Convergence of A Galerkin Method for 2-D Discontinuous Euler Flows Jian-Guo Liu 1 Institute for Physical Science andtechnology and Department of Mathe

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1 Convergence of A Galerkin Method for 2-D Discontinuous Euler Flows Jian-Guo Liu 1 Institute for Physical Science andtechnology and Department of Mathematics University of Maryland College Park, MD 2742 houping Xin 2 Courant Institute, New York University and IMS and Dept. of Math., The Chinese University of Hong Kong Shatin, N.T., Hong Kong Key Words: discontinuous Galerkin Method, discontinuous ows, incompressible Euler equations, strong convergence. AMS(MOS) subject classication: 65M6, 76M2. Abstract: We prove the convergence of a discontinuous Galerkin Method approximating the 2-D incompressible Euler equations with discontinuous initial vorticity:! 2 L 2 (). Furthermore, when! 2 L 1 (), the whole sequence is shown to be strongly convergent. This is the rst convergence result in numerical approximations of this general class of discontinuous ows. Some important ows such asvortex patches belong to this class. x1. Introduction. Numerical simulation of 2-D discontinuous incompressible ows is of considerable interests in both theoretical analysis and applications. It is believed that the Lagrangian methods such as vortex methods [5,9], or the ones based on contour 1 jliu@math.umd.edu. The research was supported in part by NSF grant DMS xinz@cims.nyu.edu. The research was supported in part by the heng Ge Ru Foundation, NSF Grant DMS , and DOE Grant De-FG2-88ER

2 dynamics [1,15] give preferable treatments for such ows especially for inviscid inter-facial ows. However, the convergence of such methods poses great diculties. Past eorts concentrate on either special ows (see [4,12,13]), or require heavy machinery (such as large derivation [14]) and yield much weak convergence results [2,3,14]. However, for more complicated ows (such as a ow mixing), such front-tracking methods are impossible to implement. Thus, grid-based methods such as nite dierence and nite elements are called for. Yet, the convergence of such methods is unknown as we know of [1, 12]. Recently, a discontinuous Galerkin method was proposed in [12] which has the main advantages that the energy is conserved even for upwind type numerical uxes, and amusingly, thenumerical enstropy is non-increasing in time. The main observation of this paper is to point out that the boundness of energy and enstropy are sucient condition for strong convergence for a class of discontinuous initial data! 2 L 2 including vortex patches. In particularly, our results imply that the discontinuous Galerkin methods in [12] do converge for such ows. x2. A Discontinuous Galerkin Method. The 2-D incompressible Euler equation in vorticity stream-function formulation reads: t! +(r? r)! = r? =(;@ x ) =! with no-ow boundary condition (2:1b) = and initial condition (2:1c)!j t= =! (x) 2 L 2 () where R 2 is a simply-connected domain with a C 2 boundary, or piecewise C 2 boundary with convex corners. Assume that is equipped with a quasi-uniform triangulation T h = fkg consisting of polygons K of maximum size (diameter) h. Denote h = [K. The vorticity! is approximated by! h in a discontinuous nite element space Vh k o = nv : v j K 2 P k (K) 8K 2T h, while the stream function is approximated by h in a 2

3 continuous one W h k = V h k \ C ( h ). Here P k (K) denotes the set of all polynomials of degree at most k on the cell K. In the following, we will also use the notations that hi stands for the standard integration over the whole domain h, while an integral over a sub-domain K is denoted by hi K. The semi-discrete discontinuous Galerkin method in [12] can be described by looking for! h 2 Vh k and h 2 W h k such that X (2:2a) h@ t! h v h i K ;h! h uh rv h i K + huh n c! h v ; h i e = e2@k 8v h 2 V k h (2:2b) ;hr h r' h i h = h! h ' h i h 8' h 2 W k h : where e is a cell boundary and n is its unit out-normal. We now explain the notations used in (2.2). First the velocity eld is given by uh = r? h. Note that even through both! h and test function v h may be discontinuous across the cell boundaries, yet the velocity eld possesses continuous normal component across each cell boundary due to the denition of the nite element space W h k. Thus the numerical ux in (2.2a) can be dened as follows: Denote by v ; h (v+ h ) the value of v h from the inside (outside) of the element K, then the upwind ux is set to be (2:3) c! h = 8 >< >:! ; h if uh n! + h if uh n < : It should be remarked that for smooth ows, we could use a central ux dened by (2:3 ) c! h = 1 2! h + +!; h : However, the up-wind uxes (2.3) are preferred since the main concerns here are discontinuous ows. The rst important property of this scheme is the conservation of (no numerical dissipation in) energy (2:4) kr? h( t)k L 2 (h ) = kr? h( )k L 2 (h ) for the upwind ux (2.3), which canbe veried directly by taking v h = h in (2.2a), summing up the resulting equations over all K in the triangulation, and using (2.2b) and the 3

4 continuity of the normal velocity across the cell boundaries. Next, taking v h =! h,integrating by parts for the second term in (2.2a), and summing up for all K and estimating the terms involving cell boundaries by using (2.3) and the continuity of the normal component of the velocity eld across the cell boundaries, one can show that the enstropy decays in the sense that (2:5) k! h ( t)k L 2 (h ) k! h ( )k L 2 (h ) k! k L 2 (h ) where the initial data! h ( ) is taken as the L 2 projection of! and hence is uniformly bounded in L 2. Furthermore, taking ' h = h in (2.2b), one derives the fact that (2:6) kr h k 2 L 2 ( h ) = ;h! h h i h : Our main observation in this paper is the fact that these three simple properties, (2.4)- (2.6), yield a strong convergence. To prove and state such a result, one needs some timeregularity estimate rst. x3. Time Regularity Estimate. In this section, we will prove the following lemma about the time regularity for the approximate solutions constructed by the discontinuous Galerkin method. Lemma 1. (time regularity) It holds that (3:1) k@ t! h k L 1 ([ T ) W ;2 q ()) + k@ t h k L 1 ([ T ) Lq ()) C for any 1 q<2 where! h and h denote respectively their natural extension or restriction from h to. Proof: We rst show that (3:2) k@ t! h k L 1 ([ T ) L 2 (h )) C h 2 For any smooth function v 2 C 1 () with zero extension to the outside, let v h be the L 2 projection of v in the space of V k h. Now taking v h as a test function in (2.2a), one gets (3:3) h@ t! h vi h = h@ t! h v h i h = X K h! h uh rv h i K ; X K 4 X e2@k huh n c! h v ; h i e I 1 + I 2

5 I 1 can be estimated directly as ji 1 jk! h k L 1 X k kuhk L 2 (K) krv h k L 2 (K) Ck! h k L 1 (h )kuhk L 2 (h )krv h k L 2 (h ) Using the inverse inequality, the L 2 estimate for uh and! h in (2.4) and (2.5), and the fact that v h is the L 2 projection of v, one has (3:4) ji 1 j C h 2 k! hk L 2 (h )kuhk L 2 (h )kv h k L 2 (h ) C h 2 kvk L 2 ( h ) Next, we proceed to estimate I 2. Since uh n is continuous and c! h take same value from both sides of a cell boundary, the contribution of the two sides will gives the jump of v h. Therefore, one can obtain (3:5) ji 2 j X K X e2@k hjuh n c! h (v + h ; v; h )ji e Thanks to the quasi-uniformly regularity in the triangulation, one can show (3:6) Hence, X X K e2@k k! h k L 2 (e)kv h k L 2 (e) C h X K k! h k L 2 (K)kv h k L 2 (K) (3:7) ji 2 j C h 2 kv hk L 2 (h ) C h 2 kvk L 2 ( h ) Combining (3.4) with (3.7) shows (3:8) jh@ t! h vi h j C h 2 kvk L 2 ( h ) Which yields the desired estimate (3.2). Since the natural extension t! h to from h gives equivalent norms. Hence we have also shown (3:2 ) k@ t! h k L 1 ([ T ) L 2 ()) C h 2 Next, let I h v be the piecewise linear interpolation of v in Vh k. Then one can decompose I 1 in (3.3) as (3:9) I 1 = X K h! h uh ri h vi K + X K h! h uh r(v h ;I h v)i K I 11 + I 12 5

6 I 11 is bounded by (3:1) ji 11 j X K k! h k L 2 (K) kuhk L 2 (K) kri h vk L 1 k! h k L 2 (h )kuhk L 2 (h )kri h vk L 1 (h ) Due to the inequality one obtains from (2.4), (2.5), and (3.1) that kri h vk L 1 (h ) Ckvk W 1 1 () (3:11) ji 11 jckvk W 1 1 () One can estimate I 12 similarly. Indeed, ji 12 j X K k! h k L 2 (K) kuhk L 2 (K) kr(v h ;I h v)k L 1 (h ) Ckr(v h ;I h v)k L 1 (h ) Using the inverse inequality kr(v h ;I h v)k L 1 (h ) C h 2 kv h ;I h vk L 2 (h ) and (3:12) kv h ;I h vk L 2 (h ) kv ;I h vk L 2 (h ) which holds true since v h is the L 2 projection of v, one gets (3:13) ji 12 j C h 2 kv ;I hvk L 2 (h ) This, together with the standard estimate for the interpolation, leads to (3:14) ji 12 jckvk H 2 () It follows from (3.11) and (3.14) that we have obtained an estimate on I 1 in (3.3) independent ofh. Now we can also derive anh-independent estimate on I 2 in (3.3) as follows. Noting that I h v is continuous at the cell boundary, one can insert it into the right hand side of (3.5) to obtain (3:15) ji 2 j X K X hjuh n c! h (v h ;I h v)ji e Ckuhk L 1 X X e2@k K e2@k 6 k! h k L 2 (e) kv h ;I h vk L 2 (e)

7 As in (3.6), one has (3:16) ji 2 j C h 2 kv h ;I h vk L 2 ( h ) C h 2 kv ;I hvk L 2 ( h ) Ckvk H 2 () Collecting all the estimates (3.11), (3.14), and (3.16), we arrive at (3:17) jh@ t! h vi h jc(kvk W 1 1 () + kvk H 2()) Ckvk W 2 p () for any p>2 : To complete the proof of the rst part of (3.1), we need to estimate the dierence for the left hand term between and h. First, jh@ t! h vi nh jk@ t! h k L 2 (nh )kvk L 1 (nh ) q jn h j Note the simple estimate jn h jch 2,and kvk L 1 (nh ) Ch 2 kvk W 1 1 () since v vanishes on the we can obtain from these and (3.2') that (3:18) jh@ t! h vi nh jchkvk W 1 1 () Chkvk W 2 q () for any 1 q<2. Consequently, (3:19) k@ t! h k L 1 ([ T ) W ;2 q ()) C for any 1 q<2 This proves the rst part of (3.1). Next, we prove the second part of inequality in (3.1). For f 2 L p () with zero extension to the outside, we let solve the following problem (3:2) ; = f in = and let h 2 W k h be the nite element solution: (3:21) hr h r' h i h = hf ' h i h for any ' h 2 W k h t h 2 W k h,wehave (3:22) h@ t h fi h = hr@ t h r h i h = ;h@ t! h h i h 7

8 where we have used (3.21) and (2.2b) after taking a time derivative. Rewrite (3.22) as (3:23) h@ t h fi h = ;h@ t! h i h ;h@ t! h ( h ; )i h Let p>2 be the dual number of q: 1=p +1=q = 1, one gets (3:24) jh@ t h fi h j k@ t! h k L 1 ([ T ) W ;2 q ())kk W 2 p () + k@ t! h k L 1 ([ T ) L 2 (h ))k h ; k L 2 (h ) Ckk W 2 p () + C h 2 k h ; k L 2 (h ) Since the domain is either C 2 or piecewise C 2 with convex corners, the following the elliptic regularity and L 2 estimate is true: (3:25) kk W 2 p () Ckfk L p () and (3:26) k h ; k L 2 (h ) Ch 2 kk H 2 () Thus, ( ) imply that Or jh@ t h fi h jckfk Lp () (3:27) k@ t h k L 1 ([ T ) Lq ( h )) C for any 1 q<2 Since natural extension of h to from h gives equivalent norms, we therefore have shown (3:27) k@ t h k L 1 ([ T ) Lq () C for any 1 q<2 This gives the second part of (3.1). The proof of the lemma is completed. x4. A Uniqueness Theorem. In this section, we generalize Yudovich's uniqueness theorem to show that weak solutions with corresponding vorticity inl 1 are unique in the wider class where the vorticities are in L 2. This generalization will be used in next section to obtain a stronger convergence theorem. The precise statement is given by: 8

9 Theorem 1: Assume that! 2 L 1. Then the weak solution (4:1)! 2 L 1 ([ T) L 1 ()) \ Lip ([ T) W ;2 r ()) to (2.1) is unique in the space L 1 ([ T) L q ()) \ Lip ([ T) W ;2 r ()) where q>4=3 and 1 r<2. Proof: Suppose that the initial boundary value problem (2.1) for the 2D Euler equations has two weak solutions with same initial vorticity! 2 L 1 and the following regularities: (4:2)! 1 2 L 1 ([ T) L 1 ()) \ Lip ([ T) W ;2 r ()) and (4:3)! 2 2 L 1 ([ T) L q ()) \ Lip ([ T) W ;2 r ()) q>4=3 and1 r<2. It suces to show that! 1 =! 2. Denote by 1 and 2 the stream functions in (2.1) corresponding to! 1 and! 2 respectively. Then by the elliptic regularity, we have (4:4) 1 2 L 1 ([ T) W 2 p ()) \ Lip ([ T) L r ()) and (4:5) 2 2 L 1 ([ T) W 2 q ()) \ Lip ([ T) L r ()) We denote the correspond velocities by u 1 = r? 1 and u 2 = r? 2. Then one can rewrite the Euler equations (2.1a) in a distribution sense: (4:6) r? (@ t u 1 + u 1 ru 1 )= 2D and (4:7) r? (@ t u 2 + u 2 ru 2 )= 2D Set (4:8) u = u 1 ; u 2 = 1 ; 2 9

10 and subtract (4.7) from (4.6) to get (4:9) r? t u + u 2 ru + uru 1 )= 2D It follows from (4.4) and (4.5) that (4:1) 2 L 1 ([ T) W 2 q ()) \ Lip ([ T) L r ()) Therefore, one can take as a test function in (4.9) to obtain integration by parts, and use fact that u = r? and vanishes on the boundary, wehave (4:11) u(@ t u + u 2 ru + uru 1 ) dx = where one has used the fact that u = r? and vanishes on the boundray. Dene (4:12) E(t) = juj 2 dx Due to the regularity assumption ( ), one can show that (4:13) Using the fact (4:14) d dt E(t) =2 u ut dx uu 2 ru dx = and the equation (4.9), we have (4:15) d dt E(t) 2 juj 2 jru 1 j dx Using the classical potential estimate (4:16) kru 1 ( t)k L p Cpk! 1 ( t)k L 1 for all 1 <p<1, where C is an constant independent onp, u 1 and! 1, together with the fact that (4:18) k! 1 ( t)k L 1 k! k L 1 1

11 one shows by the Holder inequality that d (p;1)=p dt E(t) juj dx 2p=(p;1) kru 1 k L p Cp juj 2p=(p;1) dx (p;1)=p where C is independent onp. The right hand side above can be further estimated as juj 2p=(p;1) dx = On the other hand, (juj 2 ) (p;2)=(p;1) (juj 4 ) 1=(p;1) = kuk 2(p;2)=(p;1) kuk L 4 Ckuk q=(4(q;1)) W kuk (3q;4)=(4(q;1)) 1 q L 2 L 2 kuk 4=(p;1) L 4 Since kuk W 1 q (4:19) Therefor (4:2) is bounded, we thus have shown that d E(t) CpE(t)1;(5q;4)=(4p(q;1)) dt d E(t) (5q;4)=(4p(q;1)) C dt Now one can conclude that E(t). Indeed, taking an interval [ T ] with the property that CT ( 1 ), one obtains from (4.2) and E() = that 2 (4:21) E(t) ( 1 2 )4p(q;1))=(5q;4)! as p tends to innity So E(t) for t 2 [ T ]. Repeating these arguments we conclude that E(t) =for all t<t. This completes the proof. x5. Main Convergence Theorem. Finally, we are able to state and prove our main convergence theorem. Theorem 2: Let R 2 be a simply connected domain with C 2 boundary (or piecewise smooth C 2 boundary with convex corners), and equipped with a quasi-uniform triangulation. Suppose that the initial vorticity! belongs to L 2 (). Let (! h h ) 2 Vh k W h k be the approximate solutions generated by the discontinuous Galerkin method (2.2). Then there exists a convergent subsequence of (! h h ) (for which we will still use the same notation for simplicity) such that (5:1)! h *! (star weakly) in L 1 ([ T) L 2 ()) \ Lip ([ T) W ;2 q ()) 11

12 for any 1 q<2and (5:2) h! (strongly) in L 2 ([ T) H 1 ()) and the limiting functions! have the properties that (5:3)! 2 L 1 ([ T) L 2 ()) \ Lip ([ T) W ;2 q ()) and (5:4) 2 L 2 ([ T) H 1 ( )) \ Lip ([ T) L q ()) for any 1 q<2, and for any 2 C 1 (( t) ) (5:5) T (! t +!r? r) dxdt = (5:6) =! in D In other words, (! ) isaweak solution to the Euler equation (2.1) with initial data!. Furthermore, if the initial data! 2 L 1 (), then the whole sequence of (! h h ) will converge to the unique solution of the Euler equations and the limiting vorticity! is bounded in L 1 ([ T) L 1 ()). Proof: As in the Lemma 1, we extend! h and h to form h naturally. First, (2.5) and (3.1) show that there is a subsequence of! h (for which we still use the same notation) such that (5:7)! h *! (star weakly) in L 1 ([ T) L 2 ()) \ Lip ([ T) W ;2 q ()) and the limiting function! satises (5.3). Next it follows from (2.5), (2.6), and the Poincare inequality that (5:8) k h k L 1 ([ T ) H 1 ) C: This, together with (3.1), shows that there there is a subsequence of h such that (5:9) h * (star weakly) in L 1 ([ T) H 1 ()) \ Lip ([ T) L q ()) 12

13 and 2 L 1 ([ T) H 1 ()) \ Lip ([ T) L q ()) Since the distance h is of order O(h 2 )and h vanishes h,wehave (5:1) k h k L 1 (@) Ch 2 kr h k L 1 (h ) Chkr h k L 2 (h ) Ch! Hence satises (5.4). We now show that h converges strongly. First, It follows from (5.8), (3.1) and the Lions-Aubin lemma that (5:11) h! strongly in L 2 ([ T) ) which, together with (2.6) and (5.7), yields (5:12) T T kr h k 2 L 2 dt = ; h! h h i dt!; T h! i dt: To obtain the strong convergence that (5:13) r h!r strongly in L 2 ([ T) ) it suces to show that T T kr h k 2 L 2 dt! kr k 2 L 2 dt which is a direct consequence of (5:14) ; T h! i dt = T kr k 2 L 2 dt: that will be veried below. Indeed, for any 2 C 1 (( T) ), taking ' h = I h in (2.2b) yields (5:15) ; T hr h ri h i dt = T h! h I h i dt where I h is the interpolation operator in W h k. Using the strong convergence [6], (5:16) I h! strongly in L 1 ([ T) L 2 ()) and L 1 ([ T) H 1 ()) 13

14 and weak convergences (5.7) and (5.9), one shows from (5.15) that (5:17) ; T hr ri dt = T h!i dt This immediately gives (5.14) since C 1 (( T) ) is dense in L 2 ([ T) H 1 ()). As a sequence of (5.13), one concludes that (5:18) uh! u strongly in L 2 ([ T) ) where u = r? Now we show that the limit functions (! ) are indeed a weak solution to the Euler equations. To this end, one can take v h to be I h for any 2 C 1 (( t) ) in (2.2a), sum over all the cells, and integrate in time to obtain (5:19) T (h@ t! h I h i;h! h uh ri h i) dt = where we have used the facts that the upwind uxes c! h are the same for the adjacent elements, both I h and the normal component of the velocity eld are continuous across the interior cell boundaries, and u n = on the exterior cell boundaries. Since is compactly supported, t I h = I t,wecanintegrate by parttoget (5:2) T (h! h I t i + h! h uh ri h i) dt = Using the weak convergence of! h in (5.7) and strong convergence of I h, one shows that T h! h I t i dt! T h!@ t i dt Similarly, it follows from the weak convergence of! h in (5.7), strong convergence of uh in (5.18) and strong convergence of r h that T h! h uh ri h i dt! T h! u ri dt: Hence (5:21) T h!@ t i dt + T h! u ri dt =: 14

15 This gives (5.5). Finally, (5.6) follows from (2.2b) by taking ' h = I h. Thus we have proved that (!) isaweak solution to the Euler equations (2.1). In the case that the initial data! 2 L 1 (), then the Cauchy problem for the Euler equations has a solutions! 2 L 1 ([ T) ), and from Theorem 1 we know that this solution is unique in the class of (5.3) and (5.4). Therefor every convergent subsequence has the same limit. As a consequence, the whole sequence of (! h h ) converges to the unique solution. This completes the proof of the theorem. Acknowledgment The authors would like to thank Professors Qiang Du and Jun ou for their careful reading and many useful comments on the rst draft of this paper, and thank Professor John Osborn for some helpful discussions. References [1] G.R. Baker and M.J. Shelley, On the connection between thin vortex layers and vortex sheets, J. FluidMech., 215 (199), 161{194. [2] J.T. Beale, The approximation of weak solutions to the 2-D Euler equations by vortex elements, Multidimensional Hyperbolic Problems and Computations, Edited by J. Glimm and A. Majda, IMA 29 (1991), 23{37. [3] Y. Brenier and G.H. Cottet, Convergence of particle methods with random rezoning for the 2-D Euler and Navier-Stokes equations, SIAM J. Numer. Anal., 32 (1995) 18{197. [4] R. Caisch and J. Lowengrub, Convergence of the vortex method for vortex sheets, SIAM J. Numer. Anal., 26 (1989), 16{18. 15

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