Jian-Guo Liu 1 and Chi-Wang Shu 2

Transcription

1 Journal of Computational Pysics 60, (000) doi:0.006/jcp , available online at ttp:// on Jian-Guo Liu and Ci-Wang Su Institute for Pysical Science and Tecnology and Department of Matematics, University of Maryland, College Park, Maryland 074; and Division of Applied Matematics, Brown University, Providence, Rode Island Received June 4, 999; revised January 6, 000 In tis paper we introduce a ig-order discontinuous Galerkin metod for twodimensional incompressible flow in te vorticity stream-function formulation. Te momentum equation is treated explicitly, utilizing te efficiency of te discontinuous Galerkin metod. Te stream function is obtained by a standard Poisson solver using continuous finite elements. Tere is a natural matcing between tese two finite element spaces, since te normal component of te velocity field is continuous across element boundaries. Tis allows for a correct upwinding gluing in te discontinuous Galerkin framework, wile still maintaining total energy conservation wit no numerical dissipation and total enstropy stability. Te metod is efficient for inviscid or ig Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments. c 000 Academic Press Key Words: incompressible flow; discontinuous Galerkin; ig-order accuracy.. INTRODUCTION AND THE SETUP OF THE SCHEME We are interested in solving te following D time-dependent incompressible Euler equations in vorticity stream-function formulation; u n t u 0 given on u (.) Researc supported by NSF Grant DMS Researc supported by ARO Grant DAAG , NSF Grants DMS , ECS , and INT , NASA Langley Grant NAG--070 and Contract NAS wile tis autor was in residence at ICASE, NASA Langley Researc Center, Hampton, VA , and AFOSR Grant F /00 $35.00 Copyrigt c 000 by Academic Press All rigts of reproduction in any form reserved.

2 578 LIU AND SHU were y x. Notice tat te boundary condition, plus te fact tat u n, recovers on te boundary (up to a constant) in a simple connected domain b (.) We are also interested in solving te Navier Stokes equations wit ig Reynolds numbers Re : u t u given on Re u (.3) Te boundary condition is now (.) plus te non-slip type boundary condition: n u b (.4) For simplicity, we only consider te no-flow, no-slip boundary conditions b 0 u b 0 and periodic boundary conditions. We first empasize tat, for Euler equations (.) and ig Reynolds number (Re ) Navier Stokes equations (.3), it is advantageous to treat bot te convective terms and te viscous terms explicitly. Te metods discussed in tis paper are stable under standard CFL conditions. Since te momentum equation (te first equation in (.) and (.3)) is treated explicitly in te discontinuous Galerkin framework, tere is no global mass matrix to invert, unlike conventional finite element metods. Tis makes te metod igly efficient for parallel implementation, see for example [3]. As any finite element metod, our approac as te flexibility for complicated geometry and boundary conditions. Te metod is adapted from te Runge Kutta discontinuous Galerkin metods discussed by Cockburn et al. in a series of papers [7 3, 0]. Te main difficulties in solving incompressible flows are te incompressibility condition and boundary conditions. Te incompressibility condition is global and is tus solved by te standard Poisson solver for te stream function using continuous finite elements. One advantage of our approac is tat tere is no matcing conditions needed for te two finite element spaces for te vorticity and for te stream function. Te incompressibility condition, represented by te stream function, is exactly satisfied pointwise and is naturally matced wit te convective terms in te momentum equation. Te normal velocity u n is automatically continuous along any element boundary, allowing for correct upwinding for te convective terms and still maintaining a total energy conservation and total enstropy stability. Tere is an easy proof for stability, bot in te total enstropy and in te total energy, wic does not depend on te regularity of te exact solutions. For smoot solutions error estimates can be obtained. We use te vorticity stream-function formulation of te Navier Stokes equations. Tis formulation wit te local vorticity boundary condition as been revitalized by te recent work of E and Liu [4, 5, 3]. Te main idea is to use convectively stable time-stepping procedure to overcome te cell Reynolds number constraint, explicit treatment of te viscous terms and te local vorticity boundary condition. Tis results in a decoupling of te

3 HIGH-ORDER DISCONTINUOUS GALERKIN METHOD 579 computation of stream function and vorticity at every time step. Tis metod is very efficient and accurate for moderate to ig Reynolds number flows, as demonstrated in [4, 5, 3]. Our metod, as it stands, can only compute D flows. In 3D, te normal velocity u n is no longer continuous along an element boundary, ence making te metod more complicated to design and to analyze. Similar approaces for te stream-function vorticity formulation, or for te primitive variable formulation, suitable for 3D calculations, are under investigation. We do not advocate our metod for modest or low Reynolds number flows. In suc regime viscous terms sould be treated implicitly for efficiency. Tis is a muc more callenging task in terms of space matcing caracterized by te Babu ska Brezzi Ladyzenskaja condition, projection type metods, and global vorticity boundary conditions; see, for example [4, 7 9, 5, 6, 8] etc. We remark tat te only problem of our metod for modest or low Reynolds number flows is te small time step dictated by te stability of te explicit time discretization. Of course, if te objective is to resolve te full viscous effect, ence a small time step is justified for accuracy, ten it is still adequate to use our metod. For convection-dominated flows, as we investigate in tis paper, we mention te work of Bell et al. [] for second-order Godunov-type upwinding metods; see also Levy and Tadmor [] and E and Su [6]. Tis is still an active field for researc. We now describe te setup of te sceme. We start wit a triangulation of te domain, consisting of polygons of maximum size (diameter), and te following two approximation spaces V k : K P k K K W k 0 V k C 0 (.5) were P k K is te set of all polynomials of degree at most k on te cell K. For te Euler equations (.), te numerical metod is defined as follows: find V k and W0 k, suc tat t K u K e K u n e 0 V k (.6) W0 k (.7) wit te velocity field obtained from te stream function by u (.8) Here is te usual integration over eiter te wole domain or a subdomain denoted by a subscript; similarly for te L norm. Notice tat te normal velocity u n is continuous across any element boundary e, but bot te solution and te test function are discontinuous tere. We take te values of te test function from witin te element K denoted by. Te solution at te edge is taken as a single-valued flux, wic can be eiter a central or an upwind-biased average. For example, te central flux is defined by (.9) were is te value of on te edge e from outside K, te complete upwind flux is

4 580 LIU AND SHU defined by if u n 0 if u n 0 (.0) and te Lax Friedrics upwind biased flux is defined by u n [u n ] (.) were is te maximum of u n eiter locally (local Lax Friedrics) or globally (global Lax Friedrics). We remark tat, for general boundary conditions (.), te space W0 k in (.5) sould be modified to take te boundary value into consideration. Moreover, additional pysical vorticity boundary condition for any inlet sould be given. Navier Stokes equations (.3) can be andled in a similar way, wit te additional viscous terms treated by te local discontinuous Galerkin tecnique in [3], and wit a local vorticity boundary condition in [4]. Te details are left to Section 3. Section is devoted to te discussion of stability and error estimates for te Euler equations. Accuracy ceck and numerical examples are given in Section 4. Concluding remarks are given in Section 5.. STABILITY AND ERROR ESTIMATES FOR THE EULER EQUATIONS For stability analysis, we take te test function in (.6), obtaining d dt K u K e K u n e 0 were we ave used te exact incompressibility condition satisfied by u for te second term. Performing an integration by parts for te second term, we obtain d dt K e K u n e 0 Now, using te fact tat [ ] [ ] were [ ] we obtain d dt K e K u n e e K u n[ ] e 0 Notice tat te second term is of opposite sign for adjacent elements saring a common edge e, ence it becomes zero after summing over all te elements K (using te no-flow

5 HIGH-ORDER DISCONTINUOUS GALERKIN METHOD 58 boundary condition on te pysical boundary or periodic boundary conditions). Te tird term is te numerical dissipation: wen is taken as te central flux (.9), te tird term is exactly zero; for te upwind flux (.0), te tird term becomes a positive quantity 4 e K u n [ ] e (.) wic is te total enstropy dissipation. Te effect of tis is to control te size of te jump across te element interface and essentially gluing te solution tere. Oter upwindbiased fluxes suc as te Lax Friedrics flux (.) would produce a positive term similar to tat of te total enstropy dissipation. For smoot flows tese jumps are of te order O k witin te truncation error of te sceme. We tus obtain te enstropy inequality d dt 0 (.) wic becomes an equality if te central flux (.9) is used. Te stability for te velocity field is now straigtforward: we take obtain in (.7) to C by te Poincare inequality, wic implies u C (.3) Indeed, we can obtain a total energy conservation troug te following arguments. Taking in (.6), we obtain t K u K e K u n e 0 Now te second term is zero since u 0. Te tird term vanises after summing over all elements since is continuous. Finally, noticing tat t we obtain te conservation of energy d dt d dt u d dt u 0 (.4) even for a upwind flux. Tus tere is no numerical dissipation for te energy. We remark tat te stability proof above for te total energy and total enstropy does not need any ypoteses on te regularity of te solution or te mes. We now turn to te error estimates. For tese we would need to assume tat te solution is regular ( H k for k ) and te mes is quasi-uniform. Conceptionally, since tis is a finite element metod, te exact solution of te PDE satisfies te sceme exactly. As

6 58 LIU AND SHU usual, we define te two projection operators: P is te standard L projection into te space V k ; and is te standard projection into W k 0 : Denote te error functions by 0 W k 0 and teir projections by P P We first obtain a control of in terms of, W k 0 from te sceme (.7) and te fact tat te exact solution also satisfies (.7). Now, taking, we obtain wic gives C Tis leads to a bound for te velocity field u u Since bot te numerical solution and te exact solution satisfy (.6), C (.5) t K u u K e K u n u n e 0 V k (.6) Take. Te second term becomes u u K u u K u K u P K (.7) Noticing tat u u is exactly divergence free, we may perform integration by parts to te first term on te rigt side of (.7) to obtain u u K u u K e K u u n e

7 HIGH-ORDER DISCONTINUOUS GALERKIN METHOD 583 Te second term on te rigt side of (.7) is a complete derivative and ence can be integrated to give a pure boundary term u K e K u n e Plugging all tese into (.6) wit, and collecting boundary terms, we obtain t K u u K u P K were te boundary terms e K I e 0 I e u u n e u u n ˆ n ˆ u n e e u nˆ P e e u n u n e Using te stability analysis in (.), we are left wit d dt K u u K u P K e K u nˆ P e (.8) Assuming for te moment u C (.9) we can first estimate te boundary term, using te fact tat te mes is quasi-uniform: K e K u nˆ P e K e K C [P ] e ˆ e C K e K [P ] e Using te above inequality togeter wit (.5) and (.8), we now obtain (wit te regularity assumption H k ) d dt C P H K e K [P ] e Here we understand te norms as a summation of te same norm on eac K Using te standard interpolation teory [6], we obtain d dt C C k wic yields C k

8 584 LIU AND SHU Togeter wit (.5), we ave Using an inverse inequality, we ave u u C k (.0) u u C k wit te assumption k, tis justifies te a priori assumption (.9). Te estimate (.0) is optimal in terms of te space W0 k, wic is important since te main cost for te sceme is in te Poisson solver in W0 k. Te vorticity estimate in (.0) is, owever, suboptimal wit respect to te space V k.ifw eu se W k 0 instead for te stream function and use te upwind flux (.0), ten a more detailed analysis will produce an order O k for te error in ; see [3, ] for details. However, we do not recommend tis coice in practice, as te increase of alf-order accuracy is obtained wit te price of one degree iger polynomials in te most expensive part of te algoritm, namely te Poisson solver. In our numerical experiments in Section 5, we observe tat close to k t order of accuracy is generally acieved wen kt degree polynomials are used in bot te discontinuous space for and te continuous space for, bot for uniform and for non-uniform meses. 3. THE SCHEME FOR THE NAVIER STOKES EQUATIONS For te Navier Stokes equations (.3), tere are two additional ingredients tat require our attention:. Te viscous terms cannot be directly implemented in te discontinuous space V k. Instead, te stress tensor is first obtained locally using te same discontinuous Galerkin framework.. Vorticity boundary values are not known pysically. We obtain vorticity boundary conditions locally from te stream function using te kinematic relation in (.3). We use te same finite element spaces V k and W 0 k defined in (.5) for te vorticity and stream function, respectively. Denote by V0 k k te subspace of V wit zero value at te boundary. Let W k be te finite element space extended from W 0 k wit general non-zero values at te boundary. Te numerical metod now becomes t K u K e K u n e K e K n e V k 0 (3.) Notice tat te test function is now in V0 k, (see [3]), and te stress tensor V k obtained from te vorticity by te same discontinuous Galerkin framework: is Re v K v K e K v n e v V k (3.)

9 HIGH-ORDER DISCONTINUOUS GALERKIN METHOD 585 We remark tat (3.) gives a local solution for te stress tensor, given te vorticity ; neiter a global inversion nor a global storage is needed. Te fluxes and can be cosen as central averages (3.3) or better still, as alternate one-sided fluxes, namely, at eac edge e wit an arbitrarily fixed orientation, one of and is taken as te left value and te oter taken as te rigt value. It can be verified tat, for k 0 and a rectangular triangulation, te central fluxes (3.3) produce a wide stencil central approximation to te second derivatives ( i, i and i are used for approximating xx ), wile te alternate one-sided fluxes produce a compact stencil central approximation ( i, i and i are used for approximating xx ). Also, numerical and teoretical evidence sow tat te alternate one-sided fluxes produce more accurate results [3]. In tis paper we use only te alternate one-sided fluxes for te viscous terms. For periodic boundary conditions, te sceme is now well defined. For te non-periodic case, we advocate using te approac in [3]. Altoug te basic idea in computing te vorticity boundary condition is similar to tat of te standard finite element metod in [3], tere is some difference due to te fact tat te approximation space for te vorticity in te discontinuous Galerkin metod is large tan tat in a continuous finite element metod. We outline te detailed steps ere for completeness. n Since (3.) is treated explicitly and te local discontinuous Galerkin metod is used, in te interior elements can be directly computed. However, for te boundary elements, we n need to compute in tree steps. First, for all test functions V0 k, we can compute te inner product n K directly from (3.) tanks to te explicit time stepping. Second, since W0 k k is a subspace of V0, from (3.), te inner products, n K for all W0 k as already been computed. Tis is sufficient for obtaining te stream function from n n n W k 0 (3.4) wit te velocity field obtained from te stream function by u n n (3.5) Finally, we are able to compute te vorticity n at boundary elements directly from n n for all te test functions V k tanks to te fact tat n as already been computed. For problems wit periodic boundary conditions, te formulation above admits te following stability results, d dt 0 (3.6)

10 586 LIU AND SHU wic in turn implies stability for te velocity field (.3). Te proof is similar to te Euler case; see [3] for te details. Wit te vorticity boundary condition mentioned above, we are unable to obtain a stability estimate. However, tis type of vorticity boundary treatment for conventional finite difference and finite elements is stable; see [5, 3]. Witout suc a stability estimate, tere is an issue about te uniqueness of te solution to te metod of lines discretization wit te boundary condition treatment mentioned above. However, te full discretized version works well in te numerical experiments; see Example of te next section. 4. ACCURACY CHECK AND NUMERICAL EXAMPLES We implement our metod on triangulations based on rectangles. Wen a P k Q k result is referred to it is obtained wit P k elements for te vorticity and Q k elements for te stream function, were Q k refers to te space of tensor products of D polynomials of degree up to k. We remark tat some of te teoretical results in previous sections do not apply for tese coices of mes and spaces. For example, Q k elements sould also be used for te vorticity for te exact energy conservation (.4) to old; owever, to save cost we use P k elements for te vorticity instead. Energy stability (.3) and enstropy stability (.) still old in tis case. We ave used bot te upwind flux (.0) and te (global) Lax Friedrics flux (.) for te calculations; owever, we will only sow te results obtained wit te Lax Friedrics flux to save space. Te time discretization is by te tird-order positive Runge Kutta metods in [7]. EXAMPLE. Tis example is used to ceck te accuracy of our scemes, bot for te Euler equations (.) and for te Navier Stokes equations (.3) wit Re 00, for bot te periodic and te Diriclet boundary conditions, and wit bot a uniform mes and a non-uniform mes. Te Diriclet boundary conditions use te data taken from te exact solution. Te non-uniform mes is obtained by alternating between 0 9 x and x for te mes sizes in te x direction; similarly for te mes sizes in te y direction. Te initial condition is taken as x y 0 sin x sin y (4.) wic was used in [5]. Te exact solution for tis case is known: x y t sin x sin y e t Re (4.) We use te domain [0 ] [0 ] for te periodic case and [0 ] [0 ] for te Diriclet case and compute te errors at t for te periodic case and at t for te Diriclet case. We list in Tables 5. (uniform mes) and 5. (non-uniform mes) te L and L errors, at t, measured at te center of te cells, for te periodic boundary conditions. Tables 5.3 (uniform mes) and 5.4 (non-uniform mes) contain te results wit te Diriclet boundary conditions at t. We remark tat, because of te difference in te sizes of te domains of te periodic and Diriclet cases, te errors wit te same number of cells are of different values, but te orders of accuracy are similar. We ave also computed te errors of te relevant derivatives at te centers of te cells, wic elp in giving us truly L errors trougout te domain. We will not sow tem to save space. To get an idea about te effect of te viscous terms on te time step restriction and CPU time, we point out tat for P 3 Q 3

11 HIGH-ORDER DISCONTINUOUS GALERKIN METHOD 587 TABLE 5. Accuracy Test, Uniform Meses, Periodic Boundary Conditions Euler Navier Stokes wit Re 00 Mes L error order L error order L error order L error order P Q E-03.80E E-03.8E-0 3.0E E E E E E E E E E E E P Q 6 6.6E-04.58E-03.06E E E E E E E E E E E E E E P 3 Q E-05.3E E-05.33E E E E E E E E E E E E E wit a 6 mes, te Navier Stokes code takes about twice as many time steps and about tree times as muc CPU time as te Euler code to reac te same pysical time. We ave also made several runs were te periodic and non-periodic cases ave te same pysical domain, mes, and pysical time. Te errors are very close, indicating tat te boundary effect on accuracy is small. TABLE 5. Accuracy Test, Non-uniform Meses, Periodic Boundary Conditions Euler Navier Stokes wit Re 00 Mes L error order L error order L error order L error order P Q E-03.85E E-03.80E E E E E E E E E E E E E-04.4 P Q E-04.77E E-04.8E E E E E E E E E E E E E P 3 Q E E-04.0E E E E E E E E E E E E E E

12 588 LIU AND SHU TABLE 5.3 Accuracy Test, Uniform Meses, Diriclet Boundary Conditions Euler Navier Stokes wit Re 00 Mes L error order L error order L error order L error order P Q 6 5.9E-04.3E E-04.3E E E E E E E E E E E E E-06. P Q E-05.57E-04.5E E E E E E E E E E E E E E-07.8 P 3 Q E-06.58E E-06.53E E E E E E E E E E E E E We can clearly see from tese tables tat close to k t order of accuracy is generally acieved wen kt degree polynomials are used in bot te discontinuous space for and for te Poisson solver, bot for te uniform and for te non-uniform meses. EXAMPLE. Te double sear layer problem taken from []. We solve te Euler equation (.) in te domain [0 ] [0 ] wit a periodic boundary condition and an initial TABLE 5.4 Accuracy Test, Non-uniform Meses, Diriclet Boundary Conditions Euler Navier Stokes wit Re 00 Mes L error order L error order L error order L error order P Q 6.E E E E E E E E E E E E E E E E-05.3 P Q E E-04.98E E E E E E E E E E E E E E P 3 Q E-06.49E E-06.8E E E E E E E E E E E E E

13 HIGH-ORDER DISCONTINUOUS GALERKIN METHOD 589 condition x y 0 cos x cos x sec y y sec 3 y y (4.3) were we take 5 and Te solution quickly develops into roll-ups wit smaller and smaller scales, so on any fixed grid te full resolution is lost eventually. We use fixed uniform meses of and 8 8 rectangles and perform te calculation up to t 8. We plot te time istory of te total energy (square of te L norm of velocity u) and te total enstropy (square of te L norm of vorticity ) in Fig. 5., as well as contours of te vorticity at t 6 in Fig. 5. and at t 8 in Fig. 5.3 to sow te resolution. We can see from Fig. 5. tat te numerical dissipation decreases rougly in te order of P Q -64, P Q -8, P Q -64, P 3 Q 3-64, P Q -8, and P 3 Q 3-8. We remark tat due to te dissipation from Runge Kutta time discretization and te coice of finite element spaces, te total energy in Fig. 5., left, decays rater tan stays at a constant as proved in (.4). Te decay rate of total energy or total enstropy in Fig. 5. is an indication of te actual resolution of te scemes for te given mes. Te iger-order metods ave better resolutions and in general te resolution is quite good judging from te contours. We remark tat wen te numerical viscosity becomes too small wit iger-order metods, since te scemes are linear, numerical oscillations are unavoidable wen resolution to sarp fronts is lost, leading to instability. Tis is common for all linear scemes. However, te discontinuous Galerkin metod we use ere is able to get stable solutions for muc sarper fronts wit te same mes tan central type finite difference or finite element metods. FIG. 5.. Te time istory of energy (square of te L norm of te velocity u) and total enstropy (square of te L norm of vorticity ). P Q wit 64 mes in solid line, P Q wit 8 mes in dased line, P Q wit 64 mes in das dot line, P Q wit 8 mes in dotted line, P 3 Q 3 wit 64 mes in long dased line, and P 3 Q 3 wit 8 mes in das dot dot line.

14 590 LIU AND SHU FIG. 5.. Contour of vorticity at t 6. Tirty equally spaced contour lines between 4 9 and 4 9. Left: results wit 64 mes; rigt: results wit 8 mes. Top: P Q ; middle: P Q, bottom: P 3 Q 3.

15 HIGH-ORDER DISCONTINUOUS GALERKIN METHOD 59 FIG Contour of vorticity at t 8. Tirty equally spaced contour lines between 4 9 and 4 9. Left: results wit 64 mes; rigt: results wit 8 mes. Top: P Q ; middle: P Q, bottom: P 3 Q 3.

16 59 LIU AND SHU FIG Contour of vorticity at t 8. Tirty equally spaced contour lines between 5 and 5. Left: results of te tin sear layer, 50, Re , P Q metod wit a 56 mes; rigt: results of te ultra tin sear layer, 00, Re 0 000, P Q metod wit a 5 mes. In [], sear layers are distinguised between tick and tin ones, wit te latter producing spurious vortices. Te result above corresponds to te tick sear layer in []. We sow in Fig. 5.4, left, a tin sear layer case as defined in [], corresponding to 50 wit a Reynolds number Re , simulated wit a uniform rectangular mes of cells wit P Q metod at t 8. Notice tat tis is at a muc iger Reynolds number tan tat used in [], were te Reynolds number is Re 000 and a second-order Godunov upwind projection metod wit points produces spurious non-pysical vortices. We also compute a ultratin sear layer wit 00 wit Reynolds number Re Te simulation result wit a uniform rectangular mes of 5 5 cells wit P Q metod at t 8 is sown in Fig. 5.4, rigt. More extensive numerical resolution study for tis example can be found in [4], were we explore torougly te resolution bot for te tick and for te tin sear layers. In [4] we also plot te time istory for te energy and enstropy during a mes refinement to sow tat te pysical viscosity is dominating te numerics at suc ig Reynolds numbers, according to te decay of energy and enstropy. Tis indicates tat te built-in numerical viscosity of te metods is very small. We refer te reader to [4] for details. For a comparison wit nonlinear ENO scemes, we refer to [6]. EXAMPLE 3. Te vortex patc problem. We solve te Euler equation (.) in [0 ] [0 ] wit te initial condition x y 0 x x 0 oterwise 3 4 y y (4.4) and periodic boundary conditions. Te contour plots of vorticity, wit 30 equally spaced contour lines between and, are given in Fig. 5.5 for t 5 and in Fig. 5.6 for t 0. We can see tat te sceme gives stable results for all runs, and iger-order scemes give better resolutions for vorticity.

17 HIGH-ORDER DISCONTINUOUS GALERKIN METHOD 593 FIG Contour of vorticity at t 5. Tirty equally spaced contour lines between and. Left: results wit 64 mes; rigt: results wit 8 mes. Top: P Q ; middle: P Q, bottom: P 3 Q 3.

18 594 LIU AND SHU FIG Contour of vorticity at t 0. Tirty equally spaced contour lines between and. Left: results wit 64 mes; rigt: results wit 8 mes. Top: P Q ; middle: P Q, bottom: P 3 Q 3.

19 HIGH-ORDER DISCONTINUOUS GALERKIN METHOD CONCLUDING REMARKS We ave developed a ig order discontinuous Galerkin metod for te two-dimensional incompressible Euler and Navier Stokes equations in te vorticity stream-function formulation, coupled wit a standard continuous finite element solution of te Poisson equation for te stream function. A natural matcing between te two finite element spaces allows us to obtain total energy conservation and total enstropy stability. Numerical examples are sown to demonstrate te accuracy and resolution of te metods. ACKNOWLEDGMENTS Tis work was initialized wen bot autors were visiting te Mittag-Leffler Institute in Sweden. We tank te faculty and staff tere for teir warm ospitality. REFERENCES. J. Bell, P. Colella, and H. Glaz, A second order projection metod for te incompressible Navier Stokes equations, J. Comput. Pys. 85, 57 (989).. D. L. Brown and M. Minion, Performances of under-resolved two dimensional incompressible flow simulations, J. Comput. Pys., 65 (995). 3. R. Biswas, K. D. Devine, and J. Flaerty, Parallel, adaptive finite element metods for conservation laws, Appl. Numer. Mat. 4, 55 (994). 4. A. Corin, A numerical metod for solving incompressible viscous flow problems, J. Comput. Pys., (967). 5. A. Corin, Numerical solution of te Navier Stokes equations, Mat. Comp., 745 (968). 6. P. Ciarlet, Te Finite Element Metod for Elliptic Problems (Nort-Holland, Amsterdam, 975). 7. B. Cockburn, An introduction to te discontinuous Galerkin metod for convection-dominated problems, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, edited by B. Cockburn, C. Jonson, C.-W. Su, and E. Tadmor, Lecture Notes in Matematics, Vol. 697 (Springer-Varlag, Berlin/New York, 998). 8. B. Cockburn and C.-W. Su, Te Runge Kutta local projection P -discontinuous Galerkin metod for scalar conservation laws, Mat. Model. Numer. Anal. 5, 337 (99). 9. B. Cockburn and C.-W. Su, TVB Runge Kutta local projection discontinuous Galerkin finite element metod for scalar conservation laws. II. General framework, Mat. Comp. 5, 4 (989). 0. B. Cockburn, S.-Y. Lin, and C.-W. Su, TVB Runge Kutta local projection discontinuous Galerkin finite element metod for conservation laws. III. One dimensional systems, J. Comput. Pys. 84, 90 (989).. B. Cockburn, S. Hou, and C.-W. Su, TVB Runge Kutta local projection discontinuous Galerkin finite element metod for conservation laws. IV. Te multidimensional case, Mat. Comp. 54, 545 (990).. B. Cockburn and C.-W. Su, TVB Runge Kutta local projection discontinuous Galerkin finite element metod for scalar conservation laws. V. Multidimensional systems, J. Comput. Pys. 4, 99 (998). 3. B. Cockburn and C.-W. Su, Te local discontinuous Galerkin metod for time-dependent convection diffusion systems, SIAM J. Numer. Anal. 35, 440 (998). 4. W. E and J.-G. Liu, Vorticity boundary condition and related issues for finite difference scemes, J. Comput. Pys. 4, 368 (996). 5. W. E and J.-G. Liu, Essentially compact scemes for unsteady viscous incompressible flows, J. Comput. Pys. 6, (996). 6. W. E and C.-W. Su, A numerical resolution study of ig order essentially nonoscillatory scemes applied to incompressible flow, J. Comput. Pys. 0, 39 (994). 7. V. Girault and P. A. Raviat, Finite Element Metods for Navier Stokes Equations, Teory and Algoritms (Springer-Verlag, Berlin, 986).

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