2 GUANGYU LI AND FABIO A. MILNER The coefficient a will be assume to be positive, boune, boune away from zero, an inepenent of t; c will be assume con

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1 A MIXED FINITE ELEMENT METHOD FOR A THIRD ORDER PARTIAL DIFFERENTIAL EQUATION G. Li 1 an F. A. Milner 2 A mixe finite element metho is escribe for a thir orer partial ifferential equation. The metho can be applie, for example, to the solution of a linearize Benjamin-Bona-Mahony equation for uniirectional propagation of long waves in some nonlinear ispersive systems. A time-continuous numerical metho is escribe an error estimates for its solution are emonstrate. A superconvergence estimate is prove for the auxiliary variable use in the mixe formulation. This paper is eicate to Jim Douglas, Jr. on the occasion of his 70th birthay I. Introuction. Recently the authors [2] introuce for the first time a splitting finite element metho for a thir orer Sobolev partial ifferential equation similar to the one ue to Benjamin, Bona, an Mahony (BBM) [1]. The error estimates prove in that paper are suboptimal. They are of optimal rate for the solution, but since they combine L 2 - an H 1 -errors in one boun, they are suboptimal for the erivative of the solution because of a limitation in the argument of that paper. In this paper we shall consier a more general problem resulting from the linearization of the BBM equation, an we shall how this thir orer equation can be split into a pair, one being a first orer an another a secon orer one, an how to approximate the solution of the system by a resulting mixe finite element metho in such away that optimal orer error estimates can be proven. Also, in some cases, we can show that the approximation of the auxiliary variable introuce in the mixe metho is superclose to its projection into the approximation space. We finish this section by escribing the problem we shall consier. In the next section we efine our semi-iscrete finite element approximation an in Section 3 we prove its existence an uniqueness. Finally, in Section 4 we prove the convergence of the semiiscrete approximation. We consier the numerical solution of the 1-perioic, linear, initial value problem for the following Sobolev equation (au xt ) x + cu t = (ffu x ) x + fiu x + flu; x 2 R; t 2 (0;T); (1.1) u(x; 0) = u 0 (x); x 2 R; (1.2) u(x +1;t)=u(x; t); (x; t) 2 R (0;T) (1.3) 1 Department of Mathematics, Nankai University, Tianjin, P.R.C. 2 Department of Mathematics, Purue University, West Lafayette, IN

2 2 GUANGYU LI AND FABIO A. MILNER The coefficient a will be assume to be positive, boune, boune away from zero, an inepenent of t; c will be assume constant. Then, without loss of generality, (1.1) can be rewritten as follows. (au xt ) x + u t = (ffu x ) x + fiu x + flu; x 2 R; t 2 (0;T); (1.4) The coefficients ff, fi, an fl will be C 1 functions of x an t, 1-perioic with respect to x. Moreover, ff will be assume to be boune above an below by positive constants, while fi, fl;ff t are assume to be boune. The Benjamin-Bona-Mahony equation, u t + u x + uu x u xxt = 0 correspons to the choices a = 1;ff = fl = 0 an fi = fi(u) = (1 + u) in (1.4), so that it cannot irectly use the theory evelope herein. However, the arguments presente in this work can be extene from the linear moel treate here to the semilinear case of BBM, work which will be carrie out elsewhere. II. The Mixe Finite Element Approximation. Let T > 0 an set I = (0; 1) an J = (0;T). We introuce now an auxiliary variable which will allow us to split (1.4) into two lower orer equations. Let v = au x Then (1.4) can be replace by the following system. where we have introuce the notation v xt + u t = ffv x + fiv + flu; x 2 R; t 2 J; (2.1) u x 1 v =0; x 2 R; (2.2) a ff = ff a ; fi = fi ff x Let us enote now by H r (0; 1) the stanar Sobolev space of square integrable functions with r square integrable erivatives on the unit interval, enowe with its stanar norm k k r, an let us introuce the Hilbert spaces U = L 2 p(0; 1) = ff 2 L 2 (0; 1) f(x +1)=f(x);x a.e. in Rg; V = H 1 p(0; 1) = ff 2 H 1 (0; 1) f(x +1)=f(x);x2 Rg Then, multiplying the equations (2.1) an (2.2), respectively by functions ' 2 U an ψ 2 V, an integrating with respect to x over I, we arrive at the split weak formulation we shall consier. (u; v) J! U V is the solution of the system (v xt ;')+(u t ;')=(ffv x ;')+(fiv;')+(flu;'); ' 2 L 2 p(i); (2.3) (u; ψ x )+( 1 v; ψ) =0; a ψ 2 H p(i); 1 (2.4) u( ; 0) = u 0 ; (2.5) v( ; 0) = au 0 0; (2.6)

3 A MIXED FINITE ELEMENT METHOD FOR A THIRD ORDER... 3 Now we choose a non-negative integer k, an we efine the finite element spaces we shall use for the approximation. For any non-negative integer n, let P n enote the space of polynomials of egree no greater than n. Consier now a family of partitions of I, T h, consisting of intervals of size proportional to h, an let U h = ff I! I;fj e 2 P k ;e2t h g; V h = ff 2 C 0 (0; 1); 1-perioic in x; fj e 2 P k+1 ;e2t h g Remark 2.1 Derivatives of functions in V h are in U h. Conversely, functions of U h are erivatives of functions in V h if, an only if the former have mean value zero. The semiiscrete (time-continuous) mixe finite element solution of (1.1) we shall consier is efine as (u h ;v h )J! U h V h, ((v h ) xt ;')+((u h ) t ;')=(ff (v h ) x ;')+(fiv h ;')+(flu h ;'); ' 2 U h ; (2.7) (u h ;ψ x )+( 1 a v h;ψ)=0; ψ 2 V h ; (2.8) u h ( ; 0) = P h u 0 ; (2.9) v h ( ; 0) = ß h (au 0 0); (2.10) where P h enotes the L 2 -projection of U onto U h an ß h enotes a projection of V onto V h. III. Existence an Uniqueness. We shall prove now that the system (2.7)-(2.10) amits a unique solution. Theorem 3.1. For any initial values u 0 2 Hp(I), 2 the system (2.7)-(2.10) is uniquely solvable. Proof We first show uniqueness. Assume that we have two solutions of (2.7)-(2.8) which satisfy (2.9)-(2.10), (u 1 ;v 1 ) an (u 2 ;v 2 ), say. Let y = u 1 u 2, z = v 1 v 2. Then, we have from (2.7)-(2.10), (z xt ;')+(y t ;')=(ffz x ;')+(fiz;')+(fly;'); ' 2 U h ; (3.1) (y; ψ x )+( 1 a z; ψ) =0; ψ 2 V h; (3.2) y( ; 0) = 0; (3.3) z( ; 0) = 0; (3.4) Let us take first ' = y in (3.1) an ψ = z t in (3.2). Then, aing the resulting relations it follows that t ky( ;t)k kz( ;t)k 2 0 Kfi (ffzx + fiz + fly;y)fi (3.5) Note that, since ψ 2 V h an a o not epen on t, we can ifferentiate (3.2) insie the integral to obtain the following relation (y t ;ψ x )+( 1 a z t;ψ)=0; ψ 2 V h (3.6)

4 4 GUANGYU LI AND FABIO A. MILNER Next choose ' = z x in (3.1) an ψ = z in (3.6). Then, multiplying the secon relation by c an aing this to the first, we can see that t t kz( ;t)k2 1 Kfikff 1=2 z x k 2 0 +(fiz + fly;z x )fi (3.7) Aing (3.5) an (3.7), it is immeiate to see that ky( ;t)k kz( ;t)k 2 1 K ky( ;t)k kz( ;t)k 2 1 ; (3.8) where K enotes a generic constant which may have ifferent values in ifferent occurrences, an which will always be inepenent of u, v, an h, but may epen on the coefficients of the ifferential equation (1.1). Gronwall's lemma together with (3.3) an (3.4) now yiel y =0; z =0; an we have establishe the uniqueness of solutions. As for the existence, we first observe that (2.7)-(2.8) is a linear system of first-orer ODEs for the coefficients of (u h ;v h ) in a fixe basis of U h V h. Hence, the general existence theorem for initial value problems for systems of ODEs trivially gives local-intime existence. Then, it suffices to observe that the same argument that le to (3.8) from (3.1) an (3.2) will lea, starting from (2.7) an (2.8), to the following relation t ku h ( ;t)k kv h ( ;t)k 2 1 C ku h ( ;t)k kv h ( ;t)k 2 1 Thus we obtain the following a priori estimate ku h ( ;t)k 0 + kv h ( ;t)k 1 C ku 0 k 2 ; sup 0tT which gives global existence. IV. Convergence an Error Estimation. For r a non-negative real an any function f = f(x; t), we introuce now the following notation kfk L 1 (H r ) = kfk 2 L 2 (H r ) = Z T sup fkf( ;t)k r g; 0<t<T 0 kf( ;t)k 2 r t Next we erive error estimates for the solution of (3.1)-(3.4). Theorem 4.1. If u 2 H k+3 (I) an u t 2 H k+3 (I), then the following error estimate hols ku u h k L 1 (L 2 ) + kv v h k L 1 (H 1 ) Kh k+1 kukl 1 (H k+3 ) + ku t k L 1 (H )Λ k+3

5 A MIXED FINITE ELEMENT METHOD FOR A THIRD ORDER... 5 Proof Let us introuce the errors " = v v h, ff = u u h. Then, subtracting (2.7)(2.8) from (2.3)(2.4), we arrive at the following error equations (" xt ;')+(ff t ;')=(ff" x ;')+(fi";')+(flff;'); ' 2 U h ; (4.1) (ff;ψ x )=( 1 a "; ψ); ψ 2 V h; (4.2) ff( ; 0) = u 0 P h u 0 (4.3) "( ; 0) = u 0 0 ß h u 0 0 (4.4) Let (eu;ev) = (P h u; ß h v) be the elliptic" projection of (u; v) into U h V h efine by the following equations (eu u; ') =0; ' 2 U h ; (4.5) (ev x v x ;ψ x )+ 1 a (ev v);ψ =0; ψ 2 V h (4.6) Then, we have the following stanar error estimates ku euk 0 Ch k+1 kuk k+1 ; kv evk 0 + hk(v ev) x k 0 Ch k+2 kvk k+2 ; k(v ev) t k 0 + hk(v ev) xt k 0 Ch k+2 kv t k k+2 (4.7) We now moify the error equations (4.1)-(4.2) to have a more convenient form. Let " = v ev, ρ =ev v h, ff = u eu, = eu u h, so that " = " + ρ an ff = ff +. Then, accoring to (4.1) an (4.5), (ρ xt ;')+( t ;')=(ff " x ffρ x + " xt + fi " + fiρ + flff + fl ;'); ' 2 U h (4.8) Also, since erivatives of functions in V h lie in U h,we can see from (4.2) an (4.5), that ( ; ψ x )+( 1 a ρ; ψ)=( 1 a "; ψ); ψ 2 V h (4.9) Let now ' = in (4.8) an ψ = ρ t in (4.9). Aing the resulting relations it follows that t + kρk 2 0 Kfi ( 1 a "; ρ t)+(ff " x ffρ x + " xt + fi" + fiρ + flff + fl ; )fi (4.10) Differentiation of (4.6) an (4.9) with respect to t yiel the following relations (" xt ;ψ x )+( 1 a " t;ψ)=0; ψ 2 V h ; (4.11) ( t ;ψ x )=( 1 a ρ t;ψ)+( 1 a " t;ψ); ψ 2 V h (4.12) Next, we let ' = ρ x in (4.8) an ψ = ρ in (4.12). Then, using (4.11) with ψ = ρ, we are le to the following relation t kρk2 1 Kfi (ff "x + ffρ x fi" fiρ flff fl ;ρ x )fi (4.13)

6 6 GUANGYU LI AND FABIO A. MILNER Now, let ' = t in (4.8) an ψ = ρ t in (4.12). It follows that k t k 2 0 +fl 1 p a ρ t fl 2 0 = ( 1 a " t;ρ t )+(ff " x ffρ x + " xt + fi" + fiρ + flff + fl ; t ) (4.14) Aing (4.10), (4.13) an (4.14) we see that + kρk k t k kρ t k 2 0 C t + kρk kffk k"k k" t k 2 1 Using Gronwall's lemma we see that k k L 1 (L 2 ) + kρk L 1 (H 1 ) + k t k L 2 (L 2 ) + kρ t k L 2 (L 2 ) K k ( ; 0)k kρ( ; 0)k kffk k"k k" t k 2 1 ; an using (4.3)-(4.7) we finally see that k k L 1 (L 2 ) + kρk L 1 (H 1 ) + k t k L 2 (L 2 ) + kρ t k L 2 (L 2 ) Ch k+1 kukl 1 (H k+3 ) + ku t k L 1 (H k+3 )Λ The theorem follows immeiately from this relation by writing " = " + ρ an ff = ff +, an using once more (4.7). Remark 4.1 Assume that fl 0. If we also have ff constant (that is, ff proportional to a), then we can take ψ = ρ in (4.6) to obtain the following boun Then (4.13) an (4.15) lea to the following boun j(ff " x ;ρ x )jk k"k kρk 2 0Λ (4.15) t kρk2 1 K kρk k"k0λ 2 Using Gronwall's lemma an (4.7) we are le to the following superconvergence estimate. kρk L 1 (H 1 ) Kk"k L 1 (L 2 ) Kh r kvk L 1 (H r ); 1 r k +2; (4.16) inepenently of the choice of the spaces U an U h. In particular, we obtain (using also (4.7)) the following L 2 error estimate for v h, which is of optimal orer an regularity. kv v h k L 1 (L 2 ) k"k L 1 (L 2 ) + kρk L 1 (L 2 ) Kh r kvk L 1 (H r ); 1 r k +2 If we assume that u 0 has mean value zero, then u( ;t) can be chosen with mean value zero (since a conition must be specifie in orer have uniqueness) an we can replace U h by U h =R. Then we have Vh 0 = U h (in the sense that any R function in U h is the erivative x of exactly one function of V h ). We can then let ψ(x) = 0 t(ο) ο 2 V h an use ψ x = t in (4.11). This leas to the following relation (" xt ; t )=(" xt ;ψ x )=( 1 a " t;ψ); from which, using Poincaré's lemma, we see that j(" xt ; t )jkk" t k 0 kψk 0 Kk" t k 0 kψ x k 0 k" t k 0 k t k 0 (4.17)

7 Similarly, A MIXED FINITE ELEMENT METHOD FOR A THIRD ORDER... 7 j(" xt ; )j Kk" t k 0 k k 0 (4.18) Analogously, we can take ψ(x) = R x 0 (ο) ο 2 V h an use ψ x = in (4.6) to erive, using again Poincaré's lemma, the boun Similarly, j(ff " x ; )j Kk"k 0 k k 0 (4.19) j(ff " x ; t )jkk"k 0 k t k 0 (4.20) Finally, using ψ = ρ in (4.9) an in (4.12), we immeiately see that t j(ffρ x ; )j Kkρk 0 k"k 0 ; j(ffρ x ; t )jkkρk 0 k" t k 0 (4.21) Then, combining (4.10) an (4.14) with (4.16)-(4.21) we arrive at the following estimate. + kρk k t k kρ t k 2 0 K Gronwall's lemma an (4.7) now yiel the following boun + kffk k"k k" t k 2 0 ku u h k L 1 (L 2 ) Kh k+1 kukl 1 (H k+2 ) + ku t k L 1 (H k+2 )Λ This is an error boun of optimal rate an, even though it is still suboptimal in regularity, it improves the corresponing one erive in [2]. This remark can be summarize as follows. Theorem 4.2. If u 2 H k+2 (I) an u t 2 H k+2 (I), an assume that fl = 0 an ff is proportional to a. Then the following error estimate hols ku u h k L 1 (L 2 ) Kh k+1 kukl 1 (H k+2 ) + ku t k L 1 (H )Λ k+2 If u 2 H k+3 (I) an u t 2 H k+3 (I), then we also have the following kv v h k L 1 (L 2 ) + k~v v h k L 1 (H 1 ) Kh k+2 kuk L 1 (H k+3 ); which are, respectively, an optimal an a superconvergence estimates. REFERENCES 1. T. B. Benjamin, J. L. Bona an J. J. Mahony, Moel equations for long waves in nonlinear ispersive systems, Philos. Trans. Roy. Soc. Lonon Ser. A 272 (1972), Q. Duan, G. Li an F. A. Milner, A first-secon orer splitting metho for a thir orer partial ifferential equation, Num. Meth. for Partial Diff. Eqs. 14 (1998),