ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS FOR THE NAVIER-STOKES EQUATIONS : FIRST-ORDER SCHEMES

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1 MATHEMATICS OF COMPUTATION Volume, Number, Pages S 5-578XX- ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS FOR THE NAVIER-STOKES EQUATIONS : FIRST-ORDER SCHEMES RICARDO H. NOCHETTO AND JAE-HONG PYO Abstract. The gauge formulation of the Navier-Stokes equations for incompressible fluids is a new projection method. It splits the velocity u = a+ φ in terms of auxiliary non-physical variables a and φ, and replaces the momentum equation by a heat-like equation for a and the incompressibility constraint by a diffusion equation for φ. This paper studies four time-discrete algorithms based on this splitting and the backward Euler method for a with explicit boundary conditions, and shows their stability and rates of convergence for both velocity and pressure. The analyses are variational and hinge on realistic regularity requirements on the exact solution and data. Both Neumann and Dirichlet boundary conditions are, in principle, admissible for φ but a compatibility restriction for the latter is uncovered which limits its applicability.. The Gauge or Impulse Formulation Given an open bounded polyhedral domain Ω in R d, with d = or 3, we consider the time-dependent Navier-Stokes Equations of incompressible fluids: u t + u u + p u = f, in Ω, div u =, in Ω, u, x = u, in Ω,. with vanishing Dirichlet boundary condition u = on Ω and pressure mean-value Ω p =. The primitive variables are the vector velocity u and the scalar pressure p. The viscosity = Re is the reciprocal of the Reynolds number Re. Hereafter, vectors are denoted in boldface. Pressure p can be viewed in. as a Lagrange multiplier corresponding to the incompressibility condition div u =. This coupling is responsible for compatibility conditions between the spaces for u and p, characterized by the celebrated inf-sup condition, and associated numerical difficulties [, 8]. On the other hand, projection methods were introduced independently by Chorin [3] and Temam [, 3] in the late 6 s to decouple u and p and thus reduce the computational cost. However, projection methods impose an artificial boundary condition on p, which leads to boundary layers and reduced convergence rates for p [7, 9]. Error estimates can be found in [, 7, ]. 99 Mathematics Subject Classification. 65M, 65M5, 76D5. Key words and phrases. Projection method, Gauge method, Navier-Stokes equation, incompressible fluids. c 997 American Mathematical Society

2 R.H. NOCHETTO AND J.-H. PYO The Gauge or Impulse Formulation, introduced by Oseledets [6] and E and Liu [5, 6], is a projection method especially conceived to cope with these inconsistencies. The gauge formulation consists of rewriting. in terms of two auxiliary variables, the vector field a and the scalar field φ gauge variable, which satisfy u = a + φ. Upon replacing this relation into the momentum equation in., we get Imposing a t + u u + φ t φ + p a = f. we end up with the gauge formulation p = φ t + φ,. a t + u u a = f, in Ω, a = u φ, in Ω, div u =, in Ω, p = φ t + φ, in Ω..3 The second a third equation in.3 combined yield φ = div a, and lead to the gauge formulation of. due to E and Liu [6]: a t + u u a = f, in Ω, φ = div a, in Ω, u = a + φ, in Ω, p = φ t + φ, in Ω..4 Since div u = and u = on Ω, in d there exists an unique stream function ψ such that curl ψ = u, in Ω, ψ = ν ψ =, on Ω..5 Computing the rotational in.5, and using that u = a + φ, implies ψ = rot curl ψ = rot u = rot a..6 In view of.3 and.5-.6, we get a novel gauge formulation for.: a t + u u a = f, in Ω, ψ = rot a, in Ω, ψ = ν ψ =, on Ω, u = curl ψ, in Ω, φ = u a, in Ω, p = φ t + φ, in Ω..7 Suitable boundary conditions must be given to close the above systems. A key advantage of these gauge methods is thao boundary condition is imposed on p and that we are free to choose a convenient boundary condition for the non-physical variable φ which, in view of., is expected to be smoother than p. We take a homogeneous condition either of Neumann or Dirichlet type for φ. To enforce the boundary condition u =, we can either prescribe ν φ =, a ν =, a = φ,.8 or φ =, a ν = ν φ, a =,.9

3 ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS 3 where ν and are the unit vectors in the normal and tangential directions, respectively. We call.8 the Neumann formulation and.9 the Dirichlet formulation. Wang and Liu show, for the backward Euler time discretization of.4, that the order of convergence for velocity is for the Neumann formulation and for the Dirichlet formulation [5]. Since [5] is based on asymptotic analysis, the exact solutions are assumed to be sufficiently smooth, a rather strong and unrealistic regularity requirement, particularly so for t for this entails nonlocal compatibility conditions between the initial data []; see A4 below. In addition, [5] does not address the convergence of pressure, which is the most sensitive variable. We use, instead, a variational technique to get rates of convergence for both velocity and pressure under realistic regularity assumptions on data. A distinctive aspect of our study is the assessment of pressure convergence. Since pressure is obtained through differentiation of φ, the boundary conditions.8 and.9 play a central role. The Neumann condition.8 always leads to an optimal convergence rate for velocity and a suboptimal one for pressure. In contrast, the Dirichlet condition.9 fails to yield convergence of pressure and solely gives a reduce convergence rate for velocity. The error estimates are similar to those known for the Chorin method [7, 9, ], and do not fully explain the advantages of the gauge methods [5, 6, 3, 4, 8]. They provide, however, a flexible methodology that extends to space discretization [3], a rare and fortunate situation for projection methods. The paper is organized as follows. In we formulate two semi-implicit timediscrete gauge algorithms based on Neumann condition.8. In 3 we introduce basic assumptions and recall regularity results for.. We show in 4 that the semi-implicit algorithms are unconditionally stable in energy norms, and so applicable for large Reynolds numbers; we also examine the explicit treatment of convection which requires a CFL condition. We give a priori error analyses in 5 for velocity and its time derivative via energy techniques; we refer to [, 4] for details. We present an a priori error analysis for pressure in 6. We finally conclude in 7 with a brief discussion of the Dirichlet condition.. The Gauge Methods with Neumann Condition We consider the backward Euler time discretization with uniform time step of gauge formulations.4 and.7 with Neumann.8 condition. In order to decouple the calculation of a and φ at time step n +, it is necessary to extrapolate the boundary conditions from the previous time step. This extrapolation is responsible for a boundary layer. Note that indicates a tangential unit vector to Ω whereas designates the time step. No confusion ever arises. Algorithm Gauge Formulation.4 with Neumann Condition.8. Start with initial values φ = and a = u = u, x. Repeat for n N Step : Find a as the solution of a a n + u n a + u n φ n a = ft, in Ω, a ν =, a = φ n, on Ω, Step : Find φ as the solution of φ = div a, in Ω, ν φ =, on Ω,.

4 4 R.H. NOCHETTO AND J.-H. PYO Step 3: Update u u = a + φ, in Ω. Algorithm Gauge Formulation.7 with Neumann Condition.8. Start with initial values φ = and a = u = u, x. Repeat for n N Step : Find a as the solution of. Step : Find ψ as the solution of ψ = rot a, in Ω, ψ =, Step 3: Update u and φ on Ω, u = curl ψ, in Ω, φ = u a, in Ω. Remark. Pressure. One may compute the pressure whenever necessary as p = φ φ n + φ.. Remark. Comparison of Algorithms and. The above Algorithms are equivalent. However, this equivalence is destroyed by space discretization because, in contrast to Algorithm, Algorithm is sensitive to the inf-sup condition [4, 8]. Remark.3 Velocity Boundary Condition. In view of the extrapolated boundary condition a = φ n, the boundary condition of velocity u is u ν =, u = φ φ n on Ω;.3 thus u is not zero which represents a boundary layer effect. In order to reduce the boundary layer in.3, we can use the nd order extrapolation formula a = φ n + φ n [6]. The two gauge algorithms are shown to be unconditionally stable in 4, which extend their applicability to large Reynolds numbers. We note that the equation. for a is still linear buonsymmetric. It becomes symmetric upon treating the convection term explicitly in the momentum equation [5, 6, 5], namely, a a n + u n u n a = ft..4 Gauge algorithms based on.4 are stable provided C and a rather strong, but customary, assumption is made on the discrete solution; see Preliminaries This section is mainly devoted to stating assumptions, reviewing some wellknown lemmas, and to proving basic properties of.. The basic mathematical theory summarized here can be found in the works of Constantin and Foias [4], Heywood and Rannacher [], and A. Prohl [7]. Let H s Ω be the Sobolev space with s derivatives in L Ω, L Ω = L Ω d and H s Ω = H s Ω d, where d =, 3. Let denote the L Ω norm, and, the corresponding inner product. Let s denote the norm of H s Ω for s R.

5 ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS 5 In the proof of convergence, we resort to a duality argument via the following Stokes equations: v + q = g, in Ω, div v =, in Ω, v =, on Ω. 3. We start with three basic assumptions about data Ω, u, f and solution u. Assumption A Regularity of 3.. The unique solution {v, q} of the steady Stokes equation 3. satisfies v + q C g. We remark that the validity of A is known if Ω is of class C [4], or if Ω is a twodimensional convex polygon [], and is generally believed for convex polyhedral []. Assumption A Data Regularity. The initial velocity u and the forcing term in. satisfy u H Ω {v H Ω div v = } and f, f t L, T ; L Ω. Assumption A3 Regularity of the Solution u. There exists M R such that sup ut M. t [,T ] We note that A3 is always satisfied in d, whereas it is valid in 3d provided f L,T ;L Ω and u are sufficiently small []. Let us introduce the following space which includes the solution v of the Stokes system 3.: Z := {z H Ω div z = }. 3. Then Z is a closed subspace of H Ω [8], and its dual space Z is equipped with the norm f, z f Z := sup. z ZΩ z z The following lemma is easy to prove. Lemma 3. Norm Equivalence. Let v, q, g be the functions in the Stokes system 3.. Then there exist two positive constants C, C such that C g Z v C g Z. We now define the trilinear form N associated with the convection term in. N u, v, w := u v wdx, for which the following properties are well-known [8]. Lemma 3. Properties of N. Let u, v, w H Ω and div u =. If then Ω u ν = or v = on Ω, N u, v, w = N u, w, v and N u, v, v =. Sobolev imbedding Lemma yields the following results, which will be used later in dealing with the convection term of..

6 6 R.H. NOCHETTO AND J.-H. PYO Lemma 3.3 Bounds on Trilinear Form. If d 4, then { C u v u v wdx w C u v w, and if d 3, then Ω Ω 3.3 u v wdx C u v / v / w. 3.4 Heywood and Rannacher proved the following a priori regularity estimates []. Lemma 3.4 Uniform and Weighted A Priori Estimates. Let σt be the weight function σt = min{t, }. Suppose A-3 hold, and let < T. Then the solution u, p of. satisfies T u + u t + p M, u t dt M, 3.5 and sup <t<t sup <t<t σt u t M, T σt u t + u tt + p t dt M. 3.6 The following nonlocal assumption is used to remove the weight σt in the error estimates for u t of 5 and pressure of 6. Assumption A4 Nonlocal Compatibility. Let u and f = f, be so that u t M. 3.7 Combining 3.8 with [, Corollary.], we realize that 3.7 is equivalent to the initial data u, p = p,, f satisfying the overdetermined system p = div f u u in Ω, p = u + f u u on Ω. This is true if u = f =, in which case also p = and u t =. However, u t t blows-up in general as t, thereby uncovering the practical limitations of results based on higher regularity than 3.5 and 3.6 uniformly for t. The proof of Corollary. in [], which assumes u = f = instead of 3.7, implies Lemma 3.5 below. Lemma 3.6 extends Lemma 3.5 to negative norms and is instrumental in 5. Lemma 3.5 Uniform A Priori Estimates. Suppose A-3 hold and let < T. Then 3.7 is valid if and only if T u tt t dt + sup u t t M. 3.8 <t<t Furthermore, if 3.7 holds, then T p t t + u tt dt M. 3.9 Lemma 3.6 A Priori Estimates on Z. If A-3 hold, then we have T u tt t Z dt M. 3.

7 ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS 7 Furthermore, if 3.7 hold, then sup u tt t Z M. 3. <t<t Proof. Since q Z = for all q L Ω, differentiating the momentum equation with respect to t and utilizing Lemmas yields u tt Z u t u Z + u u t Z + u t Z + f t Z C u u t + f t + C u t. Invoking Lemmas and A, we easily obtain both 3. and 3.. The following elementary but crucial relation is derived in [, 8, ]. Lemma 3.7 Div-Grad Relation. If v H Ω, then div v v Stability We prove now that Algorithms - are unconditionally stable. According to.3 the time-discrete function u does not vanish on the boundary Ω, and it is thus difficult to use u as a test function. To get around this issue, we introduce the auxiliary function û which vanishes on Ω: û := a + φ n. 4. The following useful properties of û are rather easy to show. Lemma 4. Properties of û n and u n. For all n, m non-negative integers, we have û n =, on Ω, û = a + φ n = u φ φ n, 4. u n, φ m =, and û n, u m = u n, u m. 4.3 Theorem 4. Stability. Algorithms - are unconditionally stable in the sense that for all > the following a priori bound holds: u u n + φ φ n + û n= + u N+ + φ N+ u + C ft 4.4. Proof. By definition of u and û, the momentum equation. can be rewritten as follows: û u n + u n û û φ n = ft. 4.5 We now multiply by û H Ω and use Lemmas 3. and 4. to get u un + u u n + φ φ n + û = φ n, div û + ft, û = A + A. n= 4.6

8 8 R.H. NOCHETTO AND J.-H. PYO In view of Lemmas 3.7 and 4., we have φ φ n = div û û, whence A = φ n, φ φ n φ Clearly, = φn φ φ n φ φn + û. A ft û C ft + û. Inserting A -A back into 4.6 and summing over n from to N, we get 4.4. If we treat the convection term explicitly, namely u n u n, then the momentum equation.4 of Algorithms - becomes û u n + u n u n û φ n = ft, in Ω. 4.7 In order to estimate u n u n, we need to assume that the semi-discrete solution u n is bounded in L Ω. This is a customary but rather strong assumption [5], which is not required in Theorem 4.. Remark 4.3 Stability for Explicit Schemes. Suppose max n N+ un L Ω M. 4.8 If A-3 hold, and the stability constraint M is enforced, then the following a priori estimate is valid for Algorithms - u N+ + φ N+ + û 4 u + C N ft. n= 5. Error Analysis for Velocity In this section, we carry out the error analysis for velocity of Algorithms -. We first prove that the convergence rate of velocity is of order, and then we improve the rate to order. Let ut, pt be the exact solution of. at the time step t. If û, u, p is the solution of any of the Algorithms -, then we denote the corresponding error by Ê := ut û, E := ut u, e := pt p. We observe again that û = on Ω and div û in Ω, whereas div u = in Ω and u = φ φ n on Ω. The following lemma results directly from Lemma 4.. Lemma 5. Properties of Error Functions. For all n, m non-negative integers, we have n= Ê =, on Ω, 5. div E =, Ê = E + φ φ n, 5. E n, φ m =, and Ên, E m = E n, E m. 5.3

9 ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS 9 Lemma 5. Additional Properties of Error Functions. We have φ φ n = div Ê Ê E C Ê Ê, 5.4 = E + φ φ n, φ φ n C Ê. 5.6 Proof. It is a simple consequence of A and Lemmas 3.7 and 5.. To examine Algorithms -, we first show that the semi-discrete solution u converges to ut in L, T ; L Ω with order see Theorem 5.3. We then improve the rate of convergence to order in L, T ; L Ω and L, T ; Z in Theorem 5.7. The result of Theorem 5.3 is instrumental in deriving Theorem 5.7. Theorem 5.3 Reduced Rate of Convergence for Velocity. Let A-3 hold. Then the velocity error functions of Algorithms - satisfy E E n + φ φ n + Ê + E N φ N+ C. n= Proof. By virtue of Taylor expansion for the exact velocity ut, we get ut u + ut ut + pt ut = R + ft, 5.8 where R := tu tt tdt is the truncation error. Subtracting 4.5 from 5.8 yields the error equation Ê E n Ê = R pt + φ n ut ut + u n û. 5.9 Multiplying 5.9 by Ê H Ω, and invoking Lemma 5., 5.9 becomes E En + E E n + φ φ n + Ê = R, Ê + pt, div Ê φ n, div Ê 5. N ut, ut, Ê N u n, û, Ê = We now estimate terms A to A 4 separately. Using Hölder inequality, R C t n t t dt σt u tt t dt, 4 A i. i= t u tt t dt 5.

10 R.H. NOCHETTO AND J.-H. PYO whence we deduce from 5.5 A C σt u tt t dt + C E + φ φ n t. 5. n On employing Lemma 5. and the boundary values ν φ φ n =, we obtain A C pt + φ φ n. 5.3 Making use of 5. and 5.4, we arrive at A 3 = φ n, φ φ n = φ φn + φ φ n φ φn + Ê To estimate the convection term A 4, we firsote that N u n, Ê, Ê = by Lemma 3.. We next invoke ut M, which comes from 3.5, and infer A 4 = N ut u, ut, Ê N E n, ut, Ê N u n, Ê, Ê C ut u + E n ut Ê C u t t C dt + En +. Ê Replacing A A 4 back into 5. and summing over n from to N implies E N+ + N φ N+ + φ φ n + E E n + + C n= Ê C N n= N+ n= N+ u t t dt + C En + E + pt σt u tt t dt. By the discrete Gronwall lemma and Lemma 3.4, we finally obtain Remark 5.4 Dependence on. The constant C in 5.7 depends exponentially on which is the reciprocal of the Reynolds number. This is unfortunate but customary in the error analysis of., except perhaps for the pipe flow in [9]. Remark 5.5 Estimates for Initial Errors. Choosing N = in 5.5, and realizing that E = and φ =, we readily have from Lemma 3.4 and Theorem 5.3 E + φ + φ + Ê C E + φ + C pt + C C + C u t t dt + C σt u tt t dt C, σt u tt t dt

11 ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS or C provided A4 also holds because then 3.8 applies and σt for t t. Remark 5.6 Suboptimal Order. The rate of convergence of order / of Theorem 5.3 is due to the presence of σt u tt t dt in 5. and pt in 5.3. To improve upon this we must get rid of both terms. This is precisely our next task. The main idea to obtain an error estimate in L, T ; L Ω is to invert the main elliptic operator or, equivalently, multiply by a divergence free test function satisfying the Stokes equations. For n N +, let v n, q n H Ω L Ω be the solution of v n + q n = E n, in Ω, div v n =, in Ω, 5.6 with vanishing Dirichlet boundary condition v n = on Ω. According to A, v n, q n H Ω H Ω are strong solutions of 5.6 and satisfy v n + q n C E n. 5.7 Theorem 5.7 Full Rate of Convergence for Velocity. If A-3 hold, then the velocity error functions of Algorithms - satisfy E N+ + Z E E n E + + Ê C. 5.8 Z n= n= Proof. Let v n, q n be the solution of 5.6. Then, it satisfies E E n, v = v v n, v = v vn + v v n, 5.9 as well as Ê, v Ê =, E q = E Ê, q, because of 5.3. Multiplying 5.9 by v H Ω, thus yields v vn + v v n + E = R, v Ê +, q + N u n, û, v N ut, ut, v =A + A + A We now estimate A to A 3 separately. Since v Z, defined in 3., we use 3. to find A C u tt Z dt + C v. To handle A we first recall 5. and the orthogonality E, q = for all q L Ω, which is valid for Algorithms and because div E = and E ν =. Hence, 5.7 implies A C φ φ n + E.

12 R.H. NOCHETTO AND J.-H. PYO On the other hand, the convection term A 3 can be rewritten as follows: A 3 = N E n, û, v N ut u, û, v N ut, Ê, v = A 3, + A 3, + A 3,3. 5. Since Theorem 5.3 and A yields v C E C /, and 3.5 gives ut C, appealing to 3.3 we easily deduce A 3, = N E n, Ê, v N E n, ut, v C E n Ê v + C E n ut v E 4 + C E E n + C Ê + C v. Likewise, since ut u C u t dt, we have A 3, = N ut u, Ê, v N ut u, ut, v C Ê + C v + C u t t dt. Since div ut =, we can exchange the last two arguments of A 3,3 to write A 3,3 C ut Ê v 4 Ê + C v. Inserting A 3, -A 3,3 into 5., and recalling 5.5, yields A 3 E + C E E n + C Ê + φ φ n + C v + C u t t dt. Combining 5. with A -A 3, and adding over n from to N, leads to v N+ + N v v n + N E C n= + C n= n= n= v + C N φ φ n + E E n n= Ê N+ + C u tt Z + u tt dt. Making use of 5.7, in conjunction with 3.5 and 3., we arrive at v N+ + N v v n + N E C + C n= n= 5.3 v. Applying the discrete Gronwall lemma allows us to remove the last term on the right-hand side. With the aid of Lemma 3., this implies 5.8 except for the estimate involving Ê. The latter follows from 5.5 and 5.7 together, and completes the proof. n=

13 ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS 3 We now embark on an error analysis for the time derivative of velocity. We use the notation δw := W W n. for any sequence {W n } N n= and σ n := σ for n N. We first observe an immediate consequence of Theorem 5.7. Remark 5.8 Estimate for δv. Choosing N = in 5.3, we readily get v C v φ + C + E + C t Ê + C u tt Z + u tt dt C 3 + C σt u tt + u tt Z + u tt dt C, with the aid of Remark 5.5 and Theorem 5.7. If A4 is also valid, then Remark 5.5 leads to a right-hand side C 3. Since v =, this gives δv C provided A-3 hold and δv C / provided A4 is valid as well. Lemma 5.9 Enhanced Stability. If A-3 hold, then the error functions of Algorithms - satisfy the weighted a priori bounds σ δe δe n + δφ δφ n δ + Ê +σ N+ δe N+ + σn+ 5.4 δφ N+ C. If A4 is also valid, then 5.4 becomes uniform, namely without weights. Since we intend to derive an L -based estimate, the main difficulty is to deal with pressure. We postpone the proof of Lemma 5.9 and instead apply it to derive the desired error estimate. Theorem 5. Error Estimate for Time-Derivative of Velocity. If A-3 hold, then the error functions of Algorithms - satisfy the weighted estimates σ N+ δe N+ Z + σ δe δe n + Z σ δe C. If A4 is also valid, then the following uniform error estimates hold δe N+ Z + δe δe n + Z δe C. Proof. Let v n, q n H Ω L Ω be the solution of 5.6; then δv + δq C δe. 5.5

14 4 R.H. NOCHETTO AND J.-H. PYO Subtracting two consecutive expressions 5.9, multiplying by δv H Ω, and recalling the argument leading to 5.9 and 5., yields δv δvn + δv δv n + δe = δq, δφ φ n + R R n, δv 5.6 N ut, ut, δv N u, u, δv N u n, û, δv + N u n, û n, δv = A + A + A 3. We now estimate each term A i separately. We first use 5.5 to write A δe 5 + C δφ δφ n. Since v Z, space defined in 3., we use 3. to find A = R, δv R n, δv n R n, δv δv n R, δv R n, δv n n + C u tt t Z dt + δv δv n 5.7 t. n On the other hand, an elementary manipulation of A 3 gives A 3 = N ut u + u, ut, δv N E n E n, ut, δv N u n u n, Ê, δv N u u, ut u, δv N E n, ut u, δv N u n, Ê Ên, δv =A 3, + A 3, + A 3,3 + A 3,4 + A 3,5 + A 3,6. To bound A 3, we first observe ut u + u = whence n t t n u tt tdt + ut u + u C t t u tt tdt, σt u tt t dt. This, together with ut M see 3.5, and the aid of 3.3, yields A 3, C ut u + u ut δv C σt u tt t dt + C δv, and A 3, C δe n ut δv δe 5 + δe δe n + C δv. To estimate A 3,3 A 3,6, we use to the following estimates proved in Lemma 5.9 and Theorem 5.3: σ n E n E n C, E + Ê C. 5.8

15 ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS 5 Since u u n u t t dt C 5.9 thanks to Lemma 3.4, we apply 5.5 in conjunction with 3.3 to deduce A 3,3 = N E n E n u u, Ê, δv C E n E n Ê + u u Ê δv C σ n Ê t n + u t t dt + δe t 5. n In view of 5.9, we readily obtain A 3,4 C u u ut u δv C u t t dt + C δv. With the aid of 3.3, 5.8 and 5.9, we can bound A 3,5 and A 3,6 as follows: A 3,5 C E n δut δv C and u t t dt + δe 5 A 3,6 = N E n u, δê, δv E n C δ Ê δv + u δ Ê δv C δê + C δv + δe 5 + C δφ δφ n. We now multiply both sides of 5.6 by the weight σ and sum over n for n N. We first examine the first two terms on the left-hand side of 5.6, which can be rewritten as follows: σ δv σn δv n σ σ n δv n σ N+ δv N+ σ δv N δv n σn+ δv N+ C. This is indeed a consequence of E = v =, σ =, and Remark 5.8 which give σ δv C, as well as Lemma 3. and Theorem 5.7, which imply N δvn C. Similarly, we analyze the first two terms on the right-hand side of 5.7, which become σ R, δv σ n R n, δv n σ σ n R n, δv n σ N+ R N+ Z δv N+ + R Z δv + σn+ δv N+ N+ + C + C u tt t Z dt. R n Z δv n

16 6 R.H. NOCHETTO AND J.-H. PYO Inserting the above estimates into 5.6 gives σ N+ δv N+ N + σ δv δv n + δe C + C + C + C N+ u tt Z + σt u tt + u t dt σ δv δ + Ê σ δφ δφ n N + C + δe δe n σ σ n Ê. 5.3 Since σ σ for n, recalling Lemmas 3.4, 3.6, and 5.9, and Theorems 5.3 and n 5.7, and using the discrete Gronwall lemma, the asserted weighted estimate follows. To derive the uniform error estimate, we do not multiply 5.6 by σ. Since now δv C, according to Remark 5.8, we immediately see that δv δvn C + δv N+. Likewise, using 3., we easily arrive at R, δv R n, δv n C 3/ + v. Since we can remove σ n in 5.8, and thus in the bound for A 3,3, we end up with an expression similar to 5.3 but without weights. Proceeding as before, we conclude the desired estimate via the discrete Gronwall lemma. Remark 5. Optimality. The above error estimate N n= σ δe C is consistent with the regularity T σ u tt dt of Lemma 3.4. In fact, the loss of half an order is customary unless the PDE corresponds to an angle-bounded operator [5]. This is not the case of.. Proof of Lemma 5.9. Subtracting two consecutive formulas 5.9, and multiplying by δê = δe + δφ δφ n H Ω, yields δe δen + δe δe n + δφ δφ n + δê = R R n, δê δφ n, div δê + pt p, div δê N ut, ut, δê 5.3 N u n, û, δê N u, u, δê + N u n, û n, δê = A + A + A 3 + A 4. We now estimate each term A to A 4 separately. We easily find out that A R R n Z δê 8 δê + C u tt t Z dt,

17 ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS 7 as well as A 3 pt p δ Ê 8 δê + C p t dt, the latter being a consequence of 3.6. Making use of 5. and 5.4, we arrive at A = δφ n, δφ δφ n δφ δφn + δê. On the other hand, the convection term A 4 can be rewritten as follows: A 4 = N ut u, ut, δê + N E n, ut, δê N u u, u, δê N E n, u, δê N u n, Ê, δê N u n, Ên, δê = A 4, + A 4,. We now recall that u C see 3.5, and use 3.3, to arrive at A 4, 8 δê + C E n + E n + C u t t dt. We first rewrite A 4, as follows invoking the crucial properties of N of Lemma 3.: A 4, = N un u n, Ê, Ên =N u u, δê, Ên N E n E n, δê, Ên = B n + B n. Since Theorem 5.3 yields Ên C, we obtain B n C u u δ Ê Ê n 8 δê + C n u t t dt. Instead of estimating B n, we first insert the above estimates into 5.3, multiply by the weight σ, and add over n from to N. Arguing as in Theorem 5., namely using Theorem 5.3 and Remark 5.5, the first two terms in 5.3 become σ N+ δe N+ σ δe N δe n C + σn+ δe N On the other hand, we resort to the property σ σ n σ A σ δ Ê + C 8 n= for n to write N+ Collecting all these estimates, and using Lemma 3.4, we obtain σ N+ δe N+ + N σ δe δe n + σt p t dt 5.33 t σ δ Ê +σ N+ δφ N+ N + σ δφ φ n 5.34 N C + σ B n.

18 8 R.H. NOCHETTO AND J.-H. PYO We now deal with B n in two steps via 3.4. We first use E n E n C and E n Ên C, which results from Theorem 5.3 and 5.6, to deduce B n C E n E n E n E n δê Ê n, whence σ B n 4 σ δ Ê + C / Ên. Since N Ên C according to Theorem 5.3, 5.34 gives the improvement σ E E n C 3, σ E E n C over the starting condition for the st step. Using this to estimate B n again, we discover that B n 4 δên + C σ Ê n which leads to the improved bound σ B n 4 σ δ Ê n + C N Ên C + 4 δên and proves the weighted estimate 5.4. We finally observe that if A4 is valid, then so is Lemma 3.5, thereby making unnecessary the use of weight σ in 5.3 and This leads to an inequality similar to 5.34 without weights, and implies the asserted uniform estimate. Remark 5. Convergence Rates for Velocity of Explicit Algorithms. If we treat convection explicitly, as written in 4.7, then we also need the customary but strong assumption 4.8 to obtain the results of Theorems 5.3, 5.7, and 5.., 6. Error Analysis for Pressure The goal of this section is to estimate the pressure error in L, T ; L Ω for Algorithms and, where p is computed according to., namely, p = φ φ n + φ. 6. This hinges on the error estimate for time derivative of velocity of Theorem 5.. Theorem 6. Rate of Convergence for Pressure. If A-3 hold, then the pressure error functions e of Algorithms - satisfy the weighted estimate σ e C. 6. n= If A4 is also valid, then the following uniform error estimate holds e C. 6.3 n=

19 ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS 9 Proof. We recall the existence of β > such that inf-sup condition [, 8] β q q, div w sup q L w H Ω w Ω. 6.4 Consequently, it suffices to estimate e, div w in terms of w. Multiplying 5.9 by w, and utilizing 5. and 6., we end up with e, div w = δe, w + Ê, w + N ut, ut, w N u n, û, w 6.5 φ φ n, div w R, w 5 = A i. We now proceed to estimate each term A to A 5 separately. We firsote that A C δe w C δe w, A Ê w. Term A 3 can be dealt with the aid of 3.3 and Theorem 5.3 as follows: A 3 =N ut u, ut, w + N E n, ut, w N E n, Ê, w + N u, Ê, w i= C ut u ut + E n ut + E n Ê + u Ê w C Ê + E n + ut u w, because 5.6 and 5.7 imply E n C. On the other hand, we have A 4 φ φ n w, A 5 R w. Inserting the estimates for A to A 4 back into 6.5, and employing 6.4, we obtain C e δe + Ê + E n + R + φ φ n + ut u. We now square, multiply by σ resp., and sum over n from to N. Recalling 5.9 and 5., and invoking Lemma 3.4, Theorems 5.7 and 5., and Lemma 5.9, the assertion 6. resp. 6.3 follows. Remark 6. Optimality. Both error estimates of Theorem 6. are optimal according to the regularity in time for pressure of Lemma 3.4, because the operator involved is not angle-bounded [5]. Remark 6.3 Rate of Convergence for Pressure of Explicit Algorithms. If A-3 and 4.8 hold, then the pressure error functions e of Algorithms - with explicit convection.4 satisfy 6.. Moreover, they satisfy 6.3 provided A4 is also valid.

20 Step 3: Update u u = a + φ, in Ω. R.H. NOCHETTO AND J.-H. PYO 7. The Gauge Methods with Dirichlet Condition In this section, we examine the Dirichlet condition.9 and thereby uncover a fundamental obstruction for computing pressure. This severely limits its applicability. On the other hand, the chief difficulty in deriving an error estimate for velocity is thaow u n ν. This is responsible for the reduced order O of Theorem 7.3, which is consistent with the rate obtained in [5] via asymptotics, and for the additional but realistic regularity assumption 7.4. Algorithms - can be modified as follows to account for.9: Algorithm 3 Gauge Formulation.4 with Dirichlet Condition.9. Start with initial values φ = and a = u = u, x. Repeat for n N Step : Find a as the solution of a a n + u n a + u n φ n a = ft, in Ω, a ν = ν φ n, a =, on Ω, Step : Find φ as the solution of φ = div a, in Ω, φ =, on Ω, Algorithm 4 Gauge Formulation.7 with Dirichlet Condition.9. Start with initial values φ = and a = u = u, x. Repeat for n N Step : Find a as the solution of 7.. Step : Find ψ as the solution of ψ = rot a, in Ω, ν ψ =, Step 3: Update u and φ on Ω, u = curl ψ, in Ω, φ = u a, in Ω. Remark 7. Velocity Boundary Condition. In Algorithms 3 and 4, the boundary conditions of velocity u are u ν = ψ = ν φ ν φ n, u =, on Ω. 7.3 On the other hand, φ = on Ω in Algorithm 4 upon adding a suitable constant, because φ = u = ν ψ = on Ω. Consequently, φ of Algorithm 4 satisfies 7.. Remark 7. Compatibility Condition. Upon integrating both sides of 7. and using the boundary conditions of 7., we discover the relation φ dx = ν φ dγ = a νdγ = ν φ n dγ = φ n dx. Ω Ω Ω for both Algorithms 3 and 4. This means that Ω φn dx must be constant for all time steps n, which is not true in general. So, we cannot expect the numerical Ω Ω

21 ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS solution φ n to converge to the exact solution φ, as computations corroborate [4, 8]. Since pressure p n and φ n are linked via., we cannot expect convergence of p n to p. Therefore, Algorithms 3 and 4 cannot be used for approximating p. Surprisingly, the velocity u converges to the exact solution u with a rate O under realistic regularity assumptions, which are much weaker than those in [5] for a similar rate. Theorem 7.3 Reduced Rate of Convergence for Velocity of Algorithms 3-4. Let A-3 hold as well as T p t t dt M. 7.4 Then the error functions of Algorithms 3-4 satisfy E N+ + 4 Ê + n= n= E E n + φ φ n C. 7.5 Proof. Since Ê = on Ω according to 5., the departing point is again 5.. To estimate A and A 4 we proceed as in Theorem 5.3 and thereby obtain 5. and 5.4, respectively. The remaining two terms A and A 3 are more delicate and are handled together as follows: A + A 3 = pt p, φ φ n q n, q q n + q n, pt p = B + B + B 3, where q n := p φ n. We have pt p C p t t dt, whence using Lemma 5., B C p t t dt + t 8 Ê. n For B we employ the inequality a + b + εa + + ε b for ε = 8, B = q qn + pt p φ φ n q qn + C p t t 9 dt + t 8 Ê. n Since B 3 C qn + C p t t dt, inserting these expressions into 5., and summing over n from to N, yields E N+ N + E E n + φ φ n + 4 Ê n= + q N+ C + C N+ E + C n= q n + pt n= p t t + σt u ttt + u tt dt.

22 R.H. NOCHETTO AND J.-H. PYO Finally, the discrete Gronwall lemma implies 7.5 and concludes the proof. Remark 7.4 Reduced Rate of Convergence for Velocity of Explicit Algorithms 3-4. If the assumptions A-3, 4.8 and 7.4 hold, then the error functions of Algorithms 3-4 with explicit convection.4 satisfy 7.5. References [] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag 99. [] D.L. Brown, R. Cortez, and M.L. Minion Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., [3] A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., [4] P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press 988. [5] W. E and J.-G. Liu, Gauge finite element method for incompressible flows, Int. J. Num. Meth. Fluids, 34, 7-7. [6] W. E and J.-G. Liu, Gauge method for viscous incompressible flows, Comm. Math. Sci., 3, [7] W. E and J.-G. Liu, Projection method I: Convergence and numerical boundary layers, SIAM J. Numer. Anal., 3 995, [8] V. Girault, and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer- Verlag 986. [9] C. Johnson, R. Rannacher, and M. Boman, Numerics and hydrodynamic stability: toward error control in computational fluid dynamics, SIAM J. Numer. Anal , [] J.G. Heywood and R. Rannacher, Finite element approximation of the non-stationary Navier-stokes problem. I. regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 9 98, [] R.B. Kellogg and J.E. Osborn, A regularity result for the stokes problems in a convex polygon, J. Funct. Anal., 976, [] R.H. Nochetto and J.-H. Pyo, Optimal relaxation parameter for the Uzawa method Numer. Math. to appear. [3] R.H. Nochetto and J.-H. Pyo, A finite element Gauge-Uzawa method. Part I : the Navier- Stokes equations to appear. [4] R.H. Nochetto and J.-H. Pyo, Comparing Gauge-Uzawa methods with other projection methods to appear. [5] R.H. Nochetto, G. Savaré, and C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math., 53, [6] V.I. Oseledets, A new form of writing out the Navier-Stokes equation. The Hamiltonian formalism, Russian Math. Surveys, , -.

23 ERROR ESTIMATES FOR SEMI-DISCRETE GAUGE METHODS 3 [7] A. Prohl, Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations, B.G.Teubner Stuttgart 997. [8] J.-H. Pyo The Gauge-Uzawa and related projection finite element methods for the evolution Navier-Stokes equations, Ph.D dissertation, University of Maryland. [9] R. Rannacher, On Chorin s projection method for the incompressible Navier-Stokes equations, Springer-Verlag, Lecture Notes in Mathematics 53 99, [] J. Shen, On error estimates of projection methods for Navier-Stokes equations: first order schemes, SIAM J. Numer. Anal., 9 99, [] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Siam CBMS [] R. Temam, Navier-Stokes Equations, North-Holland 984. [3] R. Temam, Sur l approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires. II. French Arch. Rational Mech. Anal., , [4] V. Thomèe, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag 997. [5] C. Wang and J.-G. Liu, Convergence of gauge method for incompressible flow, Math. Comp., 69, Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 74, USA. Partially supported by NSF Grants DMS and DMS-467. URL: address: rhn@math.umd.edu Department of Mathematics, Purdue University, West Lafayette, IN 4797, USA. Partially supported by NSF Grant DMS URL: address: pjh@math.purdue.edu