Analysis of a penalty method

Transcription

1 Analysis of a penalty metho Qingshan Chen Department of Scientific Computing Floria State University Tallahassee, FL qchen3@fsu.eu Youngjoon Hong Institute for Scientific Computing an Applie Mathematics Iniana University at Bloomington Bloomington, IN hongy@iniana.eu Roger Temam Institute for Scientific Computing an Applie Mathematics Iniana University at Bloomington Bloomington, IN temam@iniana.eu October 6, 211 Abstract In this article we stuy the simplest one-imensional transport equation u t + au x = f an stuy the implementation of the bounary conition using a penalty 1

2 metho combine with a finite element P1 iscretization. We iscuss the convergence of the metho when both the penalty parameter ɛ an the mesh size h go to zero, in sequence or simultaneously. Some numerical simulations are reporte also showing the efficiency of the metho. Numerical simulations are also mae for the similar problems in space imension 2. 1 Introuction This work results from a iscussion of the thir author (RT) with Davi Gottlieb who avocate the use of a penalty metho to implement the intricate bounary conitions that appear in the stuy of the primitive equations of the atmosphere an the oceans. Concerning the issue of the bounary conitions in the primitive equations, see e.g. [16, 17, 3, 4]. See also [18] concerning the shallow water equations on the computational sie, an [14] on the theorical sie. The bounary conitions for the primitive equations are complicate. Since an early work of Oliger an Sunström ([13]) they are known to be necessarily of a nonlocal type; see also [2]. It was shown in [16, 17] that the bounary conitions shoul be implemente moe by moe in a (special) Fourier type of ecomposition in the vertical irection an it was assume in [3] an [4] that the corresponing bounary conitions are vali as well in the nonlinear case, at least for a limite perio of time, an assumption conforme by the numerical simulations from [3] an [4] The suggestion of D. Gottlieb was to practically simplify the actual implementation of these bounary conitions by using a penalization technique, a metho he an his collaborators use successfully in the context of spectral iscretizations; see e.g. [6, 7, 8, 9]. Besies some ifferent issues concerning parabolic an hyperbolic equations, the articles [6, 7] a issues very similar to these consiere in this article in the context 2

3 of pseuo-spectral methos. Furthermore, the stuy in [6, 7] assume more regularity on the sulution than we o have. See also [21] for the implementation of a penalty metho to enforce the incompressibility conition in the Navier Stokes equations (in which case the iscretization matrix happens to be too stiff). More generally the concept of penalization which has been introuce by R. Courant ([5]) is broaly use in optimization theory (see e.g. [15]) an has been also use by J. L. Lions for various applications to nonlinear parabolic equations an to evolution variational inequalities [12]. After the passing away of D. Gottlieb, the two of authors of this article wante to pursue this iea an first to apply the penalization metho on the simple one-imensional transport equation. Although the numerical implementation is straightforwar, the stuy of the convergence raise some elicate questions an we thought that we shoul fully explore them before working on more complicate equations. We recall that this equation was the favorite moel equation of Davi Gottlieb an he use to say that when one knows how to apply a new iea to this equation, one will know how to apply it to more involve equations. This article is organize as follows. In Section 2 we present the penalty formulation of the transport equation an some a priori estimates. In Section 3 we prove some existence results through a Galerkin finite element iscretization an stuy the passage to the limit when the penalization parameters ɛ an the mesh h go to zero simultaneously or in sequence. In practice, we will let the parameters ɛ an h go to zero simultaneously, inepenently or not. The sequential stuy h then ɛ is meant to slightly simplify the presentation without aing too much to the length of the article. Finally, Section 4 presents some numerical simulations in space imension one an two showing the efficiency of the metho. Although the analysis is mae only in space imension one, we thought that, for a journal evote to scientific computing, testing the metho in the more 3

4 realistic case of space imension two woul be suitable. On the way to future application of the metho to the equation of the atmosphere an oceans, there are many theoretical or computational questions that one may or shoul aress: we o not iscuss here iscretization in time; for the experiments in Section 4, we use the classical Runge-Kutta metho of the 4th orer. Other methos of space iscretization might be consiere, higher orer methos, finite volumes, finite ifferences, etc. An other types of equations might be consiere. These equations (or at least some of them) will be aresse in future works. 2 Statement of the problem We consier the 1D transport equation u t + au x = f, < x < 1, t >, (2.1) u(, t) =, t >, (2.2) u(x, ) = u (x), < x < 1, (2.3) where a is a strictly positive constant. This moel problem is known to be well-pose an many methos of iscretization have been propose. We want here to iscuss the implementation of the bounary conition using a penalty metho, in conjunction with a iscretization by P1 finite elements. We start by iscussing the implementation of the penalty metho in the continuous (non-iscretize) context. The penalty formulation For a given ɛ >, we look for u ɛ : (, ) L 2 (, 1), such that ( ) 1 (u ɛ t, v) a(u ɛ, v x ) + au ɛ (1, t)v(1) + ɛ a u ɛ (, t)v() = (f, v), v H 1 (, 1), (2.4) 4

5 u ɛ (x, ) = u (x), (2.5) where (, ) enotes the inner prouct in L 2 (, 1). A priori estimates Assuming u ɛ is sufficiently smooth, we let v = u ɛ in (2.4) an fin 1 2 t uɛ 2 a 2 (uɛ ) au ɛ (1, t) 2 + ( ) 1 ɛ a u ɛ (, t) 2 = (f, u ɛ ), where enote the L 2 norm. Applying the Cauchy Schwarz inequality to the right han sie of equation above, an after rearrangement, we obtain t uɛ 2 u ɛ 2 + au ɛ (1, t) 2 + ( ) 2 ɛ a u ɛ (, t) 2 f 2. We classically multiply both sies by e t, an obtain ( ) 2 t (e t u ɛ 2 ) + ae t u ɛ (1, t) 2 + ɛ a e t u ɛ (, t) 2 e t f 2. Provie that (2.6) < ɛ < 2 a, (2.7) both the secon an thir terms on the left han sie of (2.6) are positive, an we infer that t (e t u ɛ 2 ) e t f 2. Integrating over (,t), an using the initial conition (2.5), we obtain u ɛ (, t) 2 e t u 2 + u ɛ (, t) 2 e T u t e t s f(, s) 2 s, e T s f(, s) 2 s. (2.8)

6 We let K T e T u 2 + e T s f(, s) 2 s. (2.9) Thus K T is a constant epening on the initial ata u, the forcing f, an the final time T. The inequality (2.8) can be written as u ɛ (, t) 2 K T for a.e. t (, T ). (2.1) The interpretation of (2.1) is the following: If u L 2 (, 1), f L 2 (, T ; L 2 (, 1)), then u ɛ, if it exists an is sufficiently smooth, belongs to a set of L (, T ; L 2 (, 1)) boune inepenently of ɛ. To obtain bouns on the bounary terms we integrate (2.6) over (, T ), ( ) 2 T e T u ɛ (, T ) 2 +a e t u ɛ (1, t) 2 t+ ɛ a e t u ɛ (, t) 2 t e t f(, t) 2 t + u 2. We multiply both sies by e T, an notice that the right han sie of the above inequality becomes exactly K T. Therefore we have ( ) 2 T u ɛ (, T ) 2 +a e T t u ɛ (1, t) 2 t+ ɛ a e T t u ɛ (, t) 2 t K T. (2.11) From the above we infer that u ɛ (1, t) 2 t 1 a K T, (2.12) u ɛ (, t) 2 t ɛ 2 aɛ K T. (2.13) We interpret (2.12) an (2.13) as follows: If u L 2 (, 1), f L 2 (, T ; L 2 (, 1)), an ɛ satisfies (2.7) 6

7 an if u ɛ exists an is sufficiently smooth, then u ɛ (1, t) an u ɛ (, t) belong to a set of L 2 (, T ) boune inepenently of ɛ. Furthermore, u ɛ (, ) L 2 (,T ) = O( ɛ) as ɛ. A priori interpretation of the bounary conitions that u ɛ satisfies. Again we assume that the penalty problem (2.4) (2.5) has a sufficiently smooth solution. We take v H 1 (, 1), then (2.4) becomes (u ɛ t, v) a(u ɛ, v x ) = (f, v), (2.14) which inicates that u ɛ satisfies (2.1) in the istribution sense. More precisely, u ɛ t + au ɛ x = f, in D ((, 1) (, T )). (2.15) We multiply (2.15) by v H 1 (, 1), integrate by parts over (, 1), an we fin (u ɛ t, v) a(u ɛ, v x ) + au ɛ (1, t)v(1) au ɛ (, t)v() = (f, v). (2.16) Comparing (2.16) with (2.4), we obtain ( 1 ɛ + a)uɛ (, t)v() =. (2.17) Since v is an arbitrary function in H 1 (, 1), we conclue that u ɛ (, t) =. (2.18) Remark 2.1 Equation (2.18) shows that the penalization of the bounary conition (2.3) in (2.4) is in fact exact in this case, that is, u ɛ oes satisfy (2.3). It is likely that this is not a generic situation, an we will not take avantage of this relatively exceptional fact. 3 Existence results We now iscuss the existence of solutions for (2.4), (2.5) using a Galerkin metho base on P1-finite elements. This iscretization is also a step towar our goal an we iscuss it in etails. 7

8 3.1 Galerkin finite element iscretization Let h = 1 N. (3.1) Let ϕ n for n =, 1,, N be the usual piecewise linear hat functions (see Fig. 3.1). Let V h be the space spanne by these ϕ ϕ 1 ϕ 2 ϕ Ν 1 ϕ Ν x = x x 1 x 3 x N 1 x 2... N Figure 3.1: The piecewise linear finite elements functions, that is, { N } V h = u n ϕ n. (3.2) n= Hence, V h is the space of piecewise linear functions over [, 1], which has ϕ n as a basis. We look for an approximate solution of (2.4) in the form of u ɛ h = u ɛ n(t)ϕ n (x). (3.3) n= We replace u ɛ by u ɛ h in (2.4), take v = v h V h, an we obtain a finite imensional Galerkin approximation of (2.4): ( t uɛ h, v h ) a(u ɛ h, x v h) + au ɛ h(1, t)v h (1) + ( ) 1 ɛ a u ɛ h(, t)v h () = (f, v h ), v h V h. (3.4) 8

9 Existence of a solution to (3.4) with prescribe initial conition is elementary; we will make this explicit below. This ientity (3.4) is require to hol for each v h V h. In particular, it will hol for every ϕ m for m N. When m =, (3.4) takes the following form: n= t uɛ n(ϕ n, ϕ ) a When 1 m N 1, n= When m = N, n= We let ( ) u ɛ 1 n(ϕ n, x ϕ ) + ɛ a u ɛ = (f, ϕ ). n= t uɛ n(ϕ n, ϕ m ) a t uɛ n(ϕ n, ϕ N ) a n= n= (3.5) u ɛ n(ϕ n, x ϕ m) = (f, ϕ m ). (3.6) u ɛ n(ϕ n, x ϕ N)+au ɛ N = (f, ϕ N ). (3.7) A h = (a mn ) N m,n=, with a mn = (ϕ n, ϕ m ), B h = (b mn ) N m,n=, with b mn = (ϕ n, ϕ mx ) except that We also let U ɛ h = (u ɛ, u ɛ 1,, u ɛ N) T, b = (ϕ, ϕ x ) ( 1 aɛ 1), b NN = (ϕ N, ϕ Nx ) 1. F ɛ h = ((f, ϕ ), (f, ϕ 1 ),, (f, ϕ N )) T. With these notations we can write the system (3.5) (3.7) in a compact matrix vector form: A h t U ɛ h ab h U ɛ h = F h. (3.8) 9

10 The solution of the ODE system (3.8) can be easily obtaine. Inee, since A h is a positive efinite matrix, it is invertible; we multiply both sies of (3.8) by A 1 h an obtain We let an rewrite (3.9) as t U h ɛ aa 1 h B huh ɛ = A 1 h F h. (3.9) B h = A 1 h B h, Fh = A 1 h F h, t U ɛ h a B h U ɛ h = F h. (3.1) The system (3.1) possesses the following unique solution U ɛ h(t) = e a B h t U ɛ h() + t e a B h (t s) Fh (s) s. (3.11) Therefore, for each h, the approximate equation (3.4) has a unique solution. 3.2 Algebraic analysis of equation (3.8) We shall later on work with the iscrete equation (3.8). Hence it will be helpful to specify the equation in full etails. The 1

11 matrices A h an B h in (3.8) are given by A h = h 3 h 6 h h h 6 h h h 6 h h 2 3 h h 6 h 6 h 3, 11

12 an aɛ B h = We also write (3.8) term by term as follows: ( h t 3 uɛ + h ) ( 1 6 uɛ 1 a 2 uɛ 1 ) 2 uɛ ɛ uɛ = (f, ϕ ), ( h t 6 uɛ + 2h 3 uɛ 1 + h ) ( 1 6 uɛ 2 a 2 uɛ 1 ) 2 uɛ 2 = (f, ϕ 1 ), ( h t 6 uɛ 1 + 2h 3 uɛ 2 + h ) ( 1 6 uɛ 3 a 2 uɛ 1 1 ) 2 uɛ 3 = (f, ϕ 2 ), ( h t 6 uɛ N 2 + 2h 3 uɛ N 1 + h ) ( 1 6 uɛ N a 2 uɛ N 2 1 ) 2 uɛ N = (f, ϕ N 1 ), ( h t 6 uɛ N 1 + h ) ( 1 3 uɛ N a 2 uɛ N 1 1 ) 2 uɛ N = (f, ϕ N ). 12

13 We ivie each sie of the equations by h, an rewrite them as: ( 2 t 3 uɛ + 1 ) 3 uɛ 1 + a uɛ 1 u ɛ + 2 h ɛh uɛ = f, ( 1 t 6 uɛ uɛ ) 6 uɛ 2 + a uɛ 2 u ɛ = f 1, 2h ( 1 t 6 uɛ uɛ ) 6 uɛ 3 + a uɛ 3 u ɛ 1 = f 2, 2h ( 1 t 6 uɛ N uɛ N ) 6 uɛ N + a uɛ N uɛ N 2 2h ( 1 t 3 uɛ N ) 3 uɛ N + a uɛ N uɛ N 1 = f N. h In the above, = f N 1, (3.12) f = 2 h (f, ϕ ), f i = 1 h (f, ϕ i), for each 1 i N 1, f N = 2 h (f, ϕ N). 13

14 We let p ɛ = 2 3 uɛ uɛ 1, g ɛ = uɛ 1 u ɛ, h p ɛ 1 = 1 6 uɛ uɛ uɛ 2, g1 ɛ = uɛ 2 u ɛ, 2h p ɛ 2 = 1 6 uɛ uɛ uɛ 3, g2 ɛ = uɛ 3 u ɛ 1, 2h p ɛ N = 1 3 uɛ N uɛ N, p ɛ N 1 = 1 6 uɛ N uɛ N uɛ N, Then (3.12) can be written as t pɛ + ag ɛ + 2 ɛh uɛ = f, t pɛ n + agn ɛ = f n, for 1 n N. gn 1 ɛ = uɛ N uɛ N 2, 2h gn ɛ = uɛ N uɛ N 1. h (3.13) (3.14) We let p ɛ h be the piecewise linear interpolation (in x) of the {p ɛ n} N n= at the gri points {x n } N n=, an let gh ɛ be the piecewise linear interpolation in x of the {gn} ɛ N n= at the gri points {x n } N n=. That is, using the piecewise linear hat functions ϕ n s, p ɛ h = g ɛ h = p ɛ nϕ n, (3.15) n= gnϕ ɛ n. (3.16) n= By interpolation, (3.12) gives an equation for p ɛ h an gɛ h : t pɛ h + ag ɛ h + 2 ɛh uɛ ϕ = f h, (3.17) 14

15 where f h = f n ϕ n n= An alternate expression of u ɛ The following is an alternate expression of u ɛ which is the iscrete analogue of (A.4) in the Appenix. We will subsequently use this expression of u ɛ. By manipulating the right column of (3.13) we fin ( ) N 1 u ɛ 1 = h 2 uɛ 1 + u ɛ n uɛ N n=2 [ ] N 1 1 h 2 gɛ + (1 (n 1)h)gn ɛ + h 1 2 gɛ N. (3.18) n=1 We then rewrite (3.18) as follows: ( ) N 1 u ɛ 1 = h 2 uɛ 1 + u ɛ n uɛ N (uɛ 1 u ɛ ) h n=2 3.3 A priori estimates (1 nh)gn ɛ n= h 2 (uɛ N + u ɛ N 1 u ɛ 1 u ɛ ) h2 2 gɛ N. (3.19) We now erive a priori bouns on the u ɛ h. We take v h = 2u ɛ h in (3.4), an integrate, t uɛ h 2 L 2 (,1) + a uɛ h(1, t) 2 + ( 2 ɛ a) uɛ h(, t) 2 = 2(f, u ɛ h) t uɛ h 2 L 2 (,1) + a uɛ h(1, t) 2 + ( 2 ɛ a) uɛ h(, t) 2 f 2 L 2 (,1) + uɛ h 2 L 2 (,1). (3.2) We recall the assumption that we mae in (2.7) concerning the smallness of ɛ. Then, from the above we obtain that t uɛ h 2 L 2 (,1) f 2 L 2 (,1) + uɛ h 2 L 2 (,1). 15

16 Applying the Gronwall inequality, we obtain u ɛ h(, t) 2 L 2 (,1) uɛ h(, ) 2 L 2 (,1) et + u 2 L 2 (,1) et + t e t s f(, s) 2 L 2 (,1) s e T s f(, s) 2 L 2 (,1) s. From above we conclue that if u L 2 (, 1), f L 2 (, T ; L 2 (, 1)), then u ɛ h 2 L (,T ;L 2 (,1)) K T u 2 L 2 (,1) et + e T s f(, s) 2 L 2 (,1) s, (3.21) where K T was efine in (2.9) an is inepenent of h an ɛ. For bouns on the bounary terms, we go back to (3.2). From there, we euce that ( u ɛh ) 2L t 2(,1) e t + [a u ɛh(1, t) 2 + ( 2ɛ ] a) uɛh(, t) 2 e t f 2 L 2 (,1) e t. (3.22) We integrate (3.22) over [, T ], an we fin u ɛ h 2 L 2 (,1) e T +a u ɛ h(1, t) 2 e t t+( 2 ɛ a) u ɛ h(, t) 2 e t t u ɛ h(, ) 2 L 2 (,1) + f(, t) 2 L 2 (,1) e t t. We multiply both sies by e T, an since u ɛ h (, ) L 2 u L 2, we infer that u ɛ h 2 L 2 (,1) + a u ɛ h(1, t) 2 e T t t + ( 2 ɛ a) u ɛ h(, t) 2 e T t t u 2 L 2 (,1) et + f(, t) 2 L 2 (,1) et t t = K T by (3.21). 16

17 From the above we euce that u ɛ h(1, t) 2 t 1 a K T, (3.23) u ɛ h(, t) 2 t ɛ 2 aɛ K T. (3.24) 3.4 The case when h only In this subsection we stuy the case when ɛ is fixe an h tens to zero. Note that this case which is not really of practical interest, is just meant to prepare the analysis in Section 3.5 when both ɛ an h go to zero, inepenently or not.we obtain the following result: Theorem 3.1 For a fixe < ɛ < 2/a, an given f L (, T ; L 2 (, 1)), u L 2 (, 1), there exists a subsequence u ɛ h, an uɛ L (, T ; L 2 (, 1)) such that, as h tens to zero, u ɛ h converges to uɛ weak-star in L (, T ; L 2 (, 1)). Moreover, u ɛ satisfies (2.4) in the istribution sense in (, T ), (2.5) in H 1 (, 1), an u ɛ satisfies in the istribution sense in (, 1) (, T ). t uɛ + a x uɛ = f (3.25) Remark 3.1 Once (3.25) has been proven we infer from (3.25) an u ɛ L 2 ((, T ) (, 1)) an using a classic result [11] recalle in the Appenix (see Theorem A.1), that the traces of u ɛ at x =, 1, t =, T are efine an respectively belong to Ht 1 (, T ) an Hx 1 (, 1). Proof of Theorem 3.1 By (3.21), (3.23) an (3.24), there exists u ɛ L (, T ; L 2 (, 1)), an χ ɛ, χ ɛ 1 L 2 (, T ), an a subsequence h such that u ɛ h uɛ weak-star in L (, T ; L 2 (, 1)), u ɛ h (1, ) χɛ 1 weakly in L 2 (, T ), (3.26) u ɛ h (, ) χɛ weakly in L 2 (, T ). 17

18 We note that χ ɛ 1 is not yet ientifie as u ɛ (1, ), nor is χ ɛ ientifie with u ɛ (, ), an furthermore u ɛ (, ), u ɛ (1, ) are not even efine at this stage since we only know that u ɛ L (, T ; L 2 (, 1)). Let {h j } be a sequence such that Then We let h j = 2 j. V hj 1 V hj V hj+1 V = j=1v hj. (3.27) The space V forms a ense subspace of H 1 (, 1). Given v V, there exists j such that for every j j, v V hj, an the following hols accoring to (3.4): ( ( ) t uɛ h j, v) a(u ɛ 1 h j, x v)+auɛ h j (1, t)v(1)+ ɛ a u ɛ h j (, t)v() = (f, v). (3.28) We let ψ be an arbitrary function in C 1 ([, T ]) with ψ(t ) =. We multiply (3.28) by ψ, integrate by parts over [, T ] an obtain (u ɛ h j, vψ ) t a (u ɛ T h j, vψ) t+a u ɛ h x j (1, t)v(1)ψ t+ ( ) T ɛ a u ɛ h j (, t)v()ψ t = (f, vψ) t + (u, vψ()). Passage to the limit We now show that u ɛ satisfies (3.25) an (2.5). Let V h = {v V h v() = v(1) = }, V = j=1v h j. (3.29) It is well-known an easy to verify that V forms a ense subspace of H 1 (, 1). 18

19 Given v V, there exists j such that v V j, an for every j j, v V h j, an the following hols, accoring to (3.4): ( t uɛ h j, v) a(u ɛ h j, v) = (f, v). (3.3) x We let ψ be an arbitrary function in C 1 ([, T ]) with ψ(t ) =. We multiply (3.3) by ψ, an integrate by parts over [, T ] to obtain (u ɛ h j, vψ ) t a (u ɛ h j, vψ) t = x (f, vψ) t+(u, vψ()). Since vψ an v x ψ L1 (, T ; L 2 (, 1)), we can pass to the limit j in (3.31), using (3.26) 1, an we fin (u ɛ, vψ ) t a (u ɛ, vψ) t = x (f, vψ) t+(u, vψ()). The equation (3.32) is true for every v V. The space V is ense in H 1 (, 1), an because each of the term in (3.32) is continuous with respect to the H 1 norm, (3.32) hols for every v H 1 (, 1). If ψ also vanishes at t =, then (3.32) reuces to (u ɛ, vψ ) t a (u ɛ, This shows that u ɛ satisfies ( t uɛ, v) a(u ɛ, vψ) t = x (3.31) (3.32) (f, vψ) t, v H 1 (, 1). x v) = (f, v), v H1 (, 1), (3.34) in the istribution sense on (, T ), an thus t uɛ + a x uɛ = f, (3.35) 19 (3.33)

20 in the istribution sense on (, 1) (, T ). Now we take ψ C 1 ([, T ]), with ψ(t ) =, ψ(), v H(, 1 1); we multiply (3.35) by v, ψ, integrate by parts over Q = (, 1) [, T ], an obtain (u ɛ, vψ ) t a (u ɛ, vψ) t = x (f, vψ) t + u ɛ (t = ), v ψ(). (3.36) As explaine in the Appenix (see Theorem A.1 an in particular (A.3)), this integration by parts formula is vali,, enoting the pairing between H 1 (, 1) an H 1 (, 1). Comparing (3.36) to (3.33) we fin that Since ψ(), this shows that u ɛ (t = ) u, v ψ() =. (3.37) u ɛ (t = ) = u in H 1 (, 1). (3.38) To pass to the limit in (3.29) we nee to be able to pass to the limit in the bounary terms u ɛ h (, t) an uɛ h (1, t). For this purpose we only have the very weak convergence result (3.26) an the equation (3.4) of which we will now take avantage. We now look at (3.19). We recall that the u ɛ j are boune accoring to (3.21) so that Therefore, as h, h h u ɛ j 2 K T for j =,, N. (3.39) h 2 (uɛ N + u ɛ N 1 u ɛ 1 u ɛ ) strongly in H 1 (, T ), h 2 (3.4) 2 gɛ N strongly in H 1 (, T ), (3.41) ( ) N uɛ 1 + u ɛ n uɛ N u ɛ (x, ) x weakly in H 1 (, T ). n=2 2 (3.42)

21 It is seen from the first equation of (3.12) that, with h small, u ɛ 1 u ɛ = 2 aɛ uɛ + O(h), with O(h) unerstoo in H 1 (, T ). (3.19) that Hence we erive from u ɛ = u ɛ (x, ) x 1 aɛ uɛ h (1 nh)gn ɛ + O(h). (3.43) For the remaining summation in (3.43), we shall make use of (3.14). It is easy to see that, like u ɛ h, pɛ h (efine in (3.13) an (3.15)) is boune in L (, T ; L 2 (, 1)), an for a convergent subsequence, p ɛ h converges to the same limit as uɛ h in the istribution sense. Hence, for an appropriate subsequence (still enote as h): p ɛ h u ɛ weakly in L 2 t (, T ; L 2 x(, 1)) = L 2 x(, 1; L 2 t (, T )). (3.44) Then we euce that n= p ɛ h t t uɛ Therefore we have weakly in L 2 x(, 1; H 1 t (, T )). (3.45) h = h = h a (1 nh)gn ɛ n= (1 nh) 1 a n= = 2 aɛ 1 a ( ) p ɛ n t + agɛ n + h (1 nh)f n + 2 aɛ uɛ + 1 a n= (1 x)f(x, ) x + 1 a 21 t t (1 nh) 1 a (1 nh)p ɛ n p ɛ n t (1 x)u ɛ (x, ) x + O(h),

22 with O(h) now unerstoo in the space of Ht 1 (, T ). Hence (3.43) becomes u ɛ = u ɛ (x, ) x 1 a + 1 a t We then introuce ũ ɛ h : (1 x)f(x, ) x (1 x)u ɛ (x, ) x + 1 aɛ uɛ + O(h). (3.46) ũ ɛ h(x, t) = u ɛ j(t), (3.47) for (j 1)h < x < jh an j = 1, N.. We observe that ũ ɛ h L (,T ; L 2 (,1)) K. (3.48) Inee it is straightforwar to verify that there exists two constants C 1 an C 2 inepenent of h such that C 1 ũ ɛ h 2 L 2 uɛ h 2 L C 2 ũ ɛ h 2 2 L2. (3.49) From this we infer that the subsequence h can be chosen so that ũ ɛ h ũɛ weak-star in L (, T ; L 2 (, 1)). (3.5) We claim that ũ ɛ = u ɛ, which is proven in the next lemma. Lemma 3.1 u ɛ h ũɛ h (, 1) (, T ). in the istribution sense on Q = Proof. We let ϕ D(, 1), ψ D(, T ). Then we fin that = = = (u ɛ h ũ ɛ h)ϕ(x) x jh j=1 (j 1)h jh j=1 (j 1)h (u ɛ h ũ ɛ h)ϕ(x) x u ɛ j u ɛ j 1 (x jh)ϕ(x) x h (u ɛ j u ɛ j 1) (x jh)ϕ j, j=1 22

23 where ϕ j = 1 h jh After re-arrangement, we obtain (u ɛ h ũ ɛ h)ϕ(x) x = N 1 j=1 (j 1)h ϕ(x) x. u ɛ j [ (x jh)ϕ j (x (j + 1)h)ϕ j+1 ]. (3.51) We easily see with Taylor s formula that (x jh)ϕ j (x (j + 1)h)ϕ j+1 h 2 C(ϕ). (3.52) Therefore we have (u ɛ h ũ ɛ h)ϕ(x) x hkc(ϕ). (3.53) Then (u ɛ h ũ ɛ h)ϕ(x)ψ(t) x t hkc(ϕ) ψ L, which shows that u ɛ h ũɛ h sense on (, T ) (, 1). as h in the istribution Remark 3.2 A stronger result is prove in [19], but it necessitates an estimate on u ɛ h that we o not have for the current case. Now from (3.21) an (3.48) we infer that h u ɛ j(t) 2 K, t, (3.54) j= 23

24 an hence, by the Cauchy Schwarz inequality, h N u ɛ j(t) h(n + 1) 1 2 ( u ɛ j(t) 2 ) K 2. (3.55) j= We now go to equations (3.12) which we will refer to as (3.12),, (3.12) N. To fix ieas, we assume that N is even, i.e. N = 2M. We a equations (3.12) 2J+1, (3.12) 2J+3,, (3.12) 2M 1 for some fixe J an we fin after multiplication by 2h: t { 2h M 1 j=j j= ( 1 6 uɛ 2j uɛ 2j ) } 6 uɛ 2j+2 + M 1 a(u ɛ 2M u ɛ 2J) = 2h f 2j+1. (3.56) By (3.55), the term insie the curly brackets in the left-han sie of (3.56) is boune in L (, T ), before applying / t, an in H 1 (, T ) after applying / t. Similarly the sum in the right-han sie of (3.56) is boune in L 2 (, T ). Finally, u ɛ 2M = uɛ h (1, t) is boune in L2 (, T ) as we learn from (2.12). Hence, for J fixe, j=j u ɛ 2J is boune in H 1 (, T ). (3.57) Similarly, starting from (3.12) 2M, we fin that, for J fixe, u ɛ 2J+1 is boune in H 1 (, T ). (3.58) We can pass to the limit in (3.56) with J such that 2Jh x (or (2J + 1)h x ). We will use the following lemma, which is easy to prove using again the Taylor formula. Lemma 3.2 The function equal to αu ɛ 2j on (2jh, (2j + 1)h) an to βu ɛ 2j+1 on ((2j + 1)h, (2j + 2)h) within the interval (, 1), weakly converges to α + β u ɛ as h tens to zero. 2 24

25 Hence the right-han sie of (3.56) converges to x f(x) x. Similarly, 2h { 2h t M 1 j=j M 1 j=j ( 1 6 uɛ 2j uɛ 2j ) 6 uɛ 2j+2 u ɛ (x, t) x, x ( 1 6 uɛ 2j uɛ 2j ) } 6 uɛ 2j+2 u ɛ (x, t) x. t x Using (3.26) 2 an (3.56), we then see that u ɛ 2J certain limit enote as l(u ɛ 2J ) an such that t x u ɛ (x, t) x + χ 1 l(u ɛ 2J) = t converges to a x f(x, t) x. (3.59) Let us now return to any of the equations (3.12) j, j = 1,, N (not j = for the moment). After multiplication by 2h, for a fixe j, we have: 2h ( 1 t 6 uɛ 2j uɛ 2j ) 6 uɛ 2j+2 in H 1 (, T ) weakly, Hence 2hf j in H 1 (, T ) weakly. u ɛ j+1 u ɛ j 1 in H 1 (, T ) weakly, j = 1,, N 1. (3.6) Consiering now J =, 1, x = (h, 2h ), we infer from (3.59) (an the same for 2J + 1) that t t Hence u ɛ (x, t) x + χ 1 l(u ɛ 1) = t u ɛ (x, t) x + χ 1 l(u ɛ 2) = t f(x, t) x. (3.61) f(x, t) x. (3.62) l(u ɛ 1) = l(u ɛ 2), (3.63) 25

26 an since l(u ɛ 2) = l(u ɛ ) by (3.6), l(u ɛ ) = l(u ɛ 1) = l(u ɛ 2). However we know by (3.26) 3 that l(u ɛ ) = χ ɛ. Hence l(u ɛ ) = l(u ɛ 1) = l(u ɛ 2) = χ ɛ. (3.64) Now we consier (3.12) multiplie by h an, letting h, we fin that u ɛ ɛ weakly in H 1 (, T ). (3.65) Finally we pass to the limit in (3.46) an we fin that χ ɛ = u ɛ (x, ) x 1 a (1 x)f(x, ) x + 1 a t which in view of (A.4) means that (1 x)u ɛ (x, ) x, (3.66) χ ɛ = u ɛ (, ) in H 1 (, T ). (3.67) Remark 3.3 We infer of course from (3.61), (3.64), (3.67), an (2.1), which has been proven, that χ ɛ 1 = u ɛ (1, ), but this result is not actually neee. Recall that we have inicate in Remark 3.1 that u ɛ (, ) an u ɛ (1, ) make sense. Now we can pass to the limit in (3.29) an we obtain (u ɛ, vψ ) t a (u ɛ T, vψ) t+a u ɛ (1, t)v(1)ψ t+ x ( ) T ɛ a u ɛ (, t)v()ψ t = (f, vψ) t + (u, vψ()). (3.68) 26

27 Assuming ψ() =, we infer from (3.68) that (u ɛ, vψ ) t a (u ɛ T, vψ) t+a u ɛ (1, t)v(1)ψ t+ x ( ) T ɛ a u ɛ (, t)v()ψ t = (f, vψ) t (3.69) which means that (2.4) hols in the istribution sense in (, T ) (an (2.5) has been proven in (3.38)). The proof of Theorem 3.1 is complete. Remark 3.4 We i not explicitly prove that u ɛ solution of (2.4) (2.5) converges to u solution of (2.1) (2.3) as ɛ. This is straightforwar using the estimates proven in Section 3.3. Another remark is in orer of which we o not make use here to maintain the generality of the approach; namely the penalization is exact in this case, that is u ɛ (, t) =, t >, ɛ >. 3.5 The case when h, ɛ simultaneously In this subsection, we stuy the case when both ɛ an h ten to zero. We shall prove the following result. Theorem 3.2 Given f L (, T ; L 2 (, 1)) an u L 2 (, 1), there exists a subsequence uh ɛ an u L (, T ; L 2 (, 1)) such that, as ɛ, h, u ɛ h converges to u in the weak-star topology in L (, T ; L 2 (, 1)). In aition, u t + a u x = f (3.7) in the istribution sense in (, 1) (, T ). The function u also satisfies the initial conition (2.3) in H 1 (, 1) an the bounary conition (2.2) in H 1 (, T ). 27

28 Proof. Consier the function space V efine in (3.27). Given v V, there exists j such that for every j j, v V hj, an the following hols accoring to (3.4): ( t uɛ h j, v) a(u ɛ h j, x v) + auɛ h j (1, t)v(1) ( ) 1 + ɛ a u ɛ h j (, t)v() = (f, v). (3.71) We let ψ be an arbitrary function in C 1 ([, T ]) with ψ(t ) =. We multiply (3.71) by ψ, an integrate by parts over [, T ] to obtain (u ɛ h j, vψ ) t a (u ɛ T h j, vψ) t+a u ɛ h x j (1, t)v(1)ψ t+ ( ) T ɛ a u ɛ h j (, t)v()ψ t = (f, vψ) t + (u, vψ()). (3.72) By (3.21), (3.23) an (3.24), there exists u L (, T ; L 2 (, 1)) an a subsequence ɛ, h such that u ɛ h u weak-star in L (, T ; L 2 (, 1)), u ɛ h (1, ) χ 1 weakly in L 2 (, T ), u ɛ h (, ) strongly in L2 (, T ), (3.73) but χ 1 is not yet ientifie as u(1, ), an (3.73) 3 oes not yet imply that u(, ) =. We procee as in Section 3.4. We first look at (3.19). We recall that the u ɛ j are boune accoring to (3.21). Therefore 28

29 we have that, as ɛ, h, h h 2 (uɛ N + u ɛ N 1 u ɛ 1 u ɛ ) strongly in H 1 (, T ), h 2 (3.74) 2 gɛ N strongly in H 1 (, T ), (3.75) ( ) N uɛ 1 + u ɛ n uɛ N u(x, ) x weakly in H 1 (, T ). n=2 It is seen from the first equation of (3.12) that u ɛ 1 u ɛ = 2 aɛ uɛ + O(h) in H 1 (, T ), as h. Hence we erive from (3.19) that u ɛ = u(x, ) x 1 aɛ uɛ h (3.76) (1 nh)gn ɛ + O(h). (3.77) in H 1 (, T ) as h. For the remaining summation in (3.43), we use (3.14) to fin h = h = h a (1 nh)gn ɛ n= (1 nh) 1 a n= = 2 aɛ 1 a n= ( ) p ɛ n t + agɛ n + h (1 nh)f n + 2 aɛ uɛ + 1 a n= (1 x)f(x, ) x + 1 a t t (1 nh) 1 a (1 nh)p ɛ n p ɛ n t (1 x)u(x, ) x + O(h). 29

30 Hence (3.43) becomes u ɛ = u(x, ) x 1 a + 1 a t We now introuce ũ ɛ h : (1 x)f(x, ) x (1 x)u(x, ) x + 1 aɛ uɛ + O(h). (3.78) ũ ɛ h(x, t) = u ɛ j(t) for (j 1)h < x < jh. (3.79) We observe as before that ũ ɛ h L (,T ; L 2 (,1)) K, (3.8) with K inepenent of ɛ an h. From this we infer that the sequences ɛ, h can be chosen so that ũ ɛ h ũ weak-star in L (, T ; L 2 (, 1)). (3.81) We claim that ũ = u, that is Lemma 3.3 As ɛ, h, u ɛ h ũɛ h sense on (, 1) (, T ). The proof is similar to that for Lemma 3.1. Now from (3.21) an (3.8) we infer that h in the istribution u ɛ j(t) 2 K, t, (3.82) j= an hence, by the Cauchy Schwarz inequality, h N u ɛ j(t) h(n + 1) 1 2 ( u ɛ j(t) 2 ) K 2. (3.83) j= The next few steps in the proof lie parallel to those in the proof of Theorem 3.1, except that in the current case ɛ an h are 3 j=

31 going to zero simultaneously. Due to the boun (3.24), instea of (3.66), we obtain the following ientity as we let ɛ an h go to zero in (3.78). = u(x, ) x 1 a (1 x)f(x, ) x + 1 a t which in view of (A.4) means that (1 x)u(x, ) x, (3.84) u(, ) =. (3.85) Remark 3.5 It is easy to infer from the above that χ 1 = u(1, ) (see also Remark 3.3), but this result is actually not neee. We now can pass to the limit in (3.72) an obtain (u, vψ ) t a T (u, vψ) t + a u(1, t)v(1)ψ t + x = Assuming ψ() =, we infer from (3.86) that (u, vψ ) t a a (u, (f, vψ) t + (u, vψ()). (3.86) vψ) t + x u(1, t)v(1)ψ t = which means that, for each v H 1 (, 1), (f, vψ) t (3.87) t (u, v) a(u, v) + au(1, t)v(1) = (f, v) (3.88) x 31

32 hols in the istribution sense on (, T ). To see the initial conition that u satisfies, we multiply (3.88) by ψ C 1 ([, T ]) with ψ(t ) = an ψ(), an integrate by parts to obtain (u, vψ ) t a = (u, Comparing (3.89) with (3.86), we have T vψ) t + a u(1, t)v(1)ψ t + x (f, vψ) t + (u(x, ), vψ()). (3.89) (u(, ) u, v) =, v H 1 (, 1). (3.9) If v H 1 (, 1), then (3.88) becomes t (u, v) a(u, v) = (f, v), (3.91) x which means that u t + a u x = f (3.92) in the istribution sense in (, 1) (, T ). From (3.9) we easily conclue that u(, ) = u in H 1 (, 1). (3.93) The proof of Theorem 3.2 is complete. 4 Numerical experiments We conuct a set of numerical experiments with the penalty formulation (2.4) (2.5) of (2.1) (2.3). The purpose of these experiments is simply to emonstrate the effectiveness of the penalty metho in enforcing the bounary conitions. A rigorous analysis of the finite element schemes, or other type of numerical schemes use to iscretize the penalty formulation (2.4) (2.5), is interesting an important, but is out of the scope of the current paper, an will be pursue in future works. Here we o 32

33 not iscuss either the time iscretization or other spatial iscretization. The time iscretization in our numerical simulation is the classical Runge-Kutta metho of the 4th orer. However, in view of getting closer to practical simulations, we present numerical simulations in both imensions 1 an 2. Furthermore, in space imension 2, we exten the metho to the case of nonhomogeneous bounary conitions(unlike the theoretical alreay performe above). 4.1 Numerical experimentation space imension 1 For the experiments, we take a = 1 an f = in (2.1). The initial state u is given by { e u (x) = 24 1 e (x.1) 2.4 if x.1 <.1, (4.1) if x.1.1, This is the classical C function with compact support. By its efinition, u (x) vanishes for x.3. At x =, u , which is close to zero, but is ifferent from zero. This mismatch/incompatibility with the homogeneous bounary conition (2.2) is known to cause singularity near the temporal-spatial corner (x =, t = ); see e.g. [2, 1]. In applications, the initial an bounary ata are taken from approximations or measurements, an the incompatibility between the initial an bounary ata is almost certain to be present. For this reason, we choose not to remove the incompatibility between the initial ata (4.1) an the bounary conition (2.2). Rather, we want to see how the penalty formulation performs in this situation. The solution of the system (2.1) (2.2) is a traveling wave whose shape is given by the initial ata function (4.1). The wave travels to the right at spee a = 1. We iscretize the penalty formulation (2.4)-(2.5) by the finite element metho with P1 piecewise linear elements. Snapshots of the solution at t =.3,.6,.9 are presente in Fig The solution is 33

34 obtaine with ɛ =.5, x = 1 2 an t = 1 4. The sequence of snapshots epicts a wave traveling to the right..3 (a) u h ε (x,t=.3) x (b) u h ε (x,t=.6) x.4.3 u h ε (x,t=.9).2.1 (c) x Figure 4.1: (a) Solution at t =.3, (b) solution at t =.6, (c) solution at t =.9. Of particular interest to us is the evolution of u ɛ h (, t) in time t. In Fig. 4.2 we plot u ɛ h (, t) against t with ɛ =.5,.1,.1. It is seen that even with ɛ as large as.5, the homogeneous bounary conition is effectively enforce for t.3. For this case, the oscillations between t = an t =.3 is apparently cause by the incompatibility between the initial ata (4.1) an the homogeneous bounary conition (2.2). As ɛ gets smaller (see panels (b) an (c) of Fig. 4.2), the oscillations become suppresse an the homogeneous bounary conition is more strictly enforce. At ɛ =.1, the homogeneous bounary conition is almost exactly impose. 34

35 (a) (b) (c) u ε h (x=,t) u ε h (x=,t) u ε h (x=,t) x t x t 1 x t Figure 4.2: The evolution of u ɛ h (, ) in time; (a) ɛ =.5, (b) ɛ =.1, (c) ɛ = Numerical experimentation space imension 2 In this section, we repeat(without the theoretical analysis), on numerical experiments performe in space imension 2. The equation is a similar wave equation. We consier two examples; one for which the bounary conitions are homogeneous an f is not zero(so-calle exact solution) an one for which f= an the bounary conitions are not homogeneous(so-calle traveling wave solution). In the first example, we solve a linear hyperbolic scalar prob- 35

36 lem: u t + u x + u y = f(x, y, t), < x, y < 1, t >, u(, y, t) =, < y < 1, t >, u(x,, t) =, < x < 1, t >, u(x, y, ) = u (x), (4.2) with f(x, y, t) = 2πcos(2πx)sin(2πy)cos(2t)+ 2πsin(2πx)cos(2πy)cos(2t) 2sin(2πx)sin(2πy)sin(2t), u (x) = sin2πxsin2πy. It is easy to check that (4.2) has an exact solution u(x, y, t) = sin(2πx)sin(2πy)cos(2t), which is also unique. This example with an exact solution allows us to compute exactly the errors in the numerical solutions an to acutely evaluate the effect of the penalization metho ((2.4) applie to the 2D case). Our main motivation in performing this experiment is to fin out how effectively the penalization metho imposes the Dirichlet bounary conitions, an how it affects the accuracy of the numerical scheme that this metho is implemente with. For proof of concept, again we implement the penalization metho with the Galerkin finite element numerical scheme with the P1 element. The penalization metho stuie in this article can be implemente with higher orer FEMs, or with Galerkin spectral methos. We sample the penalization parameter on the set {.5,.1,.1}. For each of these values for ɛ, we run the simulation with x = y =.25,.125 an.625 in orer to stuy the convergence in the errors. In Fig. 4.3, for each ɛ, we plot the L 2 errors in the solutions against the number of segments in one irection. All three curves confirm the secon orer convergence rate of 36

37 1 Rate of convergece(global) ε=.5 ε=.1 ε=.1 L 2 Error The number of segment Figure 4.3: The convergence of the L 2 errors in the solutions of (4.2) for ɛ =.5,.1,.1. the numerical scheme in use. In this example, larger values for ɛ prouce smaller errors, which seem o, but actually is in line with a conclusion rawn in [2], namely the optimal penalization parameter, in terms of global errors, is usually obtaine by trial an error. In Fig. 4.4, for each value of ɛ, we plot the L 2 errors on the bounary (x= or y=, where Dirichlet bounary conitions are impose by the original problem (4.2)) against the numbers of segments in the sie of the omain. All three curves show that the convergence rate for errors on the bounary is also the secon orer. In aition, they show that smaller values of ɛ result in smaller errors on the bounary, which means stricter enforcement of the Dirichlet bounary conitions. Fig. 4.5 reveals more etails about how ifferent values of the parameter ɛ can impact the evolution of the errors on the boun- 37

38 1 Rate of convergece on bounary x= an y= ε=.5 ε=.1 ε= L 2 Error The number of segment Figure 4.4: The convergence of the L 2 errors in the solutions of (4.2) on the bounary for ɛ =.5,.1,.1. ary. For a fixe gri resolution with x = y =.125, taking ɛ =.1 leas to a consistent an strict enforcement of the Dirichlet bounary conitions on x = or y =. Taking ɛ =.1, the exact bounary conitions appear relaxe an the error curve is oscillatory in time. Larger values for ɛ lea to much larger eviations from the exact bounary conitions an more oscillations. As the secon example, we consier the following problem. u t + u x + u y =, < x, y < 1, t > u(, y, t) = sin(2πy 4πt) < y < 1, t > (4.3) u(x,, t) = sin(2πx 4πt) < x < 1, t > u(x, y, ) = sin(2πx + 2πy) < x, y < 1 38

39 .35.3 L 2 error on bounary x= or y= (Δx=.125, Δy=.125) ε=.5 ε=.1 ε= L 2 Error t Figure 4.5: The evolution of the L 2 errors in the solutions of (4.2) on the bounary for ɛ =.5,.1,.1. It is easy to check that (4.3) has an exact traveling wave solution u(x, y, t) = sin(2πx + 2πy 4πt), which is also unique. The bounary conitions in (4.3) are non-homogeneous, unlike those in (2.1) (2.3) an those in (4.2). However, the extension of the penalty metho (2.4) to the case of non-homogeneous bounary conitions is straightforwar. In- 39

40 1 Rate of convergece(global) ε=.5 ε=.1 ε=.1 L 2 Error The number of segment Figure 4.6: The convergence of the L 2 errors in the solution of (4.3) for ɛ =.5,.1,.1. ee, the penalty formulation of (4.3) reas t (uɛ, v) + u ɛ x=1 v y u ɛ x= v y x=1 x= + u ɛ y=1 v x u ɛ y= v x (u ɛ, v x ) (u ɛ, v y ) y=1 y= + 1 (u ɛ u) v x + 1 (u ɛ u) v y = ɛ y= y= ɛ x= x= (4.4) The experiment proceure an results for (4.3) are similar to those for the example (4.2). Hence, we will just briefly state the results an point out challenges that this example poses. Fig. 4.6 shows that with the penalization technique (4.4), the numerical scheme with P1 finite elements retains approximately 4

41 1 Rate of convergece on bounary x= an y= ε=.5 ε=.1 ε=.1 L 2 Error The number of segment Figure 4.7: The convergence of the L 2 errors in the solutions of (4.3) on the bounary for ɛ =.5,.1,.1. the secon orer convergence rate. Fig. 4.7 emonstrates a secon orer convergence rate on the bounary for each of the values of ɛ. Like the previous example, this one also emonstrates, through Figures 4.6 an 4.7, that smaller values of ɛ result in smaller errors on the bounary, an thus stricter imposition of the Dirichlet bounary conitions, but that, in terms of errors in the whole omain, using smaller values of ɛ may actually lea to larger errors. The optimal parameter, in terms of global errors, nee to be foun by trial an error. Fig. 4.8 confirms that larger values for ɛ lea larger errors or the bounary an more oscillations in the error curves. 41

42 .12 L 2 Error on bounary x= or y=, Δx,Δy=.125 ε=.5 ε=.1 ε= L 2 Error t Figure 4.8: The evolution of the L 2 errors in the solutions of (4.3) on the bounary for ɛ =.5,.1,.1. A Auxiliary results A trace theorem Let X = { u L 2 (, T ; L 2 (, 1)) u t + au x L 2 (, T ; L 2 (, 1)) }. (A.1) It is easy to check that the space X, enowe with the norm u 2 X = u 2 L 2 t L2 x + u t + au x 2 L 2 t L2 x, (A.2) is a Hilbert space. Theorem A.1 If u X an a, then u is (a.e. equal to) a continuous function from [, T ] into Hx 1 (, 1) an from [, 1] into Ht 1 (, T ). In particular the traces of u at x =, 1 an 42

43 t =, T exist, belong to the corresponing H 1 spaces an the mapping from X into these spaces are linear continuous. Finally, for u X, v H 1 (, 1), the natural integration by parts formula makes sense an is vali (u t + au x )vψ x t = u(, T ), v( ) ψ(t ) u(, ), v( ) ψ() u(vψ t + av x ψ) x t, (A.3) where, is the pairing between H 1 (, 1) an H 1 (, 1). Remark A.1 The first part of this Theorem is ue to [11]; we recall its proof to establish the secon part. Proof. Since L 2 t (, T ; L 2 x(, 1)) = L 2 x(, 1; L 2 t (, T )), by Fubini s theorem, we see that if u X, then u L 2 x(, 1; L 2 t (, T )), u t L 2 x(, 1; H 1 t (, T )). Then, since u t + au x L 2 x(, 1; L 2 t (, T )), we see that from which we conclue that u x L 2 x(, 1; H 1 t (, T )), u C x ([, 1]; H 1 t (, T )). In particular, the traces of u at x = an x = 1 exist an belong to Ht 1 (, T ). We show in a totally similar manner that u C t ([, T ]; Hx 1 (, 1)) an hence the traces of u at t = an T are well efine. The continuity of the trace operators is easy. 43

44 Finally, to prove the integration by parts formula (A.3), we first observe that every term in (A.3) makes sense an that the formula makes sense for u smooth (an v, ψ as state). For an arbitrary u X, using the ensity Lemma A.2 below, we approximate u in X by a sequence of functions u j X C (Q), Q = (, 1) (, T ). For each j, (A.3) is vali with u replace by u j an we can easily pass to the limit j, using the trace properties alreay proven. The next lemma is aime at constructing the traces of u X at x =, 1. Lemma A.1 For each u X, the following hols: u(, t) = + 1 a u(x, ) x 1 a t (1 x)(u t + au x )(x, ) x (1 x)u(x, ) x in H 1 (, T ), (A.4) u(1, t) = u(x, t) x + 1 a 1 a t x(u t + au)(x, ) x xu(x, ) x in H 1 (, T ). (A.5) Remark A.2 We note that (A.4) an (A.5) make sense in H 1 t (, T ) for every u X since u an u t +au x L 2 (, 1; L 2 t (, T )). Proof. The equations (A.4) an (A.5) are elementary for u smooth. The proof consists in showing that X C (Q) is ense in X, where Q = (, 1) (, T ). Then for u X we approximate u by a sequence of functions u j X C (Q) for which (A.4) an (A.5) are true an we pass to the limit j. The ensity result use in Theorem A.1 an Lemma A.1 is the object of the next lemma. Lemma A.2 For Q = (, 1) (, T ), X C (Q) is ense in X. 44

45 Proof. Let there be given u X that we want to approximate by smooth functions. We consier four sets covering Q, esignate as B 1, B 2, B 3 an B 4, each set containing one an only one of the corners of Q. We then consier a partition of unity ϕ 1, ϕ 2, ϕ 3, ϕ 4 suborinate to this covering: 4 ϕ i = 1. 1 (A.6) It is easy to see that, for each i, ϕ i u X since ( t + a ) ( ) u (ϕ i u) = ϕ i x t + a u + u x ( ϕi t + a ϕ i x Hence it suffices to approximate each ϕ i u by smooth functions. We consier for simplicity the set B 1 containing the corner (, ). Following [1] (see also [21]), we consier a smooth function ρ, such that ρ(x, t) x t = 1, which is compactly supporte in a cone containe in the quarangle x <, t <. We then consier the function ϕ 1 u extene by zero to R 2 enote as ϕ 1 u an its regularization ρ η ϕ 1 u. As η, ρ η ϕ 1 u converges to ϕ 1 u in L 2 (R 2 ), an ρ η ϕ 1 u Q converges to ϕ 1 u in L 2 (Q). Now, ( t + a ) ϕ 1 u = g + µ, x where g is the extension by zero of ( / t+a / x)(ϕ 1 u) beyon Q, an µ is a measure supporte by x =, t =. By regularization an the choice of ρ, ρ η µ has its support outsie Q ( in x <, t < ), so that ( t + a ) ρ η ϕ 1 u Q = ρ η g Q ; x hence as η, ( / t+a / x)ρ η ϕ 1 u Q converges to ( / t+ a / x)(ϕ 1 u) in L 2 (Q). Finally, ρ η ϕ 1 u Q converges to ϕ 1 u in X an the ensity follows. 45 ).

46 Acknowlegment This work was in part supporte by the National Science Founation grant DMS-9644, the Research Fun of Iniana University an the Department of Energy grant number DE-SC2624. The authors thank the refrees an the eitors for their careful reaing of the manuscript an for their very useful remarks. References [1] Qingshan Chen, Zhen Qin, an Roger Temam, Treatment of incompatible initial an bounary ata for parabolic equations in higher imensions, Math. Comp., to appear. [2], Numerical resolution near t = of nonlinear evolution equations in the presence of corner singularities in space imension 1, Comm. Comp. Phys. 9 (211), no. 3, [3] Qingshan Chen, Ming-Cheng Shiue, an Roger Temam, The barotropic moe for the primitive equations, Journal of Scientific Computing 45 (21), [4] Qingshan Chen, Ming-Cheng Shiue, Roger Temam, an Joseph Tribbia, Numerical approximation of the invisci 3 primitive equations in a limite omain, M2AN Math. Moel. Numer. Anal., to appear. [5] R. Courant, Variational methos for the solution of problems of equilibrium an vibrations, Bull. Amer. Math. Soc. 49 (1943), MR 7838 (4,2e) [6] D. Funaro an D. Gottlieb, A new metho of imposing bounary conitions in pseuospectral approximations of hyperbolic equations, Math. Comp. 51 (1988),

47 [7], Convergence results for pseuospectral approximations of hyperbolic systems by a penalty-type bounary treatment, Math. Comp. 57 (1991), [8] D. Gottlieb an J. S. Hesthaven, Spectral methos for hyperbolic problems, J. Comput. Appl. Math. 128 (21), no. 1-2, , Numerical analysis 2, Vol. VII, Partial ifferential equations. MR MR (21m:65138) [9] JS Hesthaven, Spectral penalty methos* 1, Applie Numerical Mathematics 33 (2), no. 1-4, [1] Lars Hörmaner, L 2 estimates an existence theorems for the operator, Acta Math. 113 (1965), MR (31 #3691) [11] P. D. Lax an R. S. Phillips, Local bounary conitions for issipative symmetric linear ifferential operators, Comm. Pure Appl. Math. 13 (196), MR (22 #9718) [12] J.-L. Lions, Quelques méthoes e résolution es problèmes aux limites non linéaires, Duno, MR 41 #4326 [13] J. Oliger an A. Sunström, Theoretical an practical aspects of some initial bounary value problems in flui ynamics, SIAM J. Appl. Math. 35 (1978), no. 3, MR 58 #25389 [14] Maalina Petcu an Roger Temam, The one-imensional shallow water equations with transparent bounary conitions, Mathematical Methos in the Applie Sciences (211), n/a n/a. [15] E. Polak, Computational methos in optimization. A unifie approach, Mathematics in Science an Engineering, Vol. 77, Acaemic Press, New York, MR (43 #8222) 47

48 [16] A. Rousseau, R. Temam, an J. Tribbia, The 3D primitive equations in the absence of viscosity: bounary conitions an well-poseness in the linearize case, J. Math. Pures Appl. (9) 89 (28), no. 3, MR MR [17] Antoine Rousseau, Roger M. Temam, an Joseph J. Tribbia, Bounary value problems for the invisci primitive equations in limite omains, Hanbook of numerical analysis. Vol. XIV. Special volume: computational methos for the atmosphere an the oceans, Hanb. Numer. Anal., vol. 14, Elsevier/North-Hollan, Amsteram, 29, pp MR (21c:869) [18] M.C. Shiue, J. Laminie, R. Temam, an J. Tribbia, Bounary value problems for the shallow water equations with topography, Journal of Geophysical Research 116 (211), no. C2, C215. [19] R. Temam, Navier-Stokes equations, AMS Chelsea Publishing, Provience, RI, 21, Theory an numerical analysis, Reprint of the 1984 eition. MR MR (22j:761) [2] R. Temam an J. Tribbia, Open bounary conitions for the primitive an Boussinesq equations, J. Atmospheric Sci. 6 (23), no. 21, MR [21] Roger Temam, Sur la stabilité et la convergence e la méthoe es pas fractionnaires, Ann. Mat. Pura Appl. (4) 79 (1968), MR (39 #3175) 48