CONVERGENCE OF FINITE DIFFERENCE METHOD FOR THE GENERALIZED SOLUTIONS OF SOBOLEV EQUATIONS

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1 J. Korean Math. Soc. 34 (1997), No. 3, pp CONVERGENCE OF FINITE DIFFERENCE METHOD FOR THE GENERALIZED SOLUTIONS OF SOBOLEV EQUATIONS S. K. CHUNG, A.K.PANI AND M. G. PARK ABSTRACT. In this paper, finite difference method is applied to approximate the generalized solutions of Sobolev equations. Using the Steklov mollifier and Bramble-Hilbert Lemma, a priori error estimates in discrete L 2 as well as in discrete H 1 norms are derived first for the semidiscrete methods. For the fully discrete schemes, both backward Euler and Crank-Nicolson methods are discussed and related error analyses are also presented. 1. Introduction Let be a rectangular domain in R 2 with boundary,andt be < T <. We consider finite difference approximations for the generalized solutions of differential equations of the form (1.1a) (1.1b) (1.1c) u t + Au t + Bu = f, (x,t) (,T], u(x, ) = u (x), x, u(x, t) =, (x,t) [, T ], where f = f (x, t), A and B are of the following forms and B(x, t)u = A(x)u = (b lq (x, t) u ) + x l x q (a lq (x) u ), x l x q b l (x, t) u x l + b(x, t)u. Received April 8, Mathematics Subject Classification: 65N6. Key words and phrases: Sobolev equation, Steklov mollifier, discrete projection. The first author is supported by SNU-DaeWoo Research Fund grant and KOSEF grant

2 516 S. K. Chung, A. K. Pani and M. G. Park We now make the following assumptions. (1) The coefficients of A(x) and B(x, t), together with f, are smooth and bounded as far as the ensuring analysis demands. (2) The coefficients a lq = a ql satisfy a lq ξ l ξ q ξ 2 l, (ξ 1,ξ 2 ) R 2. (3) There exists a unique generalized solution of the problem (1.1) with smoothness corresponding to that of the generalized solution. The problem of this type arises in the study of consolidation of clay, heat conduction, homogeneous fluid flow in fissured material and shear in second order fluids. For existence, uniqueness and applications of (1.1), we refer to Ewing [5] and the extensive literatures contained therein. Finite element methods for (1.1) have been studied by Ewing [5], Ford [6], Arnold et al. [1], Lin and Zhang [1] and Nakao[11]. For the analysis of finite difference schems, Ford and Ting [7] [8] have obtained an order O(k + h 2 ) of convergence for the backward Euler method and O(k 2 + h 2 ) for the Crank-Nicolson method under the assumption that the exact solution u, u t C 4 ( ) and R. For the problem in several space variables, Ewing [4] has obtained L 2 error estimates of order O(k 2 +h 2 ) for the Crank-Nicolson method under the assumption that u, u t C 4 ( ). In all these articles [4], [7]-[8], traditional Taylor s expansion is used for convergence analysis, which imposes sever smoothness conditions on the solution. In this paper, using Steklov mollifier and a nonclassical discrete projection method we derive rates of convergences for the finite difference schemes and obtain orders of convergence compatible with the smoothness of the solution. After giving preliminaries in Section 2, we consider the semidiscrete scheme, its stability and error analysis in Section 3. In Section 4, we introduce a nonclassical discrete projection and obtain O(h 2 ) convergence in the L 2 - norm. In Section 5, we discuss fully discrete schemes which are optimal. We obtain an order O(k +h 2 ) of convergence for the backward Euler method and an oredr O(k 2 + h 2 ) for the Crank-Nicolson scheme under the assumption that u, u t H 2 ( ). 2. Preliminaries Without loss of generality, it is assumed that the domain is the unit square

3 Convergence of Sobolev equations 517 in R 2. We select a mesh of width h = 1, where M is a positive integer, and M cover = with a square grid of mesh points x ij = (ih, jh), i, j =, 1,..., M. Let h ={x ij : x ij }and h ={x ij : x ij }. Forafunctionwdefined on h, the following notations will be used: for x h and l = 1, 2, and w ±l = w(x ± he l ), w +l, q = w(x + he l he q ), l w(x) = w(x + he l) w(x) h, l w(x) = w(x) w(x he l), h where e l is the l-th unit vector in R 2. The Steklov mollifiers are defined in the following manner: S = S 2 1 S2 2 with S 2 l = S + l S l, l = 1, 2, where S + l φ(x) = 1 φ(x + she l )ds, S l φ(x) = 1 φ(x + she l )ds. The operators S ± l commute and the following relationships hold: (2.1) S + l φ x l = l φ, S l φ x l = l φ. We now introduce the discrete L 2 space, denoted by L 2 h ( h), with an inner product and the norm given by: w, v =h 2 x h w(x)v(x) and w,h = w, w 1 2, for v, w L 2 h ( h ). Further, let Hh 1 = H h 1( h) denote the discrete analogue of H 1 -Sobolev space with norm w 2 1,h = w 2,h + l w 2,h. We also introduce a discrete H 2 -Sobolev space with the following norm w 2 2,h = w 2 1,h + l q w 2,h,

4 518 S. K. Chung, A. K. Pani and M. G. Park and denote it by Hh 2 = H h 2( h). Whenever there is no confusion, we write w and w j, in place of w,h and w j,h. Throughout the paper, L 2 and H m will denote the norm in L 2 and the Sobolev space H m ( ), respectively. Further, let W m,p ( ) denote the seminorm in W m,p ( ). For functions v and w defined on h, the following identity is an easy consequence of summation (2.2) l v, w = v, l w, l = 1, 2. Along with the usual Bramble-Hilbert Lemma [2], the following bilinear version of it will be needed for our convergence analysis. For a proof, we refer the reader to Ciarlet[3]. LEMMA 2.1. Let P [r] be the set of all polynomials of degree [r], where [r] denotes the largest integer less than r >. If η is a bounded linear functional on W α,p ( ) W β,q ( ), with α, β (, ) and p, q [1, ] such that η(u,v)=, U P [α] ( ), v W β,q ( ), η(u, V) =, u W α,p ( ), V P [β] ( ), then there exists a positive constant C such that η(u,v) C u W α,p ( ) v W β,q ( ), u W α,p ( ), V P [β] ( ). In the proofs below, the inequality (2.3) ab ɛa ɛ b2, a, b R,ɛ. will be used frequently and C will denote a generic positive constant whose dependence can be easily established from the proofs. 3. Semidiscrete schemes Let A h and B h be defined for (x, t) h [, T ]as A h V = 1 [ ( l alq (x) q V ) ( + l alq (x) q V )], 2

5 Convergence of Sobolev equations 519 and of B h V = [ ( l blq (x, t) q V ) ( + l blq (x, t) q V )] b l (x, t) [ l V + l V ] + S (b(x, t)) V. Now, the semidiscrete approximation u h of (1.1) is determined as a solution (3.1a) (3.1b) (3.1c) u h,t + A h u h,t + B h u h = Sf, (x,t) h [, T ], u h (x, ) = u (x), x h, u h (x, t) =, (x,t) h (,T]. Let H,h 1 ={v H1 h :v=on h}. From the assumptions on A(x) and B(x, t), the following lemma can be easily verified using summation by parts. LEMMA 3.1. For v, w H,h 1, there exist constants C such that (1) the discrete Poincaré inequality : v 2 C 2 lv 2, (2) A h v, v C v 2 1, (3) B h v, w C v 1 w 1. For subsequent error estimates, we derive stability results for the modified semidiscrete version of (3.1); namely, (3.2) u h,t + A h u h,t + B h u h = Sf + l F, where F is a function defined on [, T ] which vanishes on h and F() =. The stability result for (3.2) is stated in the following theorem. THEOREM 3.1. Let u h be a solution of (3.2). Then there exists a constant C such that u h (t) 1 C{ u h () 1 + ( Sf(s) 2 ds) 1/2 + ( F(s) 2 ds) 1/2 }.

6 52 S. K. Chung, A. K. Pani and M. G. Park Proof. Forming the inner product between (3.2) and u h, we obtain d dt u h 2 1 C{ u h Sf u h + F u h 1 }. Integrating with respect to t, wefindthat u h (t) 2 1 C{ u h() u h (s) 2 1 ds + Sf(s) 2 ds + F(s) 2 ds}. An application of Gronwall s Lemma now completes the proof. Using the above stability result, we shall derive the following error estimate. THEOREM 3.2. Let u and u h be the solution of (1.1) and (3.1), respectively. Let u, u t H α ( ), 1 α 3, and for t (, T ]. Then there exists a constant C such that for the error e(t) = u(t) u h (t) the following estimate holds. e(t) 1 C(u, T )h α 1 Proof. From (1.1) and (3.1), we obtain e t + A h e t + B h e = (u t Su t ) + (A h u t SAu t )+(B h u SBu) = I 1 (t)+ I 2 (t)+ I 3 (t). Following Jovanović et al.[9], I 3 (t) is rewritten as I 3 (t) = l ξ lq (t) + ξ l (t) + ξ(t), where ξ lq = ξ (1) lq + ξ(2) lq + ξ(3) lq + ξ(4) lq with ξ (1) lq = S + l S3 l 2 (b u lq ) (S + l S3 l 2 x b lq)(s + l S3 l 2 q ξ (2) lq = [S + l S 2 3 l b lq 1 2 (b lq + b +l lq )](S+ l ξ (3) lq = 1 2 (b lq + b +l lq )[S+ l S 2 3 l ξ (4) lq = 1 4 (b lq b +l lq )( q u q u +l ), S 2 3 l u x q ), u ), x q u x q 1 2 ( q u + q u +l )],

7 Convergence of Sobolev equations 521 and ξ = (Sb)u S(bu). Further, we decompose ξ l as ξ l (t) = ξ (1) l (t) + ξ (2) l (t) + ξ (3) l (t) with ξ (1) l ξ (2) l ξ (3) l Similarly, we also rewrite I 2 as = b l (t)[ 1 2 ( lu + l u) S u x l ], = b l (t) S u x l S(b l (t))s u x l, = S(b l (t))s u x l S(b l (t) u x l ), I 2 (t) = l η lq (t), where η lq = η (1) lq + η(2) lq b lq are replaced by a lq. Altogether, we have + η(4) lq + η(4) lq. Here η lq are the same as ξ lq except that (3.3) e t + A h e t + B h e = I 1 (t) + ξ l + ξ + l (η lq + ξ lq ). Setting F(t) = 2 (η lq(t) + ξ lq (t)) and the first term on the right hand side of the above equation as Sf, we apply Theorem 3.1 to obtain e(t) 1 C[ e() 1 + ( I 1 (s) 2 ds) 1/2 + ( ξ(s) 2 ds) 1/2 + ( ξ l (s) 2 ds) 1/2 + {( η lq (s) 2 ds) 1/2 + ( ξ lq (s) 2 ds) 1/2 }].

8 522 S. K. Chung, A. K. Pani and M. G. Park Since I 1 (t) is a bounded linear functional on H β (D) with its kernel contained in P 1 (D), where D ={(s 1,s 2 ) R 2 : 1 s l 1,,2}. The Bramble-Hilbert Lemma, therefore, yields (3.4) I 1 (t) Ch β u t H β ( ), 1 β 2. To estimate ξ, we first note that ξ = (Sb)(u t Su t ) + (Sb)(Su t ) S(bu t ). Again a use of Lemma 2.1 yields, for 1 β 2, (3.5) ξ(s) ds C ( ) t b L (W β 1, ) h β u t (s) β ds. As in Jovanović et al. [9], we obtain an estimate for ξ lq of the form (3.6) ( ξ lq (s) 2 ds) 1/2 Ch α 1 ( u t (s) 2 α ds)1/2, where C depends on max l,q b lq L (W α 1, ) and 1 α 3. Following the estimates of ξ lq, the estimation of η lq can be easily obtained with similar bounds. Further, the estimates of ξ (i) l are similar to those of ξ (i) lq for i = 1, 2, 3. This completes the rest of the proof. REMARK. Since e e 1, we obtain from the previous Theorem e(t) C(u,T)h α 1, 1 α 3. In order to achieve an order O(h 2 ) of convergence, it is to be noted that we need u, u t L 2 (H 3 ( )). In contrast to papers [4], in which Taylor s expansion is used to derive the convergence, the above result is a substantial improvement. However, using a discrete auxiliary projection, we shall, in the next section, prove a similar result when u L (H 2 ) and u t L 2 (H 2 ).

9 Convergence of Sobolev equations Error estimates with reduced regularity In this section, we shall derive the error estimate whose order of convergence is compatible with the spatial regularity on the generalized solution u. Let us define ũ as the solution of the following auxiliary discrete problem (4.1a) (4.1b) A h ũ t + B h ũ = S( f u t ), ũ() = u (x). Since A h is positive definite, there exists a unique solution ũ of (4.1). Let ρ = u ũ. We can therefore rewrite (4.1) as (4.2) A h ρ t + B h ρ = (A h u t SAu t )+(B h u SBu) = I 2 + I 3. LEMMA 4.1. Let u, u t H α ( ), with 1 α 2 and t [, T ]. Then there exists a constant C such that ρ(t) 1, ρ t (t) 1 C(u,T)h α 1. Proof. For the estimation of ρ 1, it follows from the discrete inner product of (4.2) with ρ that d dt ρ 2 1 C{ ρ I 2 +I 3,ρ }. On integrating with respect to time and then following (3.5) (3.6), we obtain using Gronwall s Lemma ρ(t) 1 C(u,T)h α 1, 1 α 2. Similarly, forming an inner product between (4.2) and ρ t,wehave ρ t (t) 1 Ch α 1, 1 α 2. For error estimate in L 2 -norm, below we shall discuss the discrete Aubin- Nitsche duality argument, see, Pani et al. [12]-[13].

10 524 S. K. Chung, A. K. Pani and M. G. Park LEMMA 4.2. Suppose that u, u t H α ( ), for1 α 2and for t [, T ]. Then there is a constant C such that ρ(t), ρ t (t) Ch α. Proof. Let be a solution of the following second order problem (4.3) A h = ρ t, x h, =, x h. Because of the coercivity of A h, is the unique solution of (4.3) and it satisfies a discrete regularity (4.4) 2 C ρ t. Forming an inner product between (4.3) and ρ t, we obtain ρ t,ρ t = A h ρ t, = I 2 +I 3 B h ρ,. The estimates for ρ t (t) given below can be proved easily using the Steklov mollifier and the Bramble-Hilbert Lemma 2.1. For a complete proof, we refer Pani et al. [12] [13]. ρ t (t) 2 C(h α + ρ(t) ) 2, and hence, using discrete regularity, we obtain (4.5) ρ t (t) C(h α + ρ(t) ). Note that ρ(t) 2 C{ ρ() 2 + C{ ρ() 2 + h 2α + It now follows from Gronwall s Lemma that ρ t (s) 2 ds} (4.6) ρ(t) C{ ρ() +h α } Ch α. ρ(s) 2 ds}. Finally, we obtain the L 2 -error estimate for ρ t from (4.5) (4.6). Let θ(t) = u h (t) ũ(t), then the error e(t) = u(t) u h (t) = ρ(t) θ(t).

11 Convergence of Sobolev equations 525 THEOREM 4.1. Let u and u h be the solutions of (1.1) and (3.1), respectively. Further, let u L (H α ( )) and u t L 2 (H α ( )), 1 α 2. Then there exists a constant C such that e(t) C(u,T)h α. Proof. Since the estimate for ρ(t) is given in Lemmas 4.1 and 4.2, it is enough to estimate θ(t). From (1.1), (3.1) and (4.1), it follows that (4.7) θ t + A h θ t + B h θ = Su t ũ t = I 1 (t)+ρ t. It follows from (3.4) and Lemmas that θ(t) 1 C{ θ() 1 +h α u t L 2 (H α )}, 1 α 2. Because of the choice of u h (), wehaveθ() =. For L 2 -error estimate, form the inner product between (4.7) and θ and obtain θ t,θ + A h θ t,θ = I 1 +ρ t,θ B h θ,θ. Since A h is coercive, we obtain d dt θ(t) 2 C{ I 1 θ + ρ t θ + θ 2 1 }. It follows from the integration with respect to t and Gronwall s Lemma that θ(t) 2 C(T) ( I 1 (s) 2 + ρ t (s) 2 )ds. Hence, we obtain the required result from Lemma Fully discrete schemes In this section, we shall consider the stability and error analysis for the fully difference schemes which are based on the Euler and Crank-Nicolson methods. Let k = T denote the size of the time discretization for a given N

12 526 S. K. Chung, A. K. Pani and M. G. Park positive integer N and t n = nk, forn=,1,2,..., N. For any function φ, denote φ n = φ(t n ) and t φ n = φn φ n 1. k The backward Euler method. The backward Euler scheme is now defined by (5.1a) (5.1b) (5.1c) t U n + A h ( t U n) + B h U n = Sf n, x h, U n =, x h, U = u (x), x h. Below, we shall prove a stability result in discrete H 1 - norm not for (5.1 a) but for a modified equation (5.2) t U n + A h ( t U n) + B h U n = Sf n + THEOREM 5.1. Let U n be a solution of (5.1). constants C and k such that for < k k l F n. Then there are positive U J 1 C{ U 1 + (k Sf n 2 ) 1 2 }+(k J =1,2,,N. n F n 2 ) 1 2 }, Proof. Form a discrete L 2 -inner product between (5.2) and U n and then use Lemma 3.1 with summation by parts for the last term to have m=1 t U n 2 + t U n 2 C{ Sf n 2 + F n 2 + U n 2 1 }. Summing from n = 1toJ, we obtain (1 Ck) U J 2 1 C{ U k J 1 ( Sf n 2 + F n 2 )+k U }. Choose k in such a way that (1 Ck)>for<k k. An application discrete Gronwall s Lemma now completes the proof.

13 Convergence of Sobolev equations 527 LEMMA 5.1. Let u, u t, u tt L 2 ( ) for t [, T ]. Then there exists a constant C such that ũ tt (t) 1 C, t [, T ]. Proof. Forming an inner product between (4.1) and ũ, we obtain d dt ũ 1 C{ S( f u t ) ũ + ũ 1 } C{ S( f u t ) 2 + ũ 2 1 }. Integrating with respect to t and applying Gronwall s inequality, we have ũ 2 1 C{ ũ() S( f u t ) 2 ds}. For the estimate of ũ t, we obtain (5.3) ũ t 2 1 C{ S( f u t) 2 + ũ 2 1 } by taking innner product with ũ t and using (2.3). Differentiate (4.1) with respect to t and take an inner product with ũ tt,then as in (5.3) we obtain ũ tt 2 1 C{ S( f t u tt ) 2 + ũ ũ t 2 1 }. This completes the proof. It is here that we exploit the full potential of the Steklov mollification and the discrete projection. Let n = U n ũ n and E n = u n U n = ρ n n. THEOREM 5.2. Let u n and U n be the solution of (1.1) and (5.1), respectively. Further, let u, u t L (H α ( )) and u tt L (L 2 ( )) for 1 α 2. Then there are positive constants C and k such that the error E J = u(t J ) U J holds for < k k. E J C(u,T)(h α + k), J = 1, 2,,N

14 528 S. K. Chung, A. K. Pani and M. G. Park Proof. Since the estimate for ρ J can be found out from Lemma 4.1, it is sufficient to obtain an estimate for J. From (4.1) and (5.1), it follows that (5.4) t n + A h ( t n) + B h n = ( t u n t ũ n ) + (u n t t u n ) + (Su n t u n t ) + A h(ũ n t t ũ n ) = t ρ(t n ) + (ũ n t t ũ n ) I1 n + A h(ũ n t t ũ n ). Apply Theorem 5.1 to (5.4) to obtain Note that J 2 1 C{ 1 + k ( t ρ(t n ) 2 + ũ n t t ũ n 2 + I n ũ n t t ũ J 2 1 )}. k t ρ(t n ) n t n 1 ρ t ds Ch α u t L 1 (H α ), 1 α 2. Further, using Lemma 5.1, we have k I1 n Chα u t L (H α ), and k ũ n t t ũ n 1 Ck kũ tt 1 Ck ũ tt L (H 1 ). This completes the rest of the proof. The Crank-Nicolson scheme For a second order accurate in time, we consider the Crank-Nicolson scheme for (1.1). Let U n 1 2 = (U n + U n 1 )/2and f n 1 2 = f(t n 1).Define 2 the fully discrete scheme as

15 Convergence of Sobolev equations 529 (5.5a) (5.5b) (5.5c) t U n + A h t U n + B h (t n 1 )U n 1 2 = Sf n 1 2, 2 x h, U = u (x), x h, U n =, x h. Below, we shall prove a stability result in discrete H 1 - norm not for (5.5a) but for a modified equation (5.6) t U n + A h ( t U n) + B h U n 1 2 = Sf n l F n 1 2, (x,tn ) h (,T], THEOREM 5.3. Let U n be a solution of (5.5). constants C and k such that for < k k Then there are positive U J 1 C{ U 1 + (k Sf n ) 1 2 +(k F n ) 1 2 }. Proof. Forming the inner product between (5.6) and U n 1 2, it follows that t U n 2 + t U n 2 1 C{ Sfn F n U n }. Summing from n = 1toJ, we obtain (1 Ck) U J 2 1 C{ U k ( Sf n F n ) J 1 +k U n 2 1 }. m= Choosing k appropriately so that (1 Ck)>for<k k, we obtain the desired result using discrete Gronwall s Lemma. Below, we shall present an error analysis using discrete projection. Let u n and U n be the solutions of (1.1) and (5.5), respectively. Let E n = u n U n.

16 53 S. K. Chung, A. K. Pani and M. G. Park THEOREM 5.4. Let u, u t L (H α ( )) and u tt L (L 2 ( )) for 1 α 2. Then there are positive constants C and k such that the error E J = u(t J ) U J E J C{h α +k 2 }, J=1,2,,N. Proof. Since E J = ρ J J and we know the estimate for ρ J,wehave only to estimate for J = U J ũ J due to the triangle inequality. From (4.1) and (5.5), it follows that (5.6) t n + A h t m + B h (t m 1 ) m 1 2 = ( t u n t ũ n ) 2 + (u m 1 2 t t u n ) + (Su m 1 2 t u m 1 2 t ) + A h (ũ m 1 2 t t ũ m ) = t ρ(t m ) + (u m 1 2 t t u n ) I n 1 + A h(ũ m 1 2 t t ũ m ). It follows that as in Theorem 5.3 J 2 1 C{ k [ t ρ(t m ) 2 m=1 + u m 1 2 t t u n 2 + I n ũ m 1 2 t t ũ m 2 1 }. It completes the rest of proof. References [1] D. N. Arnold, J. Douglas, Jr. and V. Thomée, Superconvergence of a finite element approxiamtion to the solution of a Sobolev equation in a single space variable, Math. Comp. 36 (1981), [2] J. H. Bramble and S. R. Hilbert, Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math. 16 (1971), [3] P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, [4] R. E. Ewing, Numerical solution of Sobolev partial differential equations, SIAM J. Numer. Anal. 12 (1975), [5] R.E.Ewing,Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations, SIAM J. Numer. Anal. 15 (1978),

17 Convergence of Sobolev equations 531 [6] W. H. Ford, Galerkin approximations to non-linear pseudo-parabolic partial differential equations, Aequationes Math. 14 (1976), [7] W. H. Ford and T. W. Ting, Stability and converegence of difference approximations to pseudo-parabolic partial differential equations, Math. Comp. 27 (1973), [8] W. H. Ford and T. W. Ting, Uniform error estimates for difference approximations to nonlinear pseudo-parabolic partial differential equations, SIAM J. Numer. Anal. 11 (1974), [9] B. S. Jovanović, L. D. Ivanović, and E. E. Süli, Convergence of a finite-difference scheme for second-order hyperbolic equations with variable coefficients, IMA J. Numer. Anal. 7 (1987), [1] Y. Lin and T. Zhang, Finite element methods for nonlinear Sobolev equations with nonlinear boundary conditions, J.Math.Anal.Appl.165 (1992), [11] M. T. Nakao, Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension, Numer. Math. 47 (1985), [12] A. K. Pani, S. K. Chung, and R. S. Anderssen, On convergence of finite difference schemes for generalized solutions of parabolic and hyperbolic partialdifferential equations, CMA- MR1-91, Centre for Mathematics and its Applications, Australian National University, Canbera, Australia (1991). [13] A. K. Pani, S. K. Chung, and R. S. Anderssen, On convergence of finite difference schemes for generalized solutions of parabolic and hyperbolic integro-differential equations,cma- MR3-91, Centre for Mathematics and its Applications, Australian National University, Canbera, Australia (1991). S. K. Chung Department of Mathematics Education Seoul National University Seoul , Korea chung@plaza.snu.ac.kr A. K. Pani Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai 476, India M. G. Park Department of Mathematics Education Seoul National University Seoul , Korea