.558E-3) 3.839E-2 (9.181E-3) Table 1. Case ! = 25: Relative L2 (J ; L2( ))-Errors and Relative L2( )-Errors at time t = 1 for the solution by Frequency Domain Method. The numbers in parentheses denote the Relative L2( )Errors at time t = 1.

N! 20 30 40 50 60 70 80 160

(!) (5/2) (5/3) (5/4) (5/5) (5/6) (5/7) (5/8) (5/16)

Nx = Ny = 50 Nx = Ny = 100 Nx = Ny = 200 4.069E-3 (2.457E-3) 3.219E-3 (6.185E-4) 3.132E-3 (1.584E-4) 3.826E-3 (2.442E-3) 2.892E-3 (6.041E-4) 2.792E-3 (1.443E-4) 3.776E-3 (2.438E-3) 2.826E-3 (6.000E-4) 2.723E-3 (1.404E-4) 3.757E-3 (2.436E-3) 2.800E-3 (5.982E-4) 2.696E-3 (1.387E-4) 3.747E-3 (2.435E-3) 2.786E-3 (5.972E-4) 2.682E-3 (1.378E-4) 3.741E-3 (2.434E-3) 2.779E-3 (5.967E-4) 2.674E-3 (1.373E-4) 3.737E-3 (2.434E-3) 2.774E-3 (5.963E-4) 2.669E-3 (1.370E-4) 3.729E-3 (2.433E-3) 2.762E-3 (5.954E-4) 2.656E-3 (1.362E-4) Table 2. Case ! = 50: Relative L2 (J ; L2( ))-Errors and Relative L2( )-Errors at time t = 1 for the solution by Frequency{Domain Method. The numbers in parentheses denote the Relative L2( )Errors at time t = 1.

informations in order to extend the data smoothly to [0; 2T ] by polynomial interpolation. The data g(T + iT=Nt); i = 1; ; Nt ? 1 were then computed by using Neville's algorithm [24, 26]. For various values of !, choose N! and set !j = (j?1N=!2)! ; j = 1; ; N! . From the data g(iT=Nt); i = 0; ; Nt, the Fourier transforms bg(!j ); j = 1; ; N! ; were then calculated by using the midpoint rule. Figure 1 shows the time shape of source function g(t): 's represent the original input data g(iT=Nt); i = 0; 1; 2; ; Nt, and the solid line represents the graph of the modi ed source extended periodically and smoothly to [0; 2T ] by polynomial interpolation. We solved (3.1) for !j?1=2; j = 1; ; N! . In the calculation of nite element solutions Nx Ny uniform triangular meshes were taken for the triangulation of , and C 0 piecewise linear nite element were used. The resulting algebraic problems


N! 20 30 40 50 60 70 80 160

(!) (10/2) (10/3) (10/4) (10/5) (10/6) (10/7) (10/8) (10/16)

N! 30 40 50 60 70 80 160 320

(!) (20/3) (20/4) (20/5) (20/6) (20/7) (20/8) (20/16) (20/32)

13

Nx = Ny = 50 Nx = Ny = 100 Nx = Ny = 200 3.179E-0 (3.140E-3) 3.183E-0 (1.367E-3) 3.184E-0 (9.636E-4) 3.443E-2 (2.679E-3) 3.470E-2 (8.587E-4) 3.478E-2 (4.236E-4) 3.040E-3 (2.614E-3) 1.548E-3 (7.884E-4) 1.297E-3 (3.462E-4) 3.035E-3 (2.591E-3) 1.536E-3 (7.634E-4) 1.282E-3 (3.185E-4) 3.032E-3 (2.580E-3) 1.531E-3 (7.513E-4) 1.277E-3 (3.050E-4) 3.031E-3 (2.573E-3) 1.529E-3 (7.445E-4) 1.274E-3 (2.975E-4) 3.030E-3 (2.569E-3) 1.528E-3 (7.403E-4) 1.272E-3 (2.927E-4) 3.029E-3 (2.560E-3) 1.525E-3 (7.304E-4) 1.269E-3 (2.817E-4) Table 3. Case ! = 100: Relative L2 (J ; L2( ))-Errors and Relative L2( )-Errors at time t = 1 for the solution by Frequency{ Domain Method. The numbers in parentheses denote the Relative L2( )-Errors at time t = 1. Nx = Ny = 50 Nx = Ny = 100 Nx = Ny = 200 3.203E-0 (1.439E-2) 3.207E-0 (1.630E-2) 3.207E-0 (1.681E-2) 3.178E-0 (4.358E-3) 3.181E-0 (2.690E-3) 3.182E-0 (2.313E-3) 5.251E-1 (3.584E-3) 5.266E-1 (1.859E-3) 5.270E-1 (1.470E-3) 3.078E-2 (3.326E-3) 3.106E-2 (1.577E-3) 3.115E-2 (1.181E-3) 2.747E-3 (3.206E-3) 1.124E-3 (1.445E-3) 8.313E-4 (1.045E-3) 2.762E-3 (3.139E-3) 1.169E-3 (1.371E-3) 8.946E-4 (9.687E-4) 2.774E-3 (3.004E-3) 1.218E-3 (1.221E-3) 9.636E-4 (8.119E-4) 2.775E-3 (2.976E-3) 1.223E-3 (1.189E-3) 9.706E-4 (7.785E-4) Table 4. Case ! = 200: Relative L2 (J ; L2( ))-Errors and Relative L2( )-Errors at time t = 1 for the solution by Frequency{ Domain Method. The numbers in parentheses denote the Relative L2( )-Errors at time t = 1.

were solved by using a Gaussian elimination type solver, e.g., Yale Sparse Matrix Package [11]. Tables 1, 2, 3, and 4 show the relative L2(J ; L2( ))-errors and L2( )-errors at time t = 1 for ! = 25; 50; 100 and 200. We observe that the errors decay, but not drastically, as the number, say N! , of points in the frequency range (0; !) increases. The results convince us that we might choose reasonable number of frequencies in the frequency range (0; !), where Fourier transformed elliptic problems are solved. However, the numerical results indicate that the errors decay rather rapidly as ! increases up to certain limits. Indeed, in order to apprehend more properly

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time ! = 25 ! = 50 ! = 100 ! = 200 0.1 7.066E-2 8.358E-3 1.657E-3 3.410E-3 0.2 4.075E-2 1.013E-4 3.183E-3 2.615E-4 0.3 3.423E-2 2.902E-3 5.288E-4 1.282E-4 0.4 1.857E-2 2.705E-3 5.767E-4 8.043E-4 0.5 2.753E-2 1.750E-4 3.773E-4 1.010E-3 0.6 2.977E-2 2.003E-3 3.422E-4 3.015E-4 0.7 5.330E-2 1.285E-3 5.097E-4 5.644E-4 0.8 6.320E-2 1.350E-3 3.566E-4 5.205E-4 0.9 4.971E-2 2.361E-3 1.574E-4 3.349E-4 1.0 8.873E-2 1.370E-4 2.817E-4 7.785E-4 Table 5. Comparison of Relative L2 ( )-Errors at time t = 0:1; 0:2; ; 1 for the solutions by Frequency{Domain Method with Nx = Ny = 200 and ! = 5=8.

Nt 20 40 80 160

Nx = Ny = 50 NX = Ny = 100 NX = Ny = 200 2.338E-3 (2.447E-3) 5.505E-4 (6.140E-4) 1.462E-4 (1.552E-4) 2.375E-3 (2.445E-3) 5.867E-4 (6.122E-4) 1.421E-4 (1.535E-4) 2.382E-3 (2.445E-3) 5.940E-4 (6.118E-4) 1.467E-4 (1.530E-4) 2.384E-3 (2.445E-3) 5.960E-4 (6.117E-4) 1.485E-4 (1.529E-4) Table 6. Relative L2 (J ; L2( ))-Errors and Relative L2 ( )-Errors at time t = 1 for the solution by Crank-Nicolson Method. The numbers in parentheses denote the Relative L2( )-Errors at time t = 1.

L2(J ; L2( ))-errors in Tables 1, 2, 3, and 4, one must compare table entries for xed frequency step size ! as ! increases. We also present the relative L2( )-errors for each time step t = 0:1; 0:2; ; 1 for Nx = Ny = 200 with ! = 5=8 for the cases ! = 25; 50; 100; 200 in Table 5. The numerical data show that the relative errors behave more regularly as ! becomes larger. It is also worthwhile to observe from Table 5 that the errors are up and down but do not increase as time grows. This feature should be emphasized compared to error behaviors for traditional solvers; as time increases, errors generated by traditional solvers in the space-time formulation, such as Crank-Nicolson and backward-Euler methods, usually grow. For comparison, we provide the L2(J ; L2( ))-errors induced by Crank-Nicolson method in Table 6. The overall error behaviors of Crank-Nicolson method are a little better than those of our method. Theorem 3.3 says that the error behaviors of our method are aected by !, (!)2 and h2 while the asymptotic behaviors of the error for Crank-Nicolson method is O(h2) + O((t)2). In particular, the


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eect of ! depends upon decay rate of kfb(; !)k as ! ! 1. Hence, we expect that our method will compete with Crank-Nicolson method when ! is suciently large. Numerical results indicate that we got comparable solutions for moderate ! and !. Among all the features mentioned above, the most favorable advantage for our scheme lies in the natural parallelization when there are given massively parallel processors. [1] [2] [3] [4]

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Second International Symposium on Domain Decomposition Methods for Partial Dierential Equations, Philadelphia, 1989. SIAM. [5] T. F. Chan, R. Glowinski, J. Periaux, and O. B. Widlund, editors. Third International Symposium on Domain Decomposition Methods for Partial Dierential Equations, Phil-

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Fourth International Symposium on Domain Decomposition Methods for Partial Dierential Equations, Philadelphia, 1991. SIAM. [16] P. Grisvard. Boundary Value Problems in Non{Smooth Domains. Pitman, Boston, London,

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[18] D. E. Keyes, T. F. Chan, G. A. Meurant, J. S. Scroggs, and R. G. Voigt, editors. Fifth International Symposium on Domain Decomposition Methods for Partial Dierential Equations, Philadelphia, 1992. SIAM. [19] D. E. Keyes and J. Xu, editors. Domain Decomposition Methods in Scienti c and Engin-

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American Mathematical Society. [20] D. Kim, J. Kim, and D. Sheen. Absorbing boundary conditions for wave propagations in viscoelastic media. J. Comput. Appl. Math., 76:301{314, 1996. [21] C.-O. Lee, J. Lee, and D. Sheen. Frequency domain formulation of linearized NavierStokes equations. Technical Report 96{77, GARC, Seoul National University, Seoul 151{ 742, Korea, 1997. [22] J. L. Lions. Quelques methodes de resolution des problemes aux limites non lineaires. Dunod Gauthier-Villars, Paris, 1969. [23] J. Necas. Les Methodes Directes en Theorie des Equations Elliptiques. Masson, Paris, 1967. [24] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in Fortran, The Art of Scienti c Computing. Cambridge University Press, Cambridge New York, second edition, 1994. [25] A. Quarteroni, J. Periaux, Yu. A. Kuznetsov, and O. B. Widlund, editors. Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, volume 157 of Contemporary Mathematics, Providence, Rhode

Island, 1994. American Mathematical Society. [26] J. Stoer and R. Bulirsch. Introduction to Numerical Analysis. Springer-Verlag, New York Berlin Heidelberg, second edition, 1992. Department of Mathematics, Inha University, Inchon 402-751, Korea

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Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea

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Department of Mathematics, Seoul National University, Seoul 151{742, Korea

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