A Local Projection Stabilization/Continuous Galerkin Petrov Method for Incompressible Flow Problems

Transcription

1 A Local Projection Stabilization/Continuous Galerkin Petrov Metod for Incompressible Flow Problems Naveed Amed, Volker Jon, Gunar Matties, Julia Novo Marc 22, 218 Abstract A local projection stabilization LPS metod in space is considered to approximate te evolutionary Oseen equations. Optimal error bounds wit constants independent of te viscosity parameter are obtained in te continuous-in-time case for bot te velocity and pressure approximation. In addition, te fully discrete case in combination wit iger order continuous Galerkin Petrov cgp metods is studied. Error estimates of order k 1 are proved, were k denotes te polynomial degree in time, assuming tat te convective term is time-independent. Numerical results sow tat te predicted order is also acieved in te general case of time-dependent convective terms. AMS subject classification: 65M12, 65M15, 65M6 Keywords: Evolutionary Oseen problem, inf-sup stable pairs of finite element spaces, local projection stabilization LPS metods, continuous Galerkin Petrov cgp metods 1 Introduction Te beavior of incompressible fluid flows is modeled by te incompressible Navier Stokes equations. Analyzing numerical scemes for tese equations faces several difficulties. First, te unresolved problem of te uniqueness of te weak solution of te Navier Stokes equations in tree dimensions requires to assume uniqueness, wic is usually done by assuming sufficient regularity of te weak solution. Moreover, te estimate of te nonlinear term often uses te Gronwall lemma, suc tat an exponential factor occurs in te error bounds, depending on some norm of te velocity, e.g., on u as in [2]. As result, te obtained estimates are by far too pessimistic in practice. For tese reasons, tis paper will deal, wit respect to te numerical analysis, wit a related but simpler problem, namely te evolutionary or transient Oseen equations. Tey read in dimensionless form as follows: Department of Matematics, College of Engineering and Tecnology, American University of te Middle East, Egaila, Kuwait, naveed.amed@aum.edu.kw Weierstrass Institute for Applied Analysis and Stocastics WIAS, Morenstr. 39, 1117 Berlin, Germany, and Freie Universität Berlin, Department of Matematics and Computer Science, Arnimallee 6, Berlin, Germany, jon@wias-berlin.de Tecnisce Universität Dresden, Institut für Numerisce Matematik, 162 Dresden, Germany, gunar.matties@tu-dresden.de Departamento de Matemáticas, Universidad Autómoma de Madrid, Cantoblanco, Madrid, 2849, Spain, julia.novo@uam.es; Julia Novo s researc was financed by Spanis MICINN under grants MTM P MINECO, ES and MTM P AEI/FEDER, UE 1

2 Find ut, x :, T ] Ω R d, d {2, 3}, and pt, x :, T ] Ω R suc tat t u ν u b u σu p = f in, T ] Ω, div u = in, T ] Ω, u = on, T ] Ω, u, = u in Ω, 1 were Ω R d is a bounded domain wit polyedral Lipscitz boundary Ω, ν = Re 1 > viscosity and σ > are positive constants, bt, x is a given velocity field wit div b =, u is te initial velocity field, and T is a given final time. Witout loss of generality, one can assume σ >, since if it is not te case ten a simple cange of variable transforms te problem into 1 wit σ >, see [19, Sect. 1]. Te numerical solution of 1 requires discretizations in time and space. Concerning te temporal discretization, continuous Galerkin Petrov metods of order k 1, cgpk, will be considered. Wit respect to space, finite element metods will be studied. Since te paper will study te convection-dominated regime, were ν is smaller tan an appropriate norm of b by several orders of magnitude, a stabilization of te standard finite element discretization becomes necessary. Considering te situation tat te viscosity is muc smaller tan te convection, it is well known tat stabilized discretizations ave to be applied. Te most popular stabilized finite element metod is probably te streamline upwind Petrov Galerkin SUPG metod from [16, 22]. Often, te SUPG stabilization is used in combination wit te pressure-stabilization Petrov Galerkin PSPG metod, [29]. However, te SUPG/PSPG metod possesses some drawbacks, see [15]: it introduces a velocity-pressure coupling for wic no pysical explanation is known and te non-symmetry of te stabilization migt be of disadvantage. In te time-dependent case, te consistent application of te metod leads to a number of additional terms wic ave to be assembled, see [28, 31]. Because of te drawbacks of te SUPG/PSPG metod, we tink tat it is wort to investigate different approaces in detail, in particular suc approaces tat are symmetric and tat do not introduce an additional velocity-pressure coupling. Local projection stabilization LPS metods belong to tis class of metods and will be te topic of tis paper. A different approac from tis class was studied recently in [19], were a grad-div stabilized metod is used to discretize te evolutionary Oseen equations. Optimal bounds for te divergence of te velocity and te L 2 Ω norm of te pressure are proved for tis metod. Te LPS metod was originally proposed for te Stokes problem in [12] and it was successfully extended to transport problems in [13]. Finite element analysis for te LPS metod applied to te stationary Oseen equations can be found in [14, 34] and to convection-diffusion-reaction problems in [5, 7, 11, 35]. Te stabilization term of te LPS metod is based on a projection tat is defined on te finite element space wic approximates te solution and wic maps into a discontinuous space. Compared wit te standard Galerkin approac, te LPS metod provides additional control over parts of te fluctuation of te gradient. Te metod is weakly consistent and te construction sould lead to a consistency error tat does not spoil te optimal rate of convergence. Originally, te LPS metod was proposed as a two-level approac defining te projection spaces on coarser grids. Tis approac leads to additional couplings between neigboring mes cell and consequently, te sparsity of te matrix decreases. Tis drawback does not appear in te one-level approac, were bot spaces are defined on te same grid. In tis approac, te approximation spaces ave to be enriced in comparison wit te standard finite element spaces. Te additional degrees of freedoms tat are introduced by te enricment can be eliminated using static condensation. Altogeter, te one-level approac is, in our opinion, more appealing from te point of view of implementation and tis variant of te LPS metod 2

3 will be considered in tis paper. LPS stabilized finite element metods for te evolutionary Oseen problem were studied in [18]. In tis paper, te streamline derivative is stabilized wit te LPS metodology and an additional augmentation wit a grad-div stabilization term is used. Note tat tis metod is a different LPS metod tan studied ere. In te general case, error bounds were derived in [18] under te condition tat a certain measure of te mes size is of te same order as te square root of te viscosity. To avoid te restriction on te mes size for small viscosity, finite element spaces were ten considered tat satisfy a certain element-wise compatibility condition between te finite element velocity space and te projection space. Optimal error bounds for te pressure were not obtained in [18]. A similar LPS metod, applied to te time-dependent Navier Stokes equations, was analyzed in [8] and error estimates for te velocity in te continuous-in-time case were proved. An analysis of te fully discretized so-called ig-order term-by-term LPS metod was presented in [2]. As mentioned above, cgpk metods, wic treat te temporal derivative in a finite element manner, will be considered as temporal discretization. For incompressible flow problems, usually θ-scemes are used. Tese scemes are simple to implement, owever, tey are at most of second order, like te Crank Nicolson sceme or te fractional-step θ-sceme. In addition, tey do not allow an efficient adaptive time step control. Tere are only few studies, e.g., [24,27,3], wic consider iger order scemes, like diagonally implicit Runge Kutta DIRK metods, Rosenbrock Wanner ROW metods, or just cgp2. To te best of our knowledge, tere is no numerical analysis available for te first two classes of scemes applied to incompressible flow problems or even to convection-diffusion equations. Te situation is different for cgpk, wic are a class of finite element metods in time using discrete solution spaces wit continuous piecewise polynomials of degree less tan or equal to k and test spaces consisting of discontinuous polynomials of degree up to order k 1. Tis setup enables te performance of a standard time marcing algoritm and it does not require te solution of a global system in space and time as in space-time finite element metods. Te cgp metod in time for te eat equation was investigated in [1]. Optimal error estimates and super-convergence results were derived at te end point of te discrete time intervals. For nonlinear systems of ordinary differential equations in d space dimensions, te metods cgpk were studied in [37] even in an abstract Hilbert space setting. A-stability and optimal error estimates were proved. It was also sown tat cgpk metods ave an energy decreasing property for te gradient flow equation of an energy functional. In [5], transient convectiondiffusion-reaction equations were considered using cgpk in time combined wit LPS in space. Optimal a-priori error estimates were derived for te fully discrete sceme. It was sown numerically tat cgpk is super-convergent of order k 2 in te integrated norm and of order 2k at discrete time points. In addition, te obtained results were compared wit discontinuous Galerkin dg time stepping scemes. Numerical studies for te time-dependent Stokes equations in [23], te evolutionary Oseen equations in [4], and transient convection-diffusion-reaction equations in [5] sowed te expected orders of convergence for cgpk, k {1, 2}. Te dgk metod was analyzed for te transient Stokes equations in [1]. In addition, te iger order convergence of cgp2 compared wit te discontinuous Galerkin discretization dg1, bot metods possessing te same complexity, was demonstrated. An efficient adaptive time step control can be performed wit cgpk metods, e.g., as applied in [3] to transient convection-diffusion-reaction equations. Te adaptive time step control is based on a post-processed discrete solution, wic can be computed wit affordable costs. It was sown tat te adaptive time step control leads to lengts of te time steps tat properly reflect te dynamics of te solution. However, tere is also a certain drawback of cgpk metods for k 2: a coupled system of k equations as to be solved at eac time instance. Utilizing a clever construction proposed 3

4 in [37], te coupling is not strong, but it cannot be removed completely. Efficient solvers for tis coupled problem in case of te Navier Stokes equations were studied in [24], were a coupled multigrid metod wit Vanka-type smooters was used. Altogeter, cgpk is in our opinion an attractive alternative to θ-scemes since a iger order in time can be acieved and an efficient and inexpensive time step control is possible. Te goal of tis paper consists in studying te combination of te LPS metod in space wit te cgpk metod in time. Te numerical analysis will be performed for te transient Oseen equations 1. Tus, tis paper presents te first numerical analysis of a iger order time stepping sceme for an incompressible flow problem wit convection. In te continuousin-time case, optimal error bounds for velocity and pressure wit constants tat do not depend on te viscosity parameter ν are derived wit te assumption tat te solution is sufficiently smoot. In addition, error estimates for te fully discrete problem of order k 1 are proved, assuming, as in oter recently publised papers, tat te convective term does not depend on time. Numerical results sow tat te predicted order can be also observed in te case of timedependent convective terms. Te remainder of te paper is organized as follows: Section 2 introduces te basic notation, it presents some preliminaries, and te semi-discretization continuous-in-time of te LPS metod will be described. In Section 3, te error bounds for te semi-discrete problem are derived. Section 4 presents te error analysis of te fully discrete problem using a temporal discretization wit a cgpk metod. Numerical studies can be found in Section 5. 2 Preliminaries Trougout tis paper, standard notation and conventions will be used. For a measurable set G R d, te inner product in L 2 G, L 2 G d, and L 2 G d d will be denoted by, G. Te norm and te semi-norm in W m,p G are given by m,p,g and m,p,g, respectively. In te case p = 2, H m G, m,g, and m,g are written instead of W m,2 G, m,2,g, and m,2,g. If G = Ω, te index G in inner products, norms, and semi-norms will be omitted. Te dual pairing between a space Z and its dual Z will be denoted by,. Te temporal derivative of a function f is denoted by t f and te i-t temporal derivative by tf. i Te subspace of functions from H 1 Ω aving a vanising boundary trace is denoted by H 1 Ω wit H 1 Ω being its dual v,ϕ space wit te associated norm v 1 = sup ϕ H 1 Ω\{} ϕ. Let Z be a Banac space wit norm Z, ten te following spaces are defined L 2, t; Z := { v :, t Z : } vs 2 Z ds <, H 1, t; Z := { v L 2, t; Z : t v L 2, t; Z }, C, t; Z := {v :, t Z : v is continuous wit respect to time}, were t v is te temporal derivative of v in te sense of distributions. If t = T, ten te abbreviations L 2 Z, H 1 Z, and CZ are used and it will not be indicated weter it is a scalar-valued or vector-valued space. In order to derive a variational form of 1, te spaces V := H 1 Ω d, Q := L 2 Ω, X := { v L 2 V, t v L 2 V } and te bilinear form a u, p; v, q := ν u, v b u, v σu, v div v, p div u, q 4

5 are introduced. Ten, a variational form of 1 reads as follows: Find u X and p L 2 Q suc tat t ut, vt a ut, pt; vt, qt = ft, vt v L 2 V, q L 2 Q 2 for almost all t, T ] and u, = u. Note tat tis initial condition is well defined since functions belonging to X are continuous in time. If te initial condition u is different from, te velocity u can be decomposed in te form ut = u ψt, ψ X := {v X : v, = }. Ten for te given initial velocity field u, one as to find u = u ψt, wit ψt X, and p L 2 Q, were ψ, p is te solution of te problem wit t ψt, vt a ψt, pt; vt, qt = gt, vt g, v = f, v ν u, v b u, v σu, v. For tis reason, one can assume u =, wic will be done in te sequel. Note tat tis coice of te initial condition will result in error bounds tat do not contain contributions depending on u. Let Π : L 2 Ω d H div be te Leray projector tat maps eac function in L 2 Ω d onto its divergence-free part, were te Hilbert space H div is defined by H div = {v L 2 Ω d : v =, v n Ω = }. Te Stokes operator in Ω is defined by A : DA H div H div, A = Π, DA = H 2 Ω d V div, were te space V div = { v H 1 Ω d : v = } is equipped wit te inner product of H 1 Ω d. Let {T } be a family of quasi-uniform triangulations of Ω into compact d-simplices, quadrilaterals, or exaedra suc tat Ω = K T K. Te diameter of K T will be denoted by K and te mes size is defined by := max K. Let Y H 1 Ω be a finite element space of K T scalar, continuous, piecewise mapped polynomial functions over T. Te finite element space V for approximating te velocity field is given by V := Y d V. Te pressure is discretized using a finite element space Q Q of continuous or discontinuous functions wit respect to T. In tis paper, inf-sup stable pairs V, Q will be considered, i.e., tere is a positive constant β, independent of te triangulation, suc tat inf sup q Q \{} v V \{} div v, q v 1 q β >. 3 Since it is assumed tat te family of meses is quasi-uniform, te following inverse inequality olds v m,k C inv l m K v l,k, 4 for eac v V and l m 1, e.g., see [17, Tm ]. Te space of discretely divergence-free functions is denoted by V div = {v V : v, q = q Q }. Te linear operator A : V div V div is defined by A v, w = v, w w V div. 5 5

6 From 5, one concludes tat A 1/2 v = v, A 1/2 v = v v V div. 6 : L 2 Ω d V div, being te L2 -ortogonal projec- Te so-called discrete Leray projection Π div tion of L 2 Ω d onto V div, is given by Π div v, w = v, w w V div. 7 By definition, it follows tat tis projection is stable in te L 2 norm: Π div v v for all v L 2 Ω d. Te continuous-in-time standard Galerkin finite element metod applied to 2 consists in finding u H 1 V and p L 2 Q suc tat t u t, v a u t, p t; v, q = ft, v v V, q Q. In te convection-dominated case, it is well-known tat tis metod is unstable, unless is sufficiently small. Te use of a stabilized discretization is necessary. Tis paper concentrates on te one-level variant of te LPS metod in wic approximation and projection spaces are defined on te same mes. Let DK, K T, be local finitedimensional spaces and π K : L 2 K DK te local L 2 projection into DK. Te local fluctuation operator κ K : L 2 K L 2 K is given by κ K v := v π K v. It is applied componentwise to vector-valued and tensor-valued arguments. Te stabilization term S is defined by S u, v := κk u, κ K v K T µ K were {µ K }, K T, are non-negative constants. Tis kind of LPS metod gives additional control on te fluctuation of te gradient. Also oter variants of LPS metods are possible, e.g., by replacing in bot arguments of S, te gradient w by te derivative in te streamline direction b w or, even better [32, 33], by b K w, were b K is a piecewise constant approximation of b. But in tis metod, one as to add te grad-div term div u, div v to S, see [36]. For performing te numerical analysis, te linear operator C C v, w = te linear operator D : L 2 Ω V div te stabilized bilinear form K T µ K κ K v, κ K w K wit K, : V div V div wit v, w V div, 8 D q, w = div w, q w V div, 9 a u, p, v, q = a u, p; v, q S u, v on te product space V, Q, and te mes-dependent norm are introduced. v := {ν v 2 1 σ v 2 K T µ K κ K v 2,K } 1/2 6

7 It will be assumed tat b L L Ω H div Ω and bt = for almost all t [, T ]. Ten, a straigtforward calculation sows tat a v, q, v, q = v 2 v V, q Q. 1 Te semi-discrete LPS problem reads: Find u H 1 V and p L 2 Q suc tat for almost every t, T ] t u, v a u, p ; v, q = f, v v V, q Q. 11 For performing te analysis of LPS scemes, certain compatibility conditions between te approximation space and local projection space ave to be satisfied, see [34]. Assumption A1 Tere are interpolation operators j : H 2 Ω d V and i : H 2 Ω Q wit te approximation properties w j w,k K w j w 1,K C l K w l,k w H l K } d, 2 l r 1, q i q,k K q i q 1,K C l K q l,k q H l K, 2 l r, 12 for all K T. Te pressure interpolation operator i satisfies te ortogonality condition q i q, r K = q Q H 2 Ω, r DK. 13 Te pairs V /Q = Q r /P disc r 1 togeter wit DK = P r 1 K fulfill for r 2 assumption A1 if j is te usual Lagrangian interpolation operator and i te L 2 projection. Furter examples of inf-sup stable pairs V /Q, associated interpolation operators j and i, and projection spaces satisfying assumption A1 can be found in [36]. Assumption A2 Te fluctuation operator satisfies te approximation property κk q,k C l K q l,k K T, q H l K, l r. 14 For performing te numerical analysis, te steady-state Stokes problem ν u p = g in Ω, u = on Ω, 15 u = in Ω, will be considered. Te standard Galerkin approximation u, p V Q is te solution of te mixed finite element approximation to 15, given by ν u, v div v, p = g, v v V, 16 u, q = q Q. From [21, 25], it is known tat te following estimates old u u 1 C inf u v 1 ν 1 inf p q, 17 v V q Q p p C ν inf u v 1 inf p q, 18 v V q Q u u C inf u v 1 ν 1 inf p q. 19 v V q Q 7

8 It can be observed tat te error bounds for te velocity depend on a negative power of ν. As suggested in [19], a projection of u, p into V Q is used in te finite element analysis, were te bounds for te velocity are uniform in ν. Let u, p be te solution of 1 wit u H 1 V H l1 Ω d, p L 2 Q H l Ω, l 1, and define te rigt-and side of te Stokes problem 15 by g = f t u b u σu p, 2 suc tat u, is te solution of 15. Denoting te corresponding Galerkin approximation in V Q by s, l, one obtains from were te constant C does not depend on ν. u s u s 1 C l1 u l1, 21 l Cν l u l1, 22 Remark 1. Assuming te necessary smootness in time and considering 15 wit g = g i = i t f t u b u σu p, i 1, one can derive error bounds of form 21 and 22 also for i tu s g i and l g i, were s g i and l g i denote te solution of 16 wit rigt-and side g = g i. Hence, te estimates can be derived. i tu s g i i tu s g i 1 C l1 i tu l1, l g i Cν l i tu l1, 3 Error analysis for te continuous-in-time case In tis section, error bounds for velocity and pressure will be derived wit constants independent of ν for a sufficiently smoot solution. Te analysis follows te lines of [19]. Teorem 2. Let u, p be te solution of 2 and let u, p be te solution of 11. Assume b L L and te regularities u, p L 2 H r1 L 2 H r, t u L 2 H r. 23 Coosing te stabilization parameters of te LPS metod suc tat µ K 1 wit respect to te mes widt, ten te following error estimate olds for all t, T ] u u t 2 ν u u 2 L 2,t;L 2 σ u u 2 L 2,t;L 2 K T µ K κ K u u 2 L 2,t;L 2 K C 2r u 2 L 2,t;H r1 tu 2 L 2,t;H r p 2 L 2,t;H r were C = C σ, b L,t;L is independent of ν and., 24 Proof. Te proof of te error bound is based on te comparison of u, p wit te approximation s, l of te Stokes equations wit rigt-and side 2. Let e = u s, ten a direct calculation yields t e, v a e, p l, v, q = t u s, v b u s σu s, v p, v S s, v v V, q Q. 25 8

9 Taking v, q = e, p l in 25, one gets wit integrating by parts, using tat e as discrete divergence equal to zero, and 13 p, e = p, e = p i p, e = i p p, κ K e. Wit te Caucy Scwarz inequality and Hölder s inequality, it follows tat 1 d 2 dt e 2 ν e 2 σ e 2 µ K κ K e 2,K K T t u s e b u s e σ u s e 1/2 1/2 µ 1 K p i p 2,K µ K κ K e 2,K S s, e. K T K T Now, te term wit te stabilization as to be bounded. Te Caucy Scwarz inequality gives S s, e = S s u, e S u, e S 1/2 s u, s us 1/2 e, e S 1/2 u, us1/2 e, e. 26 Applying te stability of te fluctuation operator κ K and te coice µ K 1 of te stabilization parameters yields 1/2 S s, e C s u 1 κ K u κ K e 2,K, 27 K T suc tat 1 d 2 dt e 2 ν e 2 σ e 2 µ K κ K e 2,K K T t u s e b u s e σ u s e C 1/2 p i p s u 1 κ K u κ K e 2,K. K T Wit Young s inequality and iding terms on te left-and side, one obtains d dt e 2 2ν e 2 σ e 2 µ K κ K e 2,K K T C t u s 2 b 2 u s 2 σ 2 u s 2 C p i p 2 s u 2 1 κ K u Assuming now for t T te regularities 23, integrating 28 on, t, taking into account tat e =, since u =, and applying estimates 21, 12, and 14, one gets e t 2 2ν e 2 L 2,t;L 2 σ e 2 L 2,t;L 2 µ K κ K e 2 L 2,t;L 2 K K T C 2r u 2 L 2,t;H r1 tu 2 L 2,t;H r p 2 L 2,t;H r, 29 were C = C σ, b L,t;L is independent of ν and. Te final result is obtained by applying te triangle inequality to te left-and side of 24 and using 29 and 21. 9

10 In te next step of te error analysis, a bound for te pressure error will be derived. Teorem 3. Let te assumptions of Teorem 2 old and let ν 1, ten it olds p p L 2,t;L 2 C r t, T ], 3 were C = C β 1, u L 2,t;H r1, t u L 2,t;H r, p L 2,t;H r, σ, b L,t;L is independent of ν and. Proof. As usual, te bound is derived on te basis of te discrete inf-sup condition 3. It turns out tat te derivation requires in particular a bound for t e 1, wic will be proved first. By definition, it is t e, ϕ t e 1 = sup. ϕ H 1Ωd \{} ϕ Te first step consists in reducing te bound of t e 1 to a bound of A 1/2 t e. From [9, Lemma 3.11], it is known tat t e 1 C t e C A 1/2 Π t e, 31 were Π is te Leray projector introduced in Section 2. Applying [9, 2.15] gives A 1/2 Π t e C t e A 1/2 t e, 32 wit A defined in 5. From 31, 32, te symmetry of A, 6, and te inverse inequality 4, it follows tat t e 1 C t e C A 1/2 t e = C A 1/2 A 1/2 t e C A 1/2 t e = C A 1/2 t e C A 1/2 t e C A 1/2 t e. 33 Next, a bound for A 1/2 t e will be derived. Projecting te error equation 25 onto te discretely divergence-free space V div and using integration by parts yields t e, v ν e, v b e σe, v S e, v = t u s, v b u s σu s, v S s, v p i p, v. Utilizing 9, one finds p i p, v = D p i p, v, suc tat t e = νa e Π div b e σe C e Π div t u s b u s σu s C s 34 Π div D p i p. 1

11 Wit 8, te Caucy Scwarz inequality, 6, te L 2 stability of te fluctuation operator κ K, and µ K 1, one obtains for all v V div C v, A w C v = sup \{} w A 1/2 w V div = sup w V div \{} sup w V div \{} C sup w V div = C \{} 1/2 K T µ K κ K v, κ K A 1/2 w,k w K T µ K κ K v 2,K w 1/2 C A 1/2 w K T µ K κ K v,k 2 1/2 w w 1/2. 35 K T µ K κ K v 2,K Applying above argument to A 1/2 D p i p yields A 1/2 D p i p C p i p. 36 Definition 7 and te symmetry of A gives for any g L 2 Ω d te equality A 1/2 Π div g, v = g, A 1/2 v for all v V div. It follows wit v = A 1/2 Π div g V div and 6 tat and ence A 1/2 Π div g 2 g 1 A 1/2 In te next step, A 1/2 A 1/2 Π div g = g 1 A 1/2 Π div g A 1/2 Π div g g 1 g L 2 Ω d. 37 is applied to 34. Using 35, 36, and 37 leads to A 1/2 t e ν A 1/2 e b e σe 1 K T µ K κ K e 2,K 1/2 t u s 1 b u s σu s 1 1/2 µ K κ K s 2,K p i p. 38 K T 11

12 Taking te square of 38 and integrating on, t yields A 1/2 s e s 2 ds C ν 2 A 1/2 e s 2 ds K T µ K κ K e s 2,K ds p i ps 2 ds K T µ K κ K s s 2,K ds b e σe s 2 1 ds t u s s 2 1 ds b u s σu s s 2 1 ds. 39 It will be sown tat all te terms on te rigt-and side of 39 are of sufficiently ig asymptotic order, concretely tat tey are O 2r. Tis asymptotic beavior is obtained for te first and tird term directly from 29. Using te definition of te H 1 Ω d norm for te second term in 39, applying integration by parts, and utilizing Poincaré s inequality leads to Hence, it is b e σe 1 C b σ e. b e σe s 2 1 ds C e s 2 ds, suc tat te order of convergence O 2r can be again deduced from 29. For estimating t u s 1, te definition of te H 1 Ω d norm and Poincaré s inequality are applied, wic gives a bound of tis term by C t u s. Now, 21 is applied, see Remark 1, and using te regularity assumptions 23, te asymptotic bound for t u s 1 is O r. Consequently, te integral of its square is also bounded, giving s u s s 2 1 ds C 2r t u 2 L 2,t;H r. Te term involving te pressure is estimated wit 12. Arguing as in yields K T µ K κ K s s 2,K ds C s us 2 1 κ K us 2 ds C 2r u 2 L 2,t;H r1, were 21 and 14 were applied in te last inequality. Finally, arguing in te same way as for te second term gives from wat follows tat b u s σu s 1 C b σ u s, b u s σu s s 2 1 ds C u s s 2 ds. 12

13 Te bound for tis term is finised by applying 21. Combining te estimates for 39 wit 33, it is sown tat s e s 2 1 ds = O 2r. 4 Using now te discrete inf-sup condition 3 and 25, one obtains β p i p ν e b e σe 1 t e 1 C t u s 1 b u s σu s 1 1/2 C µ K κ K s 2,K p i p l. K T Taking te square and integrating on, t leads to β 2 p i ps 2 ds C ν 2 e s 2 ds s e s 2 1 ds s u s s 2 1 ds b e σe s 2 1 ds K T µ K κ K e s 2,K ds K T µ K κ K s s 2,K ds K T µ K κ K e 2,K 1/2 b u s s σu s s 2 1 ds p i ps 2 ds l s 2 ds Arguing exactly as in te estimates of te terms wic are on te rigt-and side of 39, using 4 for bounding se s 2 1 ds, 22 to bound te last term, and applying finally te triangle inequality, proves 3. 4 Error analysis for te fully discrete metod wit cgpk Te continuous Galerkin Petrov metod is studied as temporal discretization. Consider a partition = t < t 1 <... < t N = T of te time interval I := [, T ] and set I n = t n 1, t n ], τ n = t n t n 1, n = 1,... N, and τ := max 1nN τ n. For a given non-negative integer k, te time-continuous and time-discontinuous velocity spaces are defined as follows X c k := {u CV : u In P k I n, V }, X dc k := { u L 2 V : u In P k I n, V }, and time-continuous and time-discontinuous pressure spaces are given by Yk c := {q CQ : q In P k I n, Q }, Yk dc := { q L 2 Q : q In P k I n, Q }, for n = 1,..., N. Here, P k I n, W := { u : I n W : ut = 13 } k U i t i, t I n, U i W, i i=.

14 denotes te space of W -valued polynomials of order k in time. Te functions in te spaces Xk dc and Yk dc are allowed to be discontinuous at te nodes t n. Below, te combination of te LPS metod as spatial discretization and te cgpk time stepping sceme is denoted by LPS/cGP. Denote by Xk, c := Xc k X te subspace of Xk c wit zero initial condition and introduce a bilinear form b given by b u, p; v, q := Te LPS/cGP metod reads as follows: Find u,τ Xk, c and p,τ Yk c suc tat b u,τ, p,τ ; v,τ, q,τ = [ t u, v a u, p; v, q ] dt. f, v,τ dt v,τ X dc k 1, q,τ Y dc k 1, 41 were te index, τ refers to te discretization in space and time. Te associated continuous problem is given by: Find u X and p L 2 Q suc tat [ t ut, vt a ut, pt; vt, qt ] dt = ft, vt dt 42 for all v L 2 V, q L 2 Q. For a function w, wic is smoot on eac time interval I n, te operator π k 1 is defined by π k 1 w In t = k w t n,i L n,i t, 43 i=1 were t n,i denote te Gaussian quadrature points on I n and L n,i P k 1 I n are te associated Lagrange basis functions. From 43, it can be concluded tat π k 1 w,τ Xk 1 dc for all w,τ Xk c and π k 1 q,τ Yk 1 dc for all q,τ Yk c. Furtermore, one as for all w,τ Xk c tat w,τ t π k 1 w,τ tt j dt =, j =,..., k 1, n = 1,..., N, 44 I n were denotes te zero element in V. Te finite element analysis considers te mes-dependent norm v cgp := π k 1 v 1/2 2 dt 1 vt 2 2. It was already observed in [5] tat cgp is on Xk c Xdc k not only a semi-norm but a norm. For completeness of presentation, te corresponding arguments are repeated ere. Te first term inside te definition of v cgp guarantees tat v cgp = results in a function v wic is on eac time interval I n given by L n k tϕ x, were L n k is te transformed k-t Legendre polynomial on I n and ϕ V. Due to vt = and L N k T = 1 te function v vanises on te last time interval I N. Te continuity of v on I gives ten vt N 1 =. By recursion, one obtains v = on I and ence cgp is a norm. Te following lemma proves a property of te bilinear form b tat will be used in te derivation of te error bounds for te approximation to te velocity. 14

15 Lemma 4. Assume tat b and σ are constant wit respect to time. Ten, tere exists a constant C > independent of ν,, and τ suc tat b v,τ, q,τ ; πk 1 v,τ, π k 1 q,τ = v,τ 2 cgp v,τ, q,τ X dc k Y dc k olds true. Proof. It is b v,τ, q,τ ; πk 1 v,τ, π k 1 q,τ = [ t v,τ, π k 1 v,τ a v,τ, q,τ ; πk 1 v,τ, π k 1 q,τ ] dt. Using te fact tat te convection and reaction are time-independent functions, taking into account tat = [ q,τ, div π k 1 v,τ π k 1 q,τ, div v,τ ] dt and 1, one obtains = [ π k 1 q,τ, div π k 1 v,τ π k 1 q,τ, div π k 1 v,τ ] dt =, a v,τ, q,τ ; πk 1 v,τ, π k 1 q,τ dt T a πk 1 v,τ, π k 1 q,τ ; πk 1 v,τ, π k 1 q,τ dt = π k 1 v,τ 2 dt. Concerning te first term, it is noted tat t v,τ is a discontinuous function in time of degree k 1. Using v,τ = yields t v,τ, π k 1 v,τ dt = 1 T t v,τ, v,τ dt = 2 = 1 v,τ T 2 2. d v,τ 2 dt dt Te derivation of error bounds makes use of a time interpolation w CH of a sufficiently smoot function w, were H can be eiter a velocity space V or a pressure space Q, and w In P k I n, H, defined by wt n 1 = wt n 1, wt n = wt n, wt wt, zt dt =, 45 I n for all z P k 2 I n, H. Te standard interpolation error estimate 1/2 1/2 w w 2 m dt Cτn k1 w k1 2 m dt 46 I n I n olds true for m {, 1} and all time intervals I n, n = 1,..., N. 15

16 Teorem 5. Assume tat te spaces V, Q satisfy Assumptions A1 and A2, µ K 1 for all K T, and ν 1. Let u, p be te solution of 42 and u,τ, p,τ te solution of 41. Furter, assume tat te solution u, p is smoot enoug suc tat all te norms on te rigtand side of 47 are bounded. Ten, tere exists a positive constant C independent of ν,, and τ suc tat te error estimate u u,τ cgp C r u L2 H r1 u H 1 H r p L 2 H r ut r1 is valid. Cτ k1 u H k1 H 1 47 Proof. Te error analysis starts by decomposing te error e,τ = u,τ u into θ := s u and ξ,τ := u,τ s, wit te velocity solution s of 16, were g in 2 is defined as ν ũ. Ten, it is u,τ u = e,τ = θ ξ,τ. For te discrete error ξ,τ Lemma 4 gives ξ,τ 2 cgp = b ξ,τ, p,τ ; π k 1 ξ,τ, π k 1 p,τ. 48 Applying a straigtforward calculation yields b ξ,τ, p,τ ; π k 1 ξ,τ, π k 1 p,τ = T t u t s, π k 1 ξ,τ dt ν u s, π k 1 ξ,τ dt T b u s, π k 1 ξ,τ dt σu s, π k 1 ξ,τ dt p, π k 1 ξ,τ dt S s, π k 1 ξ,τ dt. 49 Te six terms on te rigt-and side ave to be bounded. For te first one, te error is split in two terms T t u t s, π k 1 ξ,τ dt = t u ũ, π k 1 ξ,τ dt t ũ s, π k 1 ξ,τ dt. 5 Integration by parts and using 45 yield for te first term on te rigt-and side of 5 t u ũ, π k 1 ξ,τ dt = N I n u ũ, t π k 1 ξ dt u ũ, π k 1 ξ τ tn t n 1 =. 51 For te second term on te rigt-and side of 5, te application of te Caucy Scwarz 16

17 inequality and 21 gives t ũ s, π k 1 ξ,τ dt N t ũ t s π k 1 ξ,τ dt I n N 1/2 N t ũ t s 2 dt π k 1 ξ,τ 2 dt I n N 1/2 N C r t ũ 2 r dt σ π k 1 ξ,τ 2 dt I n I n 1/2 1/2 C r u H1 H r ξ,τ cgp, 52 were in te last estimate te inequality ũ H1 H r C u H1 H r was applied. Tus, from 5, 51, and 52 one derives te bound for te first term on te rigt-and side of 49 t u t s, π k 1 ξ,τ dt C r u H1 H r ξ cgp. 53 To bound te tird term on te rigt-and side of 49, te error splitting T b u s, π k 1 ξ,τ dt = b u ũ, πk 1 ξ,τ dt b ũ s, π k 1 ξ,τ dt is used. Ten, applying 46 and 21 yields b u s, π k 1 ξ,τ dt b u ũ 1 ũ s 1 πk 1 ξ,τ dt N 1/2 N 1/2 C τn 2k2 u k1 2 1 dt 2r ũ 2 r1 dt I n I n N π k 1 ξ 2 dt I n 1/2 Cτ k1 u H k1 H 1 C r u L 2 H r1 ξ,τ cgp. 54 Arguing exactly as before gives for te second and te fourt term on te rigt-and side of 49 ν u s, π k 1 ξ,τ dt σu s, π k 1 ξ,τ dt Cν 1/2 σ 1/2 r u L 2 H r1 Cν 1/2 σ 1/2 τ k1 u H k1 H 1 ξ,τ cgp

18 To bound te fift term on te rigt-and side of 49 observe tat p, π k 1 ξ,τ dt = p, π k 1 ξ,τ dt = p, π k 1 ξ,τ dt, since te time projection π k 1 and te divergence commute. In addition, it is i p, π k 1 ξ,τ dt = π k 1 i p, π k 1 ξ,τ dt I n I n = π k 1 i p, ξ,τ dt =, 56 I n since s as discrete divergence equal to zero and te relation I n u,τ, q,τ dt = olds by definition for all q,τ Yk 1 dc. Tus, for te fift term on te rigt-and side of 49, integration by parts wit respect to space, applying te ortogonality condition 13, using 56, µ K 1, and 12 lead to = p, π k 1 ξ,τ dt i p p, π k 1 ξ,τ dt = i p p, κ K π k 1 ξ,τ K dt k T 1/2 1/2 µ 1 K i p p 2,K µ K κ K π k 1 ξ,τ 2,K dt K T K T 1/2 T C i p p 2 dt ξ,τ cgp C r p L 2 H r ξ,τ cgp. 57 Finally, to bound te last term on te rigt-and side of 49, te following decomposition is considered S s, π k 1 ξ,τ dt = S s ũ, π k 1 ξ,τ dt S ũ u, π k 1 ξ,τ dt S u, π k 1 ξ,τ dt. 58 For te first term on te rigt-and side of 58, te L 2 stability of te fluctuation operator κ K, µ K 1, and 21 are applied to obtain S s ũ, π k 1 ξ,τ dt 1/2 T 1/2 µ K κ K s ũ 2,K µ K κ K π k 1 ξ,τ 2,K dt K T K T 1/2 T 1/2 T µ K κ K s ũ 2,K dt π k 1 ξ,τ 2 dt K T C r u L 2 H r1 ξ,τ cgp

19 Applying te stability of te fluctuation operator κ K, µ K 1, and 46 gives for te second term on te rigt-and side of 58 S ũ u, π k 1 ξ,τ dt 1/2 µ K κ K ũ u 2,K µ K κ K π k 1 ξ,τ 2,K K T K T 1/2 T 1/2 T µ K κ K ũ u 2,K dt π k 1 ξ,τ 2 dt K T 1/2 dt Cτ k1 u H k1 H 1 ξ,τ cgp. 6 To finis te estimate of te last term on te rigt-and side of 49, te Caucy Scwarz inequality, te approximation properties 14 of te fluctuation operator κ K, and µ K 1 are used to get S u, π k 1 ξ,τ dt 1/2 T 1/2 µ K κ K u 2,K µ K κ K π k 1 ξ,τ 2,K dt K T K T 1/2 T 1/2 T µ K κ K u 2,K dt π k 1 ξ,τ 2 dt K T C r u L 2 H r1 ξ,τ cgp. 61 Inserting 59, 6, and 61 in 58 gives S s, π k 1 ξ,τ dt C r u L2 H r1 Cτ k1 u H k1 H 1 ξ,τ cgp. 62 Inserting 49 in 48 and utilizing 53, 54, 55, 57, and 62 lead to ξ,τ cgp C r[ ] u L2 H r1 u H1 H r p L2 H r Cτ k1 u H k1 H Applying te triangle inequality, te bound 21, and te interpolation error estimates in time gives te statement of te teorem. Arguing similarly as in [5, Tm. 4.4], one can prove te following teorem. Teorem 6. Under te assumptions of Teorem 5, te following error estimate is valid ut u,τ t 2 dt wit C independent of ν,, and τ. 1/2 C1 T 1/2 r [ ] u L 2 H r1 u H 1 H r p L 2 H r C1 T 1/2 τ k1 u H k1 H 1, 64 19

20 Proof. Denoting as before ξ,τ = u,τ s. Applying te ideas leading to 63 on [, t n ], n = 1,..., N, instead of on [, T ] gives n π k 1 ξ,τ t 2 dt 1 2 ξ,τ t n 2 C 2r[ u 2 L 2 H r1 u 2 H 1 H r p 2 L 2 H r ] Cτ 2k2 u 2 H k1 H 1, were te norms on te rigt-and side were extended from [, t n ] to [, T ] by using tat tese norms do not decrease by extending te time interval. After aving neglected te non-negative integral on te left-and side and aving multiplied by τ n, te summation over n = 1,..., N yields N N τ n ξ,τ t n 2 τ n C 2r[ ] u 2 L 2 H r1 u 2 H 1 H r p 2 L 2 H r N τ n Cτ 2k2 u 2 H k1 H Since ξ,τ is a piecewise polynomial of degree less tan or equal to k in time, te equivalence of all norms in finite-dimensional spaces gives tn tn ξ,τ 2 dt C k π k 1 ξ,τ t 2 dt τ n ξ,τ t n 2, t n 1 t n 1 were C k depends on te polynomial degree k but it is independent of τ n and. Hence, applying 65 and 63 leads to T N tn u,τ t s t 2 dt C k π k 1 ξ,τ t 2 dt τ n ξ,τ t n 2 t n 1 T N C k π k 1 ξ,τ t 2 dt τ n ξ,τ t n 2 C1 T 2r[ u 2 L 2 H r1 u 2 H 1 H r p 2 L 2 H r C1 T τ 2k2 u 2 H k1 H Now, te statement of te teorem follows by applying te triangle inequality and te time interpolation error estimates 46 togeter wit 21. Teorem 7. Let te assumptions of Teorem 5 old and let in addition u, p be smoot enoug suc tat te norms on te rigt-and side of 67 are bounded. Ten, tere exists a positive constant C independent of ν,, and τ suc tat te error estimate 1/2 T π k 1 p,τ t pt 2 dt C1 T r [ ] u H1 H r1 u H2 H r p H1 H r ] olds. C1 T τ k 1 τ u H k2 H 1 Cτ k1 p H k1 L 2 C r [ ] u L 2 H r1 u H 1 H r p H 1 H r 67 2

21 Proof. A straigtforward calculation sows tat for all v,τ Xk 1 dc and q,τ Yk 1 dc, it olds b u,τ s, p,τ ; v,τ, q,τ = = t ξ,τ, v,τ dt σξ,τ, v,τ dt t u s, v,τ dt ν ξ,τ, v,τ dt b u s, v,τ dt S s, v,τ dt From tis equation, one obtains = p,τ i p, v,τ dt p i p, v,τ dt b ξ,τ, v,τ dt t s u, v,τ dt v,τ, p,τ dt ν u s, v,τ dt p, v,τ dt. σu s, v,τ dt t ξ,τ, v,τ dt b s u, v,τ dt σξ,τ, v,τ dt ν s u, v,τ dt b ξ,τ, v,τ dt S ξ,τ, v,τ dt ν ξ,τ, v,τ dt S ξ,τ, v,τ dt σ s u, v,τ dt S s, v,τ dt. 68 To derive te error estimates, te Gaussian quadrature rule wit k points will be used for te numerical integration of te time integral. Hence, one as q 2k 1 t dt = N τ n 2 k ˆω i q 2k 1 t n,i 69 i=1 for all q 2k 1 P 2k 1 I n, were t n,i denote te corresponding quadrature points on I n and ˆω i are te weigts of te Gaussian formula on 1, 1 wic satisfy ˆω i >. Let t n, = t n 1 be an additional point. Using te discrete inf-sup condition 3, one can construct w,τ P k I n, V suc tat β π k 1 p,τ t n,i i p t n,i 2 π k 1 p,τ t n,i i p t n,i, w,τ t n,i, 7 w,τ t n,i 1 = π k 1 p,τ t n,i i p t n,i. 71 Since w,τ P k I n, V, it follows tat π k 1 w,τ P k 1 I n, V. Setting v,τ = π k 1 w,τ and 21

22 using 43, 44, one obtains p,τ i p, v,τ dt = = N p,τ i p, π k 1 w,τ dt I n N π k 1 p,τ i p, w,τ dt I n β π k 1 p,τ i p 2 dt, 72 were te exactness of te quadrature rule for polynomials of degree 2k 1, te positivity of te quadrature weigts, 69, and 7 were used. Setting v,τ = π k 1 w,τ in 68, using 72, te assumption tat b and σ are constants wit respect to time, and 43, it follows tat β π k 1 p,τ i p 2 dt = p,τ i p, π k 1 w,τ dt π k 1 p i p, π k 1 w,τ dt νπ k 1 ξ,τ, π k 1 w,τ dt σπ k 1 ξ,τ, π k 1 w,τ dt t s u, π k 1 w,τ dt π k 1 b s u,τ, π k 1 w,τ dt t ξ,τ, π k 1 w,τ dt b π k 1 ξ,τ, π k 1 w,τ dt S π k 1 ξ,τ, π k 1 w,τ dt νπ k 1 s u, π k 1 w,τ dt σπ k 1 s u, π k 1 w,τ dt S π k 1 s, π k 1 w,τ dt. 73 Te sevent term on te rigt-and side of 73 is decomposed in te form t s u, π k 1 w,τ dt = t s ũ, π k 1 w,τ dt t ũ u, π k 1 w,τ dt. For te second term on te rigt-and side, integrating by parts wit respect to time and using 45 yield t ũ u, π k 1 w,τ dt = N I n ũ u, t π k 1 w,τ dt u ũ, π k 1 w,τ tn t n 1 =. 22

23 It follows tat t s u, π k 1 w,τ dt were Poincaré s inequality was applied in te last line. Using 69, 43, and 71 gives π k 1 w,τ 2 dt = N τ n 2 C = C = C N τ n 2 N τ n 2 C t s ũ π k 1 w,τ dt t s ũ π k 1 w,τ dt, k ˆω i π k 1 w t n,i 2 i=1 k ˆω i w t n,i 2 i=1 k ˆω i π k 1 p,τ t n,i i p t n,i 2 i=1 π k 1 p,τ i p 2 dt, 74 were t n,i, i = 1,..., k, denote te node of Gaussian quadrature on I n and ˆω i, i = 1,..., k, are te corresponding weigts on [ 1, 1]. Applying 74 yields t s u, π k 1 w,τ dt C t s ũ 2 dt β π k 1 p,τ i p 2 dt. 12 Arguing in te same way for te rest of te terms on te rigt-and side of 73 leads to π k 1 p,τ i p 2 dt [ C π k 1 p i p 2 dt t ξ,τ 2 1 dt π k 1 ξ,τ 2 dt t s ũ 2 dt ν π k 1 s u 2 dt b σ π k 1 s u 2 dt ] µ K κ K π k 1 s 2,K dt. 75 K T Now, te terms on te rigt-and side of 75 need to be bounded. Te estimates for te tird term follows from Teorem 5. In te following, te L 2 stability of te projection π k 1 and te interpolation operator wit respect to time, i.e., π k 1 v C v and ṽ C v will be 23

24 often used. For te first term on te rigt-and side of 75, applying 46 and 13 gives T T π k 1 p i p 2 dt C p p 2 dt p i p 2 dt Cτ 2k2 p k1 2 dt C 2r p 2 r dt C τ 2k2 p 2 H k1 L 2 2r p 2 L 2 H r. For bounding te second term on te rigt-and side of 75, one first observes tat tξ,τ 1 dt tξ,τ dt. Now, since it is assumed tat b and σ are independent of t, te error bounds for ξ,τ can also be applied to its time derivative so tat applying 66 to t ξ,τ leads to t ξ,τ 2 dt C1 T 2r[ t u 2 L 2 H r1 tu 2 H 1 H r tp 2 L 2 H r C1 T τ 2k t u 2 H k1 H 1. For te truncation errors involving s u te last four terms, one argues as in Teorem 5 to get t s ũ 2 dt C 2r u 2 H 1 H r, ν π k 1 s u 2 dt Cν 2r u 2 L 2 H r1 τ 2k2 u 2 H k1 H 1, b σ s u 2 dt C 2r u 2 L 2 H r τ 2k2 u 2 H k1 H 1. Te bound for te last term similarly as in te estimates uses te error splitting wit respect to space and time, te L 2 stability of te fluctuation operator κ K, µ K 1, and te approximation properties of κ K. One obtains µ K κ K π k 1 s 2,K dt K T 3 κ K s ũ 2,K dt 3 κ K ũ u 2,K dt K T K T 3 κ K u 2,K dt K T C 2r u 2 L 2 H r1 τ 2k2 u 2 H k1 H r1. Te statement of te teorem follows by collecting te bounds for all terms on te rigtand side of 75, and by applying te triangle inequality and te bounds 13 and 46 for te interpolation errors in space and time. Remark 8. Instead of using tξ,τ 2 1 dt tξ,τ 2 dt, one could use t ξ,τ 2 1 dt C 24 A 1/2 t ξ,τ 2 dt ]

25 and ten argue as in te proof of Teorem 3. However, since it is assumed tat b is timeindependent, te proof presented above is sorter altoug it requires a iger regularity of te solution. 5 Numerical studies Two examples will be presented tat support te teoretical results obtained in te previous sections. In te first example, an analytical solution is considered and very small time steps are applied to support te error analysis of Section 3. In te second example te solution is polynomial in te space suc tat te approximation will be exact in te spatial part and te discretization error in time dominates. Tis example will support te analytical results from Section 4. A tird example presents a brief comparison of te metod studied in tis paper wit a different stabilized metod for te evolutionary Oseen equations. All simulations were performed on uniform quadrilateral grids were te coarsest grid level 1 is obtained by dividing te unit square into four squares. Mapped finite element spaces [17] were used, were te enriced spaces on te reference cell ˆK = [ 1, 1] 2 are given by Q bubble r ˆK := Q r ˆK span { 1 ˆx 2 11 ˆx 2 2ˆx r 1 i, i = 1, 2 }. Te combination Q bubble r ˆK wit DK = P r 1 K provides for r 2 suitable spaces for LPS metods, see [36]. Te simulations were performed wit te code MooNMD [26]. Example 9. An example wit negligible temporal error. Consider te Oseen problem 1 wit Ω =, 1 2, ν = 1 1, b = u, σ = 1, and T = 1. Te rigt-and side f and te initial condition u were cosen suc tat sinπx sinπy ut, x, y = sint, cosπx cosπy pt, x, y = sint sinπx cosπy 2 π is te solution of 1 equipped wit non-omogeneous Diriclet boundary conditions. Tis example studies te convergence order wit respect to space. To tis end, te time discretization sceme cgp2 wit te small time step lengt τ = 1/128 was used. Numerical studies concerning te coice of stabilization parameters for convection-dominated problems suggest tat a good coice is µ K, 1, e.g., see [6]. Based on tese studies and our own experience, te stabilization parameters were set to be µ K =.1. Te convergence plots for simulations wit te finite element spaces V /Q = Q bubble 3 /P disc 2 and te projection space DK = P 2 K are presented in Figure 1 and Table 1. One can see fourt order convergence for te L 2 L 2 norm and te L 2 norm at te final time. For all oter norms on te left-and side of 24 and te L 2 L 2 norm of pressure, tird order of convergence can be observed. It can be seen in Figure 1 and Table 1 tat κ K u u L 2 L 2 is te dominant term among te velocity errors on te left-and side of 24. Altogeter, te order of convergence is exactly as predicted in 24 and 3. Example 1. An example wit dominant temporal error. Let Ω =, 1 2, ν = 1 1, b = u, σ = 1, T = 1 and consider te Oseen equations 1 wit te prescribed solution sin4ty u =, pt, x, y = cos4tx.5 sin4t2y 1. costx 25

26 error u1 u 1 ν 1/2 u u L 2 L 2 u u L 2 L κ K u u L 2 L p p L 2 L level Figure 1: Example 9: Convergence of various errors wit respect to te spatial mes widt. Table 1: Example 9: Various errors wit respect to te spatial mes widt, e = u u, te order is computed from te results of te finest levels. level e 1 ν 1/2 e L 2 L 2 e L 2 L 2 κ K e L 2 L 2 p p L 2 L e e e e e e e e e e e e e e e e e e e e e e e e e-7 order In tis example, te spaces V /Q = Q bubble 2 /P disc 1 and te projection space DK = P 1 K were considered. Te mes consisted of squares. Note tat for any time t te solution can be represented exactly by functions from te finite element spaces V and Q. Hence, all occurring errors will result from te temporal discretization. Figure 2 and Table 2 report te order of convergence for te metods cgpk, k {2, 3, 4}, in combination wit te LPS metod. One can observe te predicted convergence order k 1 for te errors estimated in 47 and 64. Also for te pressure, order k 1 can be seen altoug estimate 67 predicts only order k. Example 11. Comparison wit a grad-div stabilized metod. Tis example provides a brief comparison of a LPS/cGP metod wit anoter stabilized sceme for te evolutionary Oseen problem tat was studied in te literature. Te metod for comparison is te grad-div stabilization wic is analyzed for te evolutionary Oseen equations in [19]. Te same example as in [19] was used wit te analytical solution sinπx.7 sinπy.2 u = cost, cosπx.7 cosπy.2 p = costsinx cosy cos1 1 sin1. Numerical results for ν = 1 6, b = u, σ = 1, Ω =, 1 2, and T = 5 are presented. Bot scemes were applied wit respective standard configurations tat are comparable. For te LPS/cGP metod, cgp2 was used as temporal discretization. Since tis is a tird 26

27 error time level m τ 3 τ 4 τ 5 cgp2: u u L 2 L 2 cgp2: u u cgp cgp2: p p L 2 L 2 cgp3: u u L 2 L 2 cgp3: u u cgp cgp3: p p L 2 L 2 cgp4: u u L 2 L 2 cgp4: u u cgp cgp4: p p L 2 L 2 Figure 2: Example 1: Convergence of various errors wit respect to te time step, were te time step is given by τ =.1 2 m1. Table 2: Example 1: u u cgp, te order is computed from te results of te finest levels. time level cgp2 cgp3 cgp e e e e e e e e e e e e e e e e e e e e e e e e-13 order order metod, it is natural to use also a tird order spatial discretization. Our coice was V /Q = Q bubble 3 /P disc 2, like in Example 9. Also, te same stabilization parameter µ K =.1 was utilized. Since spatial and temporal discretization are of te same order, convergence studies require te time step to be alved if te grid is refined once. Te grad-div stabilization is used in practice wit standard pairs of finite element spaces and simple temporal discretizations. Tis metod does not require suc special spaces as LPS metods. We used also a tird order metod in space, namely te Taylor Hood pair Q 3 /Q 2. Te spatial discretizations of bot stabilized metods possess a similar number of degrees of freedom on te same meses. Wit respect to te temporal discretization, te Crank Nicolson sceme was used. Tis sceme is only of second order. To account for tis order difference, te time step was refined by te factor of 8 wenever te grid was refined once factor 2 of te mes widt. Te time step on level one mes cell was set to be.4 for bot metods. Te stabilization parameter for te grad-div stabilization was cosen to be te same as in te numerical studies in [19]. Results wit respect to some standard errors are presented in Tables 3 and 4. It can be observed tat bot metods possess a similar accuracy and te pressure errors as well as te dominant velocity errors are of at least tird order. In particular, te LPS/cGP metod is competitive. A compreensive numerical assessment of stabilized metods for convection-dominated in- 27

28 Table 3: Example 11: Various errors for te LPS metod wit cgp2, e = u u, te order is computed from te results of te finest levels. level e 5 ν 1/2 e L 2 L 2 e L 2 L 2 p p L 2 L e e e e e e e e e e e e e e e e e e e e-8 order Table 4: Example 11: Various errors for te grad-div stabilization wit Crank Nicolson sceme from [19], e = u u, te order is computed from te results of te finest levels. level e 5 ν 1/2 e L 2 L 2 e L 2 L 2 p p L 2 L e e e e e e e e e e e e e e e e e e e e-8 order compressible flow problems is beyond te scope of te present paper and a future topic of researc. 6 Summary Tis paper analyzed a combination of iger order continuous Galerkin Petrov scemes in time wit te one-level variant of te LPS metod in space applied to te transient Oseen equations. Te continuous-in-time case and te fully discrete situation were considered. Optimal error bounds for velocity and pressure were obtained wit constants tat do not depend on te viscosity parameter ν. Te teoretical results were confirmed by numerical simulations. References [1] N. Amed, S. Becer, and G. Matties. Higer-order discontinuous Galerkin time stepping and local projection stabilization tecniques for te transient Stokes problem. Comput. Metods Appl. Mec. Engrg., 313:28 52, 217. [2] N. Amed, T. Cacón Rebollo, V. Jon, and S. Rubino. Analysis of a full space-time discretization of te Navier-Stokes equations by a local projection stabilization metod. IMA J. Numer. Anal., 373: , 217. [3] N. Amed and V. Jon. Adaptive time step control for iger order variational time discretizations applied to convection-diffusion-reaction equations. Comput. Metods Appl. Mec. Engrg., 285:83 11,