SIMPLE FINITE ELEMENT METHOD IN VORTICITY FORMULATION FOR INCOMPRESSIBLE FLOWS

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1 MATHEMATICS OF COMPUTATION Volume 70, Number 234, Pages S Article electronically publishe on March 3, 2000 SIMPLE FINITE ELEMENT METHOD IN VORTICITY FORMULATION FOR INCOMPRESSIBLE FLOWS JIAN-GUO LIU AND WEINAN E Abstract. A very simple an efficient finite element metho is introuce for two an three imensional viscous incompressible flows using the vorticity formulation. This metho relies on recasting the traitional finite element metho in the spirit of the high orer accurate finite ifference methos introuce by the authors in another work. Optimal accuracy of arbitrary orer can be achieve using stanar finite element or spectral elements. The metho is convectively stable an is particularly suite for moerate to high Reynols number flows. 1. Introuction In this paper, we present a very simple finite element metho for the unsteay incompressible Navier-Stokes equations NSE in vorticity-vector potential formulation. Let us first present the iea for the 2D case, where the vector potential is reuce to a stream function. The NSE rea t ω +u ω = ν ω, 1.1 ψ = ω, with no-slip bounary conition 1.2 ψ ψ =0, n =0, an the velocity is given by 1.3 u = ψ, = y, x. Aing inhomogeneous terms to the bounary conition only amounts to minor changes in what follows. For simplicity we will only consier a simply connecte omain in this paper. Finite ifference approximations of for large Reynols numbers have been stuie in etail in [9, 10, 11], where the the convection an viscous terms are both treate explicitly in the vorticity transport equation. The stream function, an hence the velocity, is then evaluate from the vorticity via the kinematic equation the secon equation in 1.1. We evelope a time-stepping proceure in which the value of the vorticity on the bounary can be obtaine explicitly from the Receive by the eitor June 8, Mathematics Subject Classification. Primary 65M60, 76M10. Key wors an phrases. Simple finite element metho, Navier-Stokes equations, vorticity bounary conitions, error analysis. JGL was supporte in parts by NSF grant DMS c 2000 American Mathematical Society

2 580 JIAN-GUO LIU AND WEINAN E stream function without any iteration, thus eliminating some traitional ifficulties associate with the vorticity formulation [21]. The classical fourth orer explicit Runge-Kutta time stepping metho was use to overcome the cell Reynols number constraint [9]. The resulting scheme is stable uner the stanar convective CFL conition. The metho is very efficient. At each time step or Runge-Kutta stage, a stanar Poisson equation is solve [9]. In a fourth orer compact ifference scheme an aitional mass-like matrix nees to be inverte [10]. Using a variational formulation an a finite element approximation gives a very general, natural, an clean way to implement the above methoology. It overcomes the ifficulty of fining an accurate local vorticity formula for a curve bounary, especially in the 3D case. The NSE can be formulate in a weak form [13]: fin ω H 1 Ω an ψ H0 1 Ω such that ϕ, t ω ϕ, ωu = ν ϕ, ω, ϕ H0 1, 1.4 ϕ, ψ = ϕ, ω, ϕ H 1. The velocity is again given by 1.3. Here, is the notation for the stanar inner prouct between functions on Ω an for the L 2 norm. In this form the first equation is taken as the evolution equation that upates ω, an the secon equation is then use to fin ψ. However we can also put 1.4 in a more symmetric form: fin ψ H0 1Ω an ω H1 Ω such that ϕ, t ψ + ϕ, ωu = ν ϕ, ω, ϕ H0 1, 1.4 ϕ, ω = ϕ, ψ, ϕ H 1. In this form the first equation is viewe as an evolution equation for upating ψ an the secon equation is use as a efinition of ω. Let Xh k be the stanar continuous finite element space [6] with kth egree polynomials on each element of a triangulation T h = {K}. Denotebyh the maximum size iameter of the elements. Let X0,h k be the subspace of Xk h with zero bounary values. The finite element approximation, treating time as continuous for now, is given as follows. Fin ω h Xh k an ψ h X0,h k such that ϕ, t ω h ϕ, ω h u h = ν ϕ, ω h, ϕ X 1.5 0,h k, ϕ, ψ h = ϕ, ω h, ϕ Xh k, an the velocity fiel is obtaine from the stream function via 1.6 u h = ψ h. Clearly the velocity u h satisfies the ivergence free conition everywhere, an the normal velocity u h n is continuous across any element bounary. The treatment of the convection term in 1.5 is legitimate even though the tangential velocity is not continuous across the element bounaries. Finite element approximations of the form 1.5 have been stuie for the case of steay state flow. However, the computation of the vorticity an stream function are fully couple, cf. [14]. See [7, 12] for the case of the biharmonic equation an [2, 1, 5, 18, 25] for the steay Navier-Stokes equation. The main contribution of this paper is an efficient time stepping proceure of 1.5 which will be escribe in Section 2. All the nice features mentione before in the finite ifference setting will be maintaine for the finite element metho. At the same time the metho is generalize in an extremely simple an clean way for the general omains in both 2D an 3D, with arbitrary orer of accuracy. The high orer spectral element metho can be easily aapte into our scheme. More complicate physics, such as

3 SIMPLE FINITE ELEMENT METHOD 581 stratifie flow, gravitation an centrifugal forces, turbulence moels, etc., can be incorporate into the scheme. There is also a simple an natural generalization of the above methoology to 3D, which will be iscusse in Section 3. It is well known that there is a major ifference between two an three imensions for vorticity-base numerical methos. Most apparent of all is the fact that both the vorticity an stream function become vector instea of scalar fiels in 3D. At the same time, the stream function changes its name to vector potential. Along with this is the necessity to enforce the solenoial conitions for the vorticity an vector potential. This turns out to be a major problem in esigning efficient numerical methos in 3D base on this formulation. In Section 4, we will present an error analysis for both 2D an 3D. A near optimal estimate of h k 1/2 in L 2 -norm for velocity an vorticity will be prove for the stanar kth orer finite element metho. Systemic numerical experiments of the methos propose in this paper will be reporte in [20] for the stanar finite element metho, in [19] for iscontinuous Galerkin metho. 2. Explicit time-stepping proceure Since finite element methos amount to centere schemes, a high orer time iscretization metho must be use in orer to avoi cell Reynols number constraints. This is iscusse in etail in [9]. In practice we use the classical fourth orer Runge-Kutta metho, which can essentially be written as four forwar Euler type steps [9]. For example, the intermeiate stage u in RK4 for the ODE u = fu can be formulate as u = u n tfu. Therefore, we will illustrate the time iscretization of 1.5 using forwar Euler. Suppose we know the values of ω n, ψ n an u n at t n. Wefirstcomputean auxiliary term ϕ, ω n+1 for any ϕ X0,h k from I ϕ, ω n+1 = ϕ, ω n + t ϕ, ω n u n ν t ϕ, ω n. Using this auxiliary term, we can solve for stream function ψ n+1 X0,h k from II ϕ, ψ n+1 = ϕ, ω n+1, ϕ X0,h k. From ψ n+1, we can obtain the vorticity ω n+1 by inverting a mass matrix from III ϕ, ω n+1 = ϕ, ψ n+1, ϕ Xh k. The right han sie of the above equation oes not have to be compute again for each test function ϕ X0,h k since it is equal to the auxiliary term from II, which has alreay been compute in I. Finally we compute the velocity 2.1 u n+1 = ψ n+1. We shoul emphasize that in the above time stepping proceure, the momentum equation I is completely ecouple from the kinematic equation II. There is no iteration require between the vorticity an stream function to recover the bounary values for the vorticity. Many existing methos require this, for example [21, 2, 1, 22, 16, 14]. At first sight, this proceure seems circular since II an III seem to use the same equation, but are use to compute ifferent things. So some comments are in orer:

4 582 JIAN-GUO LIU AND WEINAN E 1. Step I only computes ϕ, ω n+1 forϕ X0,h k. To completely etermine ωn+1 we nee to compute ϕ, ω n+1 for all ϕ Xh k. The computation of ϕ, ωn+1 for the egrees of freeom associate with the bounary noes is split into two steps. First in Step II we compute the stream-function ψ n+1. Fortunately, an this is very important for the success of this proceure, knowing ϕ, ω n+1 for ϕ X0,h k is enough to compute ψn+1. This is the same reason why the explicit methos work so well in the finite ifference setting. Having ψ n+1, we then compute ϕ, ω n+1 forϕ Xh k \ Xk 0,h using step III. However, III is also vali for ϕ X0,h k because of II. 2. The equation ϕ, ω n+1 = ϕ, ψ n+1 forϕ Xh k \ Xk 0,h can be thought of as the vorticity bounary conition. This is a natural generalization of Thom s formula. The above proceure can be escribe more clearly with matrix notation. Let φ i,i =1, 2,,N 0, be a basis corresponing to the interior noes, an φ j,j = N 0 +1,N 0 +2,,N 0 + N b, a basis corresponing to the bounary noes. Denote ω n =ω i anψ n =ψ i, vectors of size N 0 + N b ann 0,representingthe vorticity an stream function N 0+N b ωh n = N 0 ω i φ i, ψh n = 2.2 ψ i φ i, i=1 respectively. For the Lagrangian finite element space, the number of noes is equal to the imension of the finite element space, i.e., 2.3 N 0 = imension of X0,h k, N 0 + N b = imension of Xh k. Denote by M = φ i,φ j the stanar N 0 + N b N 0 + N b mass matrix an by A = φ i, φ j then 0 + N b N 0 + N b stiffness matrix, partitione, accoring to the imensions N 0 an N b,as A0,0 A 2.4 A = 0,b. A b,0 A b,b The auxiliary term in I is enote by ω n+1 =ω i, a vector of size N 0 an having values 2.5 ω l = N 0+N b i=1 i=1 φ l,φ i ω i, l =1, 2,,N 0. We enote the nonlinear term by N u n,ω n =N i, a vector of size N 0 an having values 2.6 N l = N 0+N b i=1 N 0 j=1 φ l,φ i φ j ω i ψ j, l =1, 2,,N 0. Then the momentum equation I can be written as ω n+1 ω n I + N u n,ω n =νa 0,0,A 0,b ω n. t We can use the auxiliary variable ω n+1 to solve for the stream function ψ n+1 from II A 0,0 ψ n+1 = ω n+1.

5 SIMPLE FINITE ELEMENT METHOD 583 We can solve for ω n+1 by inverting the mass matrix M from Mω n+1 ω n+1 III = A b,0 ψ n+1. The velocity is again given by 2.1. The above time stepping proceure is very similar to the one use in an essentially compact fourth orer finite ifference scheme EC4 for short propose in [10]. The vorticity bounary value on the left sie of III is relate to the stream function ψ n+1 through the local kinematic relation on the right han sie of III. The bounary values of ω n+1 are obtaine along with the interior values in III by inverting a mass matrix. This is quite general an natural an oes not result in any iteration between the vorticity an stream function. The simplicity an efficiency of the above scheme is obvious. The main computation involves solving a stanar Poisson equation II an inverting a stanar mass matrix III. Stanar FEM packages with a Poisson solver can easily be moifie to compute the unsteay viscous incompressible flow. Alternatively, the finite element iscretization of 1.4 results in a symmetric Galerkin form: fin ψ h X0,h k an ω h Xh k such that ϕ, t ψ h + ϕ, ω h u h = ν ϕ, ω h, ϕ X 1.5 0,h k, ϕ, ω h = ϕ, ψ h, ϕ Xh k. In a matrix notation as efine before, 1.5 becomes A 0,0 t ψ h + N u h,ω h = νa 0,0,A 0,b ω h, 1.5 A0,0 Mω h = ψ A h. b,0 Explicit time iscretization of above equation is equivalent to the time-stepping I III. Equation 1.5 can be viewe as an ODE system for ψ h after eliminating ω h. 3. Finite element metho in the 3D vorticity-vector potential formulation Now we present the finite element metho in the 3D vorticity-vector potential formulation. A corresponing finite ifference version was stuie in [11]. We will use the following 3D Navier-Stokes equation in the vorticity-vector potential formulation: t ω + ω u = ν ω, 3.1 ψ = ω, with six bounary conitions n ψ =0, ψ =0, 3.2 ω =0, n ψ =n u b. The velocity is given by 3.3 u = ψ. It is easy to verify that imply 3.4 ω =0, u =0, ψ =0. As a consequence of 3.4, are equivalent to the Navier-Stokes equation in primitive variables. See [23, Theorem 3.1] for a more etaile iscussion. For

6 584 JIAN-GUO LIU AND WEINAN E simplicity we assume u b = 0. General inhomogeneous bounary conitions can be treate similarly. The ifficulty in esigning an efficient numerical metho for arises from the nee to enforce the solenoial conitions 3.4 numerically. We will use a iv-curl type stiffness matrix see 3.6 below in our finite element approximation to overcome this ifficulty. Define the space Y =H 1 3 an the subspace 3.5 Y τ = {ψ Y : n ψ =0 onγ}. It was prove in [8] that the H 1 norm in Y τ is equivalent to the iv-curl norm 3.6 φ [φ, φ], [φ, ψ] φ, ψ + φ, ψ, an that the following Poincaré inequality hols: 3.7 φ C φ, φ Y τ. Inee, it was shown in [8] that for any φ, ψ in Y τ, φ, ψ =[φ, ψ]+ φ ψ γ, Γ R where R is the mean raius of the curvature of Γ. R>0when Ω is convex. A variational form for 3.1 an 3.2 is given as follows. Fin ω Y an ψ Y τ such that 3.9 φ, t ω + φ, ω u = ν[φ, ω], φ Y τ, [φ, ψ] = φ, ω, φ Y, or alternatively in a more symmetric way, fin ψ Y τ an ω Y such that 3.9 [φ, t ψ]+ φ, ω u = ν[φ, ω], φ Y τ, φ, ω =[φ, ψ], φ Y. Again, the velocity is given by 3.3. It is easy to verify that 3.9 is equivalent to See [3] for iscussions on the variational formulation in multiconnecte omains. Let Yh k =Xk h 3 be the stanar continuous finite element space with egree k an 3.10 Yτ,h k = {ψ Yh k : n ψ =0 onγ}. In general, the bounary conition in 3.10 is unerstoo to be vali for all the bounary noes. The finite element approximation of 3.9 is given as follows. Fin ω h Yh k 3.11 an ψ h Yτ,h k such that φ, t ω h + φ, ω h u h = ν[φ, ω h ], φ Yτ,h k [φ, ψ h ] = φ, ω h, φ Yh k, or, written in a symmetric Galerkin form, fin ψ h Y k τ,h an ω h Y k h [φ, t ψ h ]+ φ, ω h u h = ν[φ, ω h ], φ Yτ,h 3.11 k, φ, ω h =[φ, ψ h ], φ Yh k, an the velocity is given by 3.12 u h = ψ h. such that

7 SIMPLE FINITE ELEMENT METHOD 585 Explicit time-stepping proceure. As in 2D, we now present an efficient time stepping scheme for Again, we use forwar Euler as an illustration. Suppose we know the values of ω n, ψ n an u n at t n. Wefirstcomputean auxiliary term φ, ω n+1 for any φ Yτ,h k from A φ, ω n+1 = φ, ω n t φ, ω n u n ν t[φ, ω n ]. Using this auxiliary term we can solve for the vector potential ψ n+1 Y K τ,h from B [φ, ψ n+1 ]= φ, ω n+1, φ Y k τ,h. From ψ n+1, we can obtain the vorticity ω n+1 by inverting a mass matrix from C φ, ω n+1 =[φ,ψ n+1 ], φ Y k h. The right han sie of the above equation oes not have to be compute again for each test function φ Yτ,h k since it is equal to the auxiliary term from B, which has alreay been compute in A. Finally we compute the velocity 3.13 u n+1 = ψ n+1. Remark 1. When the bounary of the omain consists of flat surfaces, i.e., a polygon, then the mean raius of curvature R =. From 3.8, we know that for any φ, ψ Y τ, φ, ψ =[φ, ψ]. The stiffness matrix in B can be replace by the stanar stiffness matrix for the vector Poisson equation. In the case when the bounary surfaces are parallel to the coorinate planes, the three components of the vector potential ecouple, an the bounary conitions in B become homogeneous Dirichlet bounary conitions for the tangential components of the vector potential, an Neumann bounary conition for the normal component. This special geometric property has been explore before in a finite ifference approximation [11]. Designing simple an efficient finite ifference schemes for 3D in vorticity-vector potential formulations have been attacke in [11] with the same strategy: treat the viscous term explicitly an use a local vorticity bounary formula. A secon orer an fourth orer compact ifference scheme on a nonstaggere gri was propose in [11], an a local vorticity bounary conition, analogous to Thom s formula in 2D, is erive. 4. Stability an error estimates for the semi-iscrete case The stability analysis an error estimates for the finite element approx -imations in 2D an in 3D are rather stanar. For completeness, we give a etaile account here. For the stability analysis, we have the following conservation of energy: Theorem 1. Let u h an ω h be the solution of the finite approximation for the 2D Navier Stokes equation Then we have t 4.1 u h,t 2 L +2ν ω 2 h,s 2 L s = u h, L. 2 0

8 586 JIAN-GUO LIU AND WEINAN E Let u h, ω h an ψ h be the solution of the finite element approximation for 3D Navier Stokes equation Then we have t u h,t 2 L + ψ h,t 2 2 L +2ν ω 2 h,s 2 L 4.2 s 2 0 = u h, 0 2 L + ψ h, L. 2 Proof. We first prove 4.1. Take ϕ = ψ h in the first equation of 1.5 to obtain 4.3 ψ h, t ω h ψ h,ω h u h + ν ψ h, ω h =0. The secon term is zero since u h ψ h =0. Takingϕ = ω h in the secon equation of 1.5, we have 4.4 ψ h, ω h + ω h,ω h =0. Take the time erivative of the secon equation of 1.5, an replace ϕ by ψ h to obtain ψ h, t ω h = ψ h, t ψ h = 1 t 2 ψ h 2 = t 2 u h 2. We obtain from 4.3 an 4.4 the conservation of energy 4.6 t u h 2 +2ν ω h 2 =0. Integration in time gives 4.1. The proof for the 3D problem is similar. We take φ = ψ h in the first equation of 3.11 to obtain 4.7 ψ h, t ω h + ψ h, ω h u h + ν[ψ h, ω h ]=0. The secon term is zero since ψ h ω h u h =0. Takingφ = ω h in the secon equation of 3.11, we have 4.8 [ψ h, ω h ]= ω h, ω h. As in 4.6, we have from the secon equation of 3.11 that t ω h, ψ h = 1 t 2 ψ h 2. We obtain from 4.7 an 4.8 the conservation of energy 4.9 t ψ h 2 +2ν ω h 2 =0. Integrating in time an using the fact that u h = ψ, we obtain 4.2. This completes the proof of Theorem 1. Layzhenskaya-Babuska-Brezzi LBB conition. The finite element approximations 1.5 an 3.11 can be viewe as a mixe approximation of the Navier- Stokes equations an , respectively. The LBB conition plays an important role in the error analysis of mixe type problems [4, 13, 16]. We shoul emphasize that in the fully iscrete scheme , the momentum equation 1.7 is completely ecouple from the kinematic equation 1.8. Nevertheless the LBB conition is still important when choosing the iscrete spatial spaces. The same is true for the 3D problem. The LBB conition for 1.5 is 4.10 inf sup ψ h X0,h k ϕ h Xh k ϕ h, ψ h ϕ h ψ h C>0,

9 SIMPLE FINITE ELEMENT METHOD 587 an the LBB conition for 3.11 is 4.11 inf sup ψ h Yτ,h k φ h Yh k [φ h, ψ h ] φ h ψ h C>0. Using the Poincaré inequality for ψ h we can obtain 4.10 irectly from ϕ h, ψ h sup ψ h, ψ h C ψ h. ϕ h X ϕ h k h ψ h Using the Poincaré inequality 3.7, estimation of 4.11 is similar. Directly using the LBB conition will give an h k 1 estimate for ω an u for the biharmonic equation or the steay Stokes equation, cf. [4]. A more careful analysis using an L -estimate or uality argument will give an h k 1/2 accuracy cf. [24, 12]. The time epenence an the nonlinear term will contribute some more complications. We will give a etaile account of the error estimates for the full time epenent NSEinboth2Dan3Dbelow. Some estimates for projection operators. We first efine two projection operators for the 2D finite element spaces. P is the stanar L 2 projection into the space X k h 4.12 ϕ, ω Pω =0, ϕ X k h ; an Π is the stanar projection into X k 0,h 4.13 ϕ, ψ Πψ =0, ϕ X k 0,h. We have the stanar estimates for P an Π [6] 4.14 φ P φ + φ Πφ Ch k φ H k+1, an a maximum estimate [6] 4.15 ψ Πψ 1, Ch k ln h α ψ W k+1,, where α =1ifk =1;otherwiseα =0. Similarly, we efine the two projection operators for the 3D finite element spaces. P is the stanar L 2 projection into the space Y k h 4.16 φ, ω P ω =0, φ Y k h ; an we have the stanar estimates [6] 4.17 φ P φ Ch k φ H k+1, an Π is the projection into Y k τ,h 4.18 [φ, ψ Πψ] =0, φ Yτ,h. k It was prove in [8] that 4.19 φ Πφ Ch k φ H k+1 for any φ Y H k+1 3 an φ =0.

10 588 JIAN-GUO LIU AND WEINAN E Error estimate for 2D case. We now procee to estimate the error for the finite element approximation 1.5 for the Navier-Stokes equation 1.4. Let ω an ψ be the exact solution of 1.4. We enote the error functions by 4.20 ε = ω ω h, δ = ψ ψ h. Since both the numerical solution an the exact solution satisfy 1.5, one has after subtracting one from another, ϕ, t ε ϕ, ωu ω h u h + ν ϕ, ε =0 ϕ X0,h k 4.21, ϕ, δ + ϕ, ε =0 ϕ Xh k. Following the stanar strategy for estimating the error in a finite element approximation for the evolution type equation [17], we will first estimate the following projections of the error terms in 4.20: 4.22 ε h = Pε= Pω ω h, δ h =Πδ =Πψ ψ h. Using 4.12 we obtain from 4.21 ϕ, t ε h ϕ, ωu ω h u h + ν ϕ, ε =0 ϕ X ,h k, ϕ, δ + ϕ, ε h =0 ϕ Xh k. Take ϕ = δ h in the first equation of The first term can be estimate in a manner similar to that use to estimate 4.6 δ h, t ε h = t 2 δ h 2. The secon term becomes δ h, ωu ω h u h = δ h,ωu u h + δ h,εu h = δ h,ω δ δ h + δ h,εu h = δ h,ω ψ Πψ + δ h,εu h. In the secon equality above we have use the fact that δ h δ h = 0. Assuming for the moment that 4.25 u h C, the secon orer term can be estimate as δ h, ωu ω h u h C 1 δ h ψ Πψ + ε, where 4.26 C 1 = C C + ω L. Together with 4.23 an 4.24 we have 4.27 t δ h 2 2ν δ h, ε + C 1 δ h ψ Πψ + ε. To estimate the first term on the right han sie of 4.27, we take ϕ = ε h in the secon equation of Direct computation leas to δ h, ε = ε h,ε h + δ h, ε ε h ε h, δ δ h = ε h,ε h + δ h, ω Pω ε h, ψ Πψ.

11 SIMPLE FINITE ELEMENT METHOD 589 Plugging back into 4.27, we obtain t δ h 2 +2ν ε h 2 C 1 δ h ψ Πψ + ω Pω + ε h ν ε h, ψ Πψ C 1 δ h ε h + h k ψ H k+1 + ω H k+1 2ν ε h, ψ Πψ. In the last inequality, we have use the estimate To estimate the last term on the right han sie of 4.28, we enote by ε h,0 Xo,h k the interior components of ε h an by ε h,b the bounary components of ε h, i.e., ε h = ε h,0 + ε h,b. Using the projection 4.13, we have ε h, ψ Πψ = ε h,b, ψ Πψ. Direct estimation gives ε h,b, ψ Πψ Ch ψ Πψ 1, ε h,b 4.29 Ch k+1 ψ W k+1, ε h,b with a ln h factor on the right han sie of 4.29 if k = 1. Here the summation is over the bounary noes. The bounary noes on the right han sie of 4.29 can be irectly estimate as follows: ε h,b Ch 1/2 ε h,b 2 Ch 3/2 ε h,b Ch 3/2 ε h. Together with 4.28 we have t δ h 2 +2ν ε h 2 C 1 h k δ h + ε h ψ H k+1 + ω H k+1 C 1 δ h + Ch k 1/2 ψ W k+1,. Using Höler s inequality, we have 4.30 t δ h 2 + ν ε h 2 C2 1 ν δ h 2 + Ch2k 1 ν ψ 2W k+1, + ω 2Hk+1. The Grownwall inequality gives δ h,t L 0,T ];L 2 + ε h,t L 2 0,T ];L 2 C 2 h k 1/2 where C 2 = ψ L 0,T ];W k+1, + ω L 0,T ];H k+1 e C2 1 T/ν /C 1. Therefore u u h = ψ ψ h = ψ ψ h 4.31 ψ Πψ + δ h C 2 h k 1/2. Using an inverse inequality, we have 4.32 u u h C 2 h k 3/2.

12 590 JIAN-GUO LIU AND WEINAN E This justifies the aprioriassumption 4.25 when k 2bytaking C = u + C 2 h k 3/2. Therefore we have prove the the following theorem. Theorem 2. Let ψ,ω,u be an exact solution to the Navier-Stokes equation 1.4 an ψ h,ω h, u h be the numerical solution of the finite element approximation with stanar kth orer finite element space Xh k, k 2. Then we have u u h L 0,T ];L 2 + ω ω h L 2 0,T ];L Ch ψ k 1/2 L 0,T ];W k+1, + ω L 0,T ];H k+1 exp CT ω 2 + u 2 ν, where C is a constant that oes not epen on h or the solution. Error estimate for 3D case. As in 2D, we let ω an ψ be the exact solution to 3.9 an efine the error functions by 4.34 ɛ = ω ω h, δ = ψ ψ h, an their projections by 4.35 ɛ h = P ɛ = P ω ω h, δ h =Πδ =Πψ ψ h. Since both the numerical solution an the exact solution satisfy 3.11, we have φ, t ɛ h + φ, ω u ω h u h + ν[φ, ɛ] = 0 φ Yτ,h k 4.36, [φ h, δ]+ φ h, ɛ h =0 φ Yh k. Above we have use the property of the projection Take φ = δ h in the first equation of The first term becomes t 2 δ h 2 = δ h, t ω h, an the secon term becomes δ h, ω u ω h u h = δ h, ω u u h + δ h, ɛ u h = δ h, ω δ δ h + δ h, ɛ u h. Assuming for the moment that 4.38 we have 4.39 uh C, δ h, ω u ω h u h C δ h ψ Πψ + ɛ. From 4.36, 4.37 an 4.39 we have 4.40 t δ h 2 2ν[δ h, ɛ]+c δ h ψ Πψ + ɛ. To estimate the first term on the right han sie of 4.40, we take φ = δ h in the secon equation of Direct computation leas to 4.41 [δ h, ɛ] = ɛ h, ɛ h +[δ h, ω Πω h ] [ɛ h, ψ Πψ]. The last term can be estimate by the inverse inequality, 4.42 [ɛ h, ψ Πψ] C h ɛ h ψ Πψ.

13 SIMPLE FINITE ELEMENT METHOD 591 Plugging 4.42 an 4.41 back to 4.40, we have t δ h 2 +2ν ɛ h 2 C δ h ψ Πψ + ω P ω + ɛ h ν C h ɛ h ψ Πψ. Using the estimates 4.17 an 4.19 for the first two terms on the right han sie of 4.43 we have 4.44 t δ h 2 + ν ɛ h 2 C δ h 2 + Ch 2k 2. This gives 4.45 δ h Ch k 1. Using an inverse inequality, we are able to justify the aprioriassumption 4.38 when k 2. We now summarize the above arguments in the following theorem. Theorem 3. Let ψ, ω, u be an exact solution to the Navier-Stokes equation 3.9 an ψ h, ω h, u h be the numerical solution of the finite element approximation with stanar kth orer finite element space Xh k, k 3. Then we have u u h L 0,T ];L 2 + ω ω h L2 0,T ];L 2 + ψ h L 0,T,L 2 Ch ψ k 1 L 0,T ];H k+1 + ω 4.46 L 0,T ];H k+1 exp CT ω 2 + u 2. ν Here C is a constant that oes not epen on h or the solution. We remark that an h 1/2 orer shaper estimate can be improve in 4.46, provie the following maximum norm estimate for the projection of Π in 4.18 is true: ψ Πψ Ch k ln h α ψ W k+1,, where ψ = ψ + ψ, α =1ifk =1;otherwiseα =0. 5. Concluing remarks We presente a very simple, efficient an accurate finite element metho for the unsteay incompressible Navier-Stokes equation NSE in vorticity-vector potential stream function in 2D formulation. The stanar continuous finite element space with kth egree polynomials is use for both the vorticity an vector potential. A near optimal error estimate of h k 1/2 is obtaine for both velocity an vorticity. The efficiency of the metho results from the explicit time stepping proceure that we escribe in Section 1 for 2D an in Section 2 for 3D. The viscous term is treate explicitly an the bounary vorticity is etermine by a local explicit formula from the vector potential stream function via the kinematic relation at no aitional cost. The resulting momentum equation is completely ecouple from the kinematic equation. There is no iteration between the vorticity an the vector potential stream function. At each time step or Runge-Kutta stage the main

14 592 JIAN-GUO LIU AND WEINAN E computation involves solving a stanar Poisson equation an inverting a stanar mass matrix. In the 3D case, the stiffness matrix is of iv-curl form an is use to achieve the solenoial conitions for the vorticity an vector-potential. This is a key ifficulty in esigning efficient numerical methos for NSE for 3D. In the case when the omain is a polygon, the iv-curl form stiffness matrix is reuce to the stanar stiffness matrix for the vector Poisson equation. Stanar FEM packages with a Poisson solver can be easily moifie to compute the unsteay viscous incompressible flow. A finite ifference version of the above scheme has been stuie in etail an successfully use in many applications [9, 10, 11]. The time stepping follows exactly the same steps in the essentially compact fourth ifference scheme [10]. A variational formulation an finite element approximation give a very general, natural, an clean way to implement the above methoology. It overcomes the ifficulty of fining an accurate local vorticity formula for a curve bounary, especially in the 3D case. References 1. E. Barragy an G.F. Carey, Stream function vorticity solution using high-p element-byelement techniques, Commun. in Applie Numer. Methos, K.E. Barrett, A variational principle for the stream function-vorticity formulation of the Navier-Stokes equations incorporating no-slip conitions, J. Comput. Phys MR 57: A. Benali, J.M. Dominguez an S. Gallic, A variational approach for the vector potential formulation of the Stokes an Navier-Stokes problems in three imensional omains, J.Math. Anal. Appl MR 86k: F. Brezzi an M. Fortin, Mixe an hybri finite element methos, Springer-Verlag, New York, MR 92: A. Campion-Renson an M. J. Crochet, On the stream function-vorticity finite element solutions of Navier-Stokes equations, Int. J. Num. Meth. Engng P. Ciarlet, The Finite Element Metho for Elliptic Problems, North-Hollan, Amsteram, MR 58: P. Ciarlet an P. Raviart, A mixe finite element metho for the biharmonic equation, in Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, C. e Boor, e, Acaemic Press, New York, MR 58: F. El Dabaghi an O. Pironneau, Stream vector in three imensional aeroynamics, Numer. Math MR 87h: Weinan E an J.-G. Liu, Vorticity bounary conition an relate issues for finite ifference schemes, J. Comput. Phys., , MR 97a: Weinan E an J.-G. Liu, Essentially compact schemes for unsteay viscous incompressible flows, J. Comput. Phys., , MR 97e: Weinan E an J.-G. Liu, Finite ifference methos in vorticity-vector potential formulation on 3D non-staggere gris, J. Comput. Phys., R.S. Falk an J.E. Osborn, Error estimates for mixe methos, R.A.I.R.O. Anal. Numer MR 82j: V. Girault an P.A. Raviart, Finite Element Methos for Navier-Stokes Equations, Theory an Algorithms, Springer-Verlag, Berlin, MR 88b: P.M. Gresho, Incompressible flui ynamics: some funamental formulation issues, Ann. Rev. Flui Mech , MR 92e: P.M. Gresho, Some interesting issues in incompressible flui ynamics, both in the continuum an in numerical simulation, Avances in Applie Mechanics, , MR 93e: M. Gunzburger, Finite element methos for viscous incompressible flows, Acaemic Press, Boston, 1989 MR 91: C. Johnson an V. Thomée, Error estimates for some mixe finite element methos for parabolic type problems, R.A.I.R.O. Anal. Numer MR 83c:65239

15 SIMPLE FINITE ELEMENT METHOD M. Ikegawa, A new finite element technique for the analysis of steay viscous flow problems, Int. J. Num. Meth. Engng J.-G. Liu an C.-W. Shu, A high orer iscontinuous Galerkin metho for incompressible flows, J. Comput. Phys., to appear. 20. J.-G. Liu an J. Xu, An efficient finite element metho for viscous incompressible flows, in preparation. 21. S.A. Orszag an M. Israeli, Numerical simulation of viscous incompressible flows, Ann. Rev. Flui Mech., , O.A. Pironneau, Finite Element Methos for Fluis, John Wiley an Sons 1989 MR 90j: L. Quartapelle, Numerical Solution of the Incompressible Navier-Stokes Equations, Birkhäuser, Berlin, MR 95i: R. Schulz, A mixe metho for fourth orer problems using linear elements, R.A.I.R.O. Anal. Numer W.N.R. Stevens, Finite element, stream function-vorticity solution of steay laminar natural convection, Int. J. Num. Meth. Fluis Institute for Physical Science an Technology an Department of Mathematics, University of Marylan, College Park, MD aress: jliu@math.um.eu Courant Institute of Mathematical Sciences, New York, NY aress: weinan@cims.nyu.eu