Postprocessing finite-element methods for the Navier–Stokes Equations: the Fully discrete case
Abstract
An accuracy-enhancing postprocessing technique for finite-element discretizations of the Navier–Stokes equations is analyzed. The technique had been previously analyzed only for semidiscretizations, and fully discrete methods are addressed in the present paper. We show that the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising from any convergent time-stepping procedure. Further refined bounds are obtained when the timestepping procedure is either the backward Euler method or the two-step backward differentiation formula.
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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. NUMER. ANAL.c 2008 Society for Industrial and Applied Mathematics Vol. 47, No. 1, pp. 596–621 POSTPROCESSING FINITE-ELEMENT METHODS FOR THE NAVIER–STOKES EQUATIONS: THE FULLY DISCRETE CASE∗ JAVIER DE FRUTOS†,BOSCOGARC ´ IA-ARCHILLA‡,AND JULIA NOVO§ Abstract. An accuracy-enhancing postprocessing technique for finite-element discretizations of the Navier–Stokes equations is analyzed. The technique had been previously analyzed only for semidiscretizations, and fully discrete methods are addressed in the present paper. We show that the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising from any convergent time-stepping procedure. Further refined bounds are obtained when the timestepping procedure is either the backward Euler method or the two-step backward differentiation formula. Key words. Navier–Stokes equations, mixed finite-element methods, time-stepping methods, optimal regularity, error estimates, backward Euler method, two-step BDF AMS subject classifications. 65M60, 65M20, 65M15, 65M12 DOI. 10.1137/070707580 1. Introduction. The purpose of the present paper is to study a postprocessing technique for fully discrete mixed finite-element (MFE) methods for the incompressible Navier–Stokes equations ut−Δu+(u·∇)u+∇p=f,(1.1) div(u)=0,(1.2) in a bounded domain Ω ⊂Rd(d=2,3) with smooth boundary subject to homogeneous Dirichlet boundary conditions u=0on∂Ω. In (1.1), uis the velocity field, p the pressure, and fa given force field. We assume that the fluid density and viscosity have been normalized by an adequate change of scale in space and time. For semidiscrete MFE methods the postprocessing technique has been studied in [2, 3, 18] and is as follows. In order to approximate the solution uand pcorresponding to a given initial condition u(·,0) = u0,(1.3) at a time t∈(0,T], T>0, consider first standard MFE approximations uhand phto the velocity and pressure, respectively, solutions at time t∈(0,T] of the corresponding discretization of (1.1)–(1.3). Then compute MFE approximations ˜uhand ˜phto the ∗Received by the editors November 8, 2007; accepted for publication (in revised form) September 5, 2008; published electronically December 31, 2008. http://www.siam.org/journals/sinum/47-1/70758.html †Departamento de Matem´atica Aplicada, Universidad de Valladolid, 47011 Valladolid, Spain ([email protected]). This author’s research was financed by DGI-MEC under project MTM2007- 60528 (cofinanced by FEDER funds) and Junta de Castilla y Le´on under grants VA079A06 and VA045A06. ‡Departamento de Matem´atica Aplicada II, Universidad de Sevilla, 41002 Sevilla, Spain (bosco@ esi.us.es). This author’s research was supported by DGI-MEC under project MTM2006-00847. §Departamento de Matem´aticas, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain (julia. nov[email protected]). This author’s research was financed by DGI-MEC under project MTM2007-60528 (cofinanced by FEDER funds) and Junta de Castilla y Le´on under grants VA079A06 and VA045A06. 596 Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. POSTPROCESSING FINITE-ELEMENT METHODS 597 solution ˜uand ˜pof the following Stokes problem, −Δ˜u+∇˜p=f−d dtuh−(uh·∇)uhin Ω,(1.4) div (˜u)=0 in Ω,(1.5) ˜u=0 on∂Ω.(1.6) The MFE on this postprocessing step can be either the same MFE over a finer grid or a higher-order MFE over the same grid. In [2, 18] it is shown that if the errors in the velocity (in the H1norm) and the pressure of the standard MFE approximations uhand phare O(t−(r−2)/2hr−1), r=2,3,4, for t∈(0,T], then those of the postprocessed approximations ˜uhand ˜phare O(t−(r−1)/2hr|log(h)|), that is, an O(h|log(h)|) improvement with respect the standard MFE error bound) (see precise statement on Theorem 2.2 below), and if r≥3 (finite elements of degree at least two), the O(h|log(h)|) improvement is also obtained in the L2norm of the velocity. In practice, however, the finite-element approximations uhand phcan rarely be computed exactly, and one has to compute approximations U(n) h≈uh(tn)and P(n) h≈ph(tn)atsometimelevels0=t0<t 1···<t N=T,bymeansofatime integrator. Consequently, instead of the postprocessed approximations ˜uh(tn)and ˜ph(tn), one obtains ˜ U(n) hand ˜ P(n) has solutions of a system similar to (1.4)–(1.6) but with uhon the right-hand side of (1.4) replaced by U(n) hand ˙uhreplaced by an appropriate approximation d∗ tU(n) h. In the present paper we analyze the errors u(tn)−˜ U(n) hand p(tn)−˜ P(n) h.We show that, if any convergent time stepping procedure is used to integrate the standard MFE approximation, then the error of the fully discrete postprocessed approximation, u(tn)−˜ U(n) h, is that of the semidiscrete postprocessed approximation u(tn)−˜uh plus a term ˜enwhose norm is proportional to that of the time-discretization error en=uh(tn)−U(n) hof the MFE method, and, furthermore, we show ˜en=enplus higher-order terms for two particular time integration methods, the backward Euler method and the two-step backward differentiation formula (BDF) [9] (see also [25, section III.1]). We remark that the fact that ˜enis asymptotically equivalent to en has proved its relevance when developing a posteriori error estimators for dissipative problems [17] (see also [15, 16]). To prove ˜en≈enwe perform first a careful error analysis of the backward Euler method and the two-step BDF. This allows us to obtain error estimates for the pressure that improve by a factor of the time step k those in the literature [10, 34]. It must be noticed that the backward Euler method and the two-step BDF are the only G-stable methods (see, e.g., [26, section V.6]) in the BDF family of methods. G-stability makes it easier the use of energy methods in the analysis, and this has proved crucial in obtaining our error bounds. At present we ignore if error bounds similar to that obtained in the present paper can be obtained without resource to energy methods, so that error bounds for higher-order methods in the BDF family of methods can be proved. The analysis of fully discrete postprocessed methods may be less trivial than it may seem at first sight, since although many results for postprocessed semidiscrete methods can be found in the literature (see next paragraph) as well as numerical experiments (carried out with fully discrete methods) showing an increase in accuracy similar to that theoretically predicted in semidiscrete methods [3, 11, 14, 12, 20, 22], the only analysis of postprocessed fully discrete methods is that by Yan [44]. There, for a semilinear parabolic equation of the type ut−Δu=F(u), Δ being the Laplacian Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 598 J.DEFRUTOS,B.GARC ´ IA-ARCHILLA, AND J. NOVO operator and Fa smooth and bounded function, the postprocess of a finite-element (FE) approximation when integrated in time with the backward Euler method with fixed stepsize kis analyzed (higher-order time-stepping methods are also considered, but only for linear homogeneous parabolic equations). Error estimates are obtained where an O(k(1 + h2)) term is added to the bounds previously obtained for the postprocessed semidiscrete approximation. It must be remarked, though, that in [44] no attempt is made to analyze methods for equations with convective terms. In fact, in [44], it is stated that “It is not quite clear how it is possible to generalize our method to deal with a nonlinear convection term”. This is precisely what we do in the present paper. The postprocess technique considered here was first developed for spectral methods in [20, 21]. Later it was extended to methods based on Chebyshev and Legendre polynomials [11], spectral element methods [12, 13], and finite element methods [22, 14]. In these works, numerical experiments show that, if the postprocessed approximation is computed at the final time T, the postprocessed method is computationally more efficient than the method to which it is applied. Similar results are obtained in the numerical experiments in [2, 3] for MFE methods. Due to this better practical performance, the postprocessing technique has been applied to the study of nonlinear shell vibrations [37], as well as to stochastic differential parabolic equations [38]. Also, it has been effectively applied to reduce the order of practical engineering problems modeled by nonlinear differential systems [42, 43]. The postprocess technique can be seen as a two-level method, where the postprocessed (or fine-mesh) approximations ˜uhand ˜phare an improvement of the previously computed (coarse mesh) approximations uhand ph. Recent research on two-level finite-element methods for the transient Navier–Stokes equations can be found in [23, 27, 28, 40] (see also [30, 29, 36, 39] for spectral discretizations), where the fully nonlinear problem is dealt with on the coarse mesh, and a linear problem is solved on the fine mesh. The rest of the paper is as follows. In section 2 we introduce some standard material and the methods to be studied. In section 3 we analyze the fully discrete postprocessed method. In section 4 we prove some technical results and, finally, section 5 is devoted to analyze the time discretization errors of the MFE approximation when integrated with the backward Euler method or the two-step BDF. 2. Preliminaries and notations. 2.1. The continuous solution. We will assume that Ω is a bounded domain in Rd,d=2,3, of class Cm,form≥2, and we consider the Hilbert spaces H=u∈L2(Ω)d|div(u)=0,u·n|∂Ω=0 , V=u∈H1 0(Ω)d|div(u)=0 , endowed with the inner product of L2(Ω)dand H1 0(Ω)d, respectively. For l≥0 integer and 1 ≤q≤∞, we consider the standard Sobolev spaces, Wl,q(Ω)d, of functions with derivatives up to order lin Lq(Ω), and Hl(Ω)d=Wl,2(Ω)d. We will denote by ·l the norm in Hl(Ω)d,and·−lwill represent the norm of its dual space. We consider also the quotient spaces Hl(Ω)/Rwith norm pHl/R=inf{p+cl|c∈R}. Let Π : L2(Ω)d−→ Hbe the L2(Ω)dprojection onto H. WedenotebyAthe Stokes operator on Ω: A:D(A)⊂H−→ H, A =−ΠΔ,D(A)=H2(Ω)d∩V. Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. POSTPROCESSING FINITE-ELEMENT METHODS 599 Applying Leray’s projector to (1.1), the equations can be written in the form ut+Au +B(u, u)=Πfin Ω, where B(u, v)=Π(u·∇)vfor u,vin H1 0(Ω)d. We shall use the trilinear form b(·,·,·) defined by b(u, v, w)=(F(u, v),w)∀u, v, w ∈H1 0(Ω)d, where F(u, v)=(u·∇)v+1 2(∇·u)v∀u, v ∈H1 0(Ω)d. It is straightforward to verify that benjoys the skew-symmetry property (2.1) b(u, v, w)=−b(u, w, v)∀u, v, w ∈H1 0(Ω)d. Let us observe that B(u, v)=ΠF(u, v)foru∈V, v ∈H1 0(Ω)d. We shall assume that uis a strong solution up to time t=T,sothat (2.2) u(t)1≤M1,u(t)2≤M2,0≤t≤T, for some constants M1and M2. We shall also assume that there exists another constant ˜ M2such that (2.3) f1+ft1+ftt1≤˜ M2,0≤t≤T. Let us observe, however, that if for k≥2 sup 0≤t≤T ∂k/2 tf k−1−2k/2+ (k−2)/2 j=0 sup 0≤t≤T ∂j tf k−2j−2<+∞, and if Ω is of class Ck, then, according to Theorems 2.4 and 2.5 in [32], there exist positive constants Mkand Kksuch that the following bounds hold: u(t)k+ut(t)k−2+p(t)Hk−1/R≤Mkτ(t)1−k/2,(2.4) t 0 σk−3(s)u(s)2 k+us(s)2 k−2+p(s)2 Hk−1/R+ps(s)2 Hk−3/Rds≤K2 k,(2.5) where τ(t)=min(t, 1) and σn=e−α(t−s)τn(s)forsomeα>0. Observe that for t≤T<∞,wecantakeτ(t)=tand σn(s)=sn. For simplicity, we will take these values of τand σn. We note that although the results in the present paper require only (2.2) and (2.3) to hold, those in [18] that we summarize in section 2.3 require that for r=3,4, (2.4)– (2.5) hold for k=r+2. 2.2. The spatial discretization. Let Th=(τh i,φ h i)i∈Ih,h>0 be a family of partitions of suitable domains Ωh,wherehis the maximum diameter of the elements τh i∈T h,andφh iare the mappings of the reference simplex τ0onto τh i.Werestrict ourselves to quasi-uniform and regular meshes Th. Let r≥3, we consider the finite-element spaces Sh,r =χh∈CΩh|χh|τh i◦φh i∈Pr−1(τ0)⊂H1(Ωh),S 0 h,r =Sh,r ∩H1 0(Ωh), Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 600 J.DEFRUTOS,B.GARC ´ IA-ARCHILLA, AND J. NOVO where Pr−1(τ0) denotes the space of polynomials of degree at most r−1onτ0.Since we are assuming that the meshes are quasi-uniform, the following inverse inequality holds for each vh∈(S0 h,r)d(see, e.g., [7, Theorem 3.2.6]) vhWm,q(τ)d≤Chl−m−d1 q−1 qvhWl,q(τ)d,(2.6) where 0 ≤l≤m≤1, 1 ≤q≤q≤∞,andτis an element in the partition Th. We shall denote by (Xh,r,Q h,r−1) the so-called Hood–Taylor element [5, 35], when r≥3, where Xh,r =S0 h,rd,Q h,r−1=Sh,r−1∩L2(Ωh)/R,r≥3, and the so-called mini-element [6] when r=2,whereQh,1=Sh,2∩L2(Ωh)/R,and Xh,2=(S0 h,2)d⊕Bh. Here, Bhis spanned by the bubble functions bτ,τ∈T h, defined by bτ(x)=(d+1) d+1λ1(x)···λd+1(x), if x∈τand 0 elsewhere, where λ1(x),...,λ d+1(x) denote the barycentric coordinates of x.Fortheseelementsa uniform inf-sup condition is satisfied (see [5]), that is, thereexistsaconstantβ>0 independent of the mesh grid size hsuch that (2.7) inf qh∈Qh,r−1 sup vh∈Xh,r (qh,∇·vh) vh1qhL2/R≥β. The approximate velocity belongs to the discretely divergence-free space Vh,r =Xh,r ∩χh∈H1 0(Ωh)d|(qh,∇·χh)=0 ∀qh∈Qh,r−1, which is not a subspace of V. We shall frequently write Vhinstead of Vh,r whenever the value of rplays no particular role. Let Πh:L2(Ω)d−→ Vh,r be the discrete Leray’s projection defined by (Πhu, χh)=(u, χh)∀χh∈Vh,r. We will use the following well-known bounds (2.8) (I−Πh)uj≤Chl−jul,1≤l≤2,j=0,1. We will denote by Ah:Vh→Vhthe discrete Stokes operator defined by (∇vh,∇φh)=(Ahvh,φ h)=A1/2 hvh,A 1/2 hφh∀vh,φ h∈Vh. Let (u, p)∈(H2(Ω)d∩V)×(H1(Ω)/R) be the solution of a Stokes problem with right-hand side g, we will denote by sh=Sh(u)∈Vhthe so-called Stokes projection (see [33]) defined as the velocity component of solution of the following Stokes problem: find (sh,q h)∈(Xh,r,Q h,r−1) such that (∇sh,∇φh)+(∇qh,φ h)=(g,φh)∀φh∈Xh,r,(2.9) (∇·sh,ψ h)=0 ∀ψh∈Qh,r−1.(2.10) Obviously, sh=Sh(u). The following bound holds for 2 ≤l≤r: (2.11) u−sh0+hu−sh1≤Chlul+pHl−1/R. Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. POSTPROCESSING FINITE-ELEMENT METHODS 601 The proof of (2.11) for Ω = Ωhcan be found in [33]. For the general case, Ωhmust be such that the value of δ(h)=max x∈∂Ωhdist(x, ∂Ω) satisfies (2.12) δ(h)=Oh2(r−1). This can be achieved if, for example, ∂Ω is piecewise of class C2(r−1), and superparametric approximation at the boundary is used [1]. Under the same conditions, the bound for the pressure is [24] (2.13) p−qhL2/R≤Cβhl−1ul+pHl−1/R, where the constant Cβdepends on the constant βin the inf-sup condition (2.7). In the sequel we will apply the above estimates to the particular case in which (u, p) is the solution of the Navier–Stokes problem (1.1)–(1.3). In that case shis the discrete velocity in problem (2.9)–(2.10) with g=f−ut−(u·∇u). Note that the temporal variable tappears here merely as a parameter and then, taking the time derivative, the error bounds (2.11) and (2.13) can also be applied to the time derivative of shchanging u,pby ut,pt, respectively. Since we are assuming that Ω is of class Cmand m≥2, from (2.11) and standard bounds for the Stokes problem [1, 19], we deduce that (2.14) A−1Π−A−1 hΠhf j≤Ch2−jf0∀f∈L2(Ω)d,j=0,1. In our analysis we shall frequently use the following relation, which is a consequence of (2.14) and the fact that any fh∈Vhvanishes on ∂Ω. For some c≥1, (2.15) 1 c As/2 hfh 0≤fhs≤c As/2 hfh 0∀fh∈Vh,s=1,−1. Finally, we will use the following inequalities whose proof can be obtained applying [32, Lemma 4.4] vh∞≤CAhvh0∀vh∈Vh,(2.16) ∇vhL3≤C∇vh1/2 0Ahvh1/2 0∀vh∈Vh.(2.17) We consider the finite-element approximation (uh,p h)to(u, p), solution of (1.1)– (1.3). That is, given uh(0) = Πhu0, we compute uh(t)∈Xh,r and ph(t)∈Qh,r−1, t∈(0,T], satisfying (˙uh,φ h)+(∇uh,∇φh)+b(uh,u h,φ h)+(∇ph,φ h)=(f,φh)∀φh∈Xh,r,(2.18) (∇·uh,ψ h)=0 ∀ψh∈Qh,r−1.(2.19) For convenience, we rewrite this problem in the following way, (2.20) ˙uh+Ahuh+Bh(uh,u h)=Π hf, uh(0) = Πhu0, where Bh(u, v)=Π hF(u, v). For r=2,3,4,5, provided that (2.11)–(2.13) hold for l≤r, and (2.4)–(2.5) hold for k=r,thenwehave (2.21) u(t)−uh(t)0+hu(t)−uh(t)1≤Chr t(r−2)/2,0≤t≤T, Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 602 J.DEFRUTOS,B.GARC ´ IA-ARCHILLA, AND J. NOVO (see, e.g., [18, 32, 33]), and also, (2.22) p(t)−ph(t)L2/R≤Chr−1 t(r−2)/2,0≤t≤T, where r=rif r≤4andr=r+1ifr= 5. Results in [18] hold for hsufficiently small. In the rest of the paper we assume hto be small enough for (2.21)–(2.22) to hold. Observe that from (2.21) and (2.2) it follows that uh(t)1is bounded for 0 ≤ t≤T. However, further bounds for uh(t) will be needed in the present paper, so we recall the following result, which, since we are considering finite times 0 <T <+∞, it is a rewriting of [34, Proposition 3.2]. Proposition 2.1. Let the forcing term fin (1.1) satisfy (2.3). Then, there exists a constant ˜ M3>0, depending only on ˜ M2,Ahuh(0)0and sup0≤t≤Tuh(t)1,such that the following bounds hold for 0≤t≤T: F0,2(t)≡Ahuh(t)2 0≤˜ M2 3,(2.23) F1,r(t)≡trAr/2 h˙uh(t)2 0≤˜ M2 3,r=0,1,2,(2.24) F2,r(t)≡tr+2Ar/2 h¨uh(t)2 0≤˜ M2 3,r=−1,0,1,(2.25) I1,r(t)≡t 0 sr−1Ar/2 h˙uh(s)2 0ds≤˜ M2 3,r=1,2,(2.26) I2,r(t)≡t 0 sr+1Ar/2 h¨uh(s)2 0ds≤˜ M2 3,r=−1,0,1.(2.27) 2.3. The postprocessed method. This method obtains for any t∈(0,T] an improved approximation by solving the following discrete Stokes problem: find (˜uh(t),˜ph(t)) ∈( X, Q) satisfying ∇˜uh(t),∇˜ φ+∇˜ph(t),˜ φ=f, ˜ φ−buh(t),u h(t),˜ φ (2.28) −˙uh(t),˜ φ∀˜ φ∈ X, ∇·˜uh(t),˜ ψ=0 ∀˜ ψ∈ Q,(2.29) where ( X, Q)iseither: (a) The same-order MFE over a finer grid. That is, for h<h,wechoose ( X, Q)=(Xh,r,Q h,r−1). (b) A higher-order MFE over the same grid. In this case we choose ( X, Q)= (Xh,r+1,Q h,r). In both cases, we will denote by Vthe corresponding discretely divergence-free space that can be either V=Vh,r or V=Vh,r+1 depending on the selection of the postprocessing space. The discrete orthogonal projection into Vwill be denoted by Πh,and we will represent by Ahthe discrete Stokes operator acting on functions in V.Notice then that from (2.28) it follows that ˜uh(t)∈˜ Vand it satisfies (2.30) ˜ Ah˜uh(t)=˜ Πhf−F(uh(t),u h(t)) −˙uh(t). In [18] the following result is proved. Theorem 2.2. Let (u, p)be the solution of (1.1)–(1.3) and for r=3,4,let(2.4)– (2.5) hold with k=r+2 and let (2.11) hold for 2≤l≤r.Then,thereexistsa Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. POSTPROCESSING FINITE-ELEMENT METHODS 603 positive constant Csuch that the postprocessed MFE approximation to u,˜uhsatisfies the following bounds for r=3,4and t∈(0,T]: (i) if the postprocessing element is ( X, Q)=(Xh,r,Q h,r−1),then u(t)−˜uh(t)j≤C t(r−2)/2(h)r−j+C t(r−1)/2hr+1−j|log (h)|,j=0,1,(2.31) p(t)−˜ph(t)L2/R≤C t(r−2)/2(h)r−1+C t(r−1)/2hr|log (h)|,(2.32) (ii) if the postprocessing element is ( X, Q)=(Xh,r+1,Q h,r),then u(t)−˜uh(t)j≤C t(r−1)/2hr+1−j|log (h)|,j=0,1,(2.33) p(t)−˜ph(t)L2/R≤C t(r−1)/2hr|log (h)|.(2.34) Since the constant Cdepends on the type of element used, the result is stated for a particular kind of MFE methods, but it applies to any kind of MFE method satisfying the LBB condition (2.7), the approximation properties (2.11)–(2.13), as well as negative norm estimates, that is, u−sh−m≤Chl+min (m,r−2)(ul+pHl−1/R) for m=1,2and1≤l≤r. For these negative norm estimates to hold, it is necessary on the one hand that Ω is of class C2+m, and, on the other hand, that Xh,r ⊂H1 0(Ω)d, so that Xh,r consists of continuous functions vanishing on ∂Ω (i.e., discontinuous elements are excluded). As pointed out in [18, Remark 4.2], with a much simpler analysis than that needed to prove Theorem 2.2, together with results in [2], the previous result applies to the so-called mini element (r= 2) but excluding the case j=0(L2errors) in (2.31) and (2.33). 3. Analysis of fully discrete postprocessed methods. 3.1. The general case. As mentioned in the Introduction, in practice, it is hardly ever possible to compute the MFE approximation exactly, and, instead, some time-stepping procedure must be used to approximate the solution of (2.18)–(2.19). Hence, for some time levels 0 = t0<t 1<··· <t N=T, approximations U(n) h≈ uh(tn)andP(n) h≈ph(tn) are obtained. Then, given an approximation d∗ tU(n) hto ˙uh(tn), the fully discrete postprocessed approximation ( ˜ U(n) h,˜ P(n) h) is obtained as the solution of the following Stokes problem: ∇˜ U(n) h,∇˜ φ+∇˜ P(n) h,˜ φ=f, ˜ φ−bU(n) h,U(n) h,˜ φ−d∗ tU(n) h,˜ φ∀˜ φ∈ X, (3.1) ∇·˜ U(n) h,˜ ψ=0 ∀˜ ψ∈ Q,(3.2) where ( X, Q) is as in (2.28)–(2.29). Notice then that ˜ U(n) h∈˜ Vand it satisfies (3.3) ˜ Ah˜ U(n) h=˜ Πhf−FU(n) h,U(n) h−d∗ tU(n) h. Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 604 J.DEFRUTOS,B.GARC ´ IA-ARCHILLA, AND J. NOVO For reasons already analyzed in [17] and confirmed in the arguments that follow, we propose (3.4) d∗ tU(n) h=Π hf−AhU(n) h−BhU(n) h,U(n) h as an adequate approximation to the time derivative ˙uh(tn), which is very convenient from the practical point of view. We decompose the errors u(t)−˜ U(n) hand p(t)−˜ P(n) has follows, u(t)−˜ U(n) h=(u(t)−˜uh(tn)) + ˜en,(3.5) p(tn)−˜ P(n) h=(p(tn)−˜ph(tn)) + ˜πn,(3.6) where ˜en=˜uh(tn)−˜ U(n) hand ˜πn=˜ph(tn)−˜ Pnare the temporal errors of the fully discrete postprocessed approximation ( ˜ U(n) h,˜ P(n) h). The first terms on the right-hand sides of (3.5)–(3.6) are the errors of the (semidiscrete) postprocessed approximation whose size is estimated in Theorem 2.2. In the present section we analyze the time discretization errors ˜enand ˜πn. To estimate the size of ˜enand ˜πn, we bound them in terms of en=uh(tn)−U(n) h, the temporal error of the MFE approximation. We do this for any time-stepping procedure satisfying the following assumption (3.7) lim k→0max 0≤n≤Nen0=0,and lim sup k→0 max 0≤n≤Nen1=O(1), where k=max{tn−tn−1|1≤n≤N}. Bounds for en0and en1of size O(k2/tn) and O(k/t1/2 n), respectively, have been proven for the Crank–Nicolson method in [34] (see also [41]). The arguments in [10] can be adapted to show that, for the two-step BDF, enj≤Ck2−j/2/tn,for2≤n≤N,j=0,1 (although in section 5 we shall obtain sharper bounds of en1). For problems in two spatial dimensions, bounds for a variety of methods can be found in the literature (see a summary in [31]). In the arguments in the present section we use the following inequalities [34, (3.7)] which hold for all vh,w h∈Vhand φ∈H1 0(Ω)d: |b(vh,v h,φ)|≤cvh3/2 1Ahvh1/2 0φ0,(3.8) |b(vh,w h,φ)|+|b(wh,v h,φ)|≤cvh1Ahwh0φ0,(3.9) |b(vh,w h,φ)|+|b(wh,v h,φ)|≤vh1wh1φ1.(3.10) Proposition 3.1. Let (2.11) hold for l=2. Then, there exists a positive constant C=C(max0≤t≤TAhuh(t)0)such that ˜en−enj≤Ch2−jen1+en3 1+Ahen0,j=0,1,1≤n≤N,(3.11) ˜πnL2(Ω)/R≤C˜en1+en1+en2 1,1≤n≤N.(3.12) Proof. From (3.4) and (2.20) it follows that (3.13) ˙uh(tn)−d∗ tU(n) h=−Ahen+Π hFU(n) h,U(n) h−F(uh(tn),u h(tn)), Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. POSTPROCESSING FINITE-ELEMENT METHODS 611 Integrating with respect to tin (4.15) and taking into account (2.3), (2.25) with r=1, and (4.16), the bound (4.10) follows. To prove (4.12), we take derivatives twice with respect to tin (2.18) and then we set φ=t4Ah... uh,sothat t4 A1/2 h ... uh 2 0+1 2t4d dtAh¨uh2 0=t4ftt −d2 dt2Bh(uh,u h),A h... uh, and, since A1/2 hΠhftt0≤Cftt1, t4 A1/2 h ... uh 2 0+d dt t4Ah¨uh2 0≤4t3Ah¨uh2 0 +2t4C2ftt2 1+ A1/2 h d2 dt2Bh(uh,u h) 2 0. (4.17) Applying (4.5) to bound the third term on the right-hand side above we have t4 A1/2 h d2 dt2Bh(uh,u h) 2 0≤Ct4Ahuh2 0Ah¨uh2 0+Ah˙uh4 0 ≤Ct ˜ K1+ ˜ M2 4t3Ah¨uh2 0+tAh˙uh2 0, (4.18) the last inequality being a consequence of (2.23) and (2.24) with r=2. Thus,integrating with respect to tin (4.17) and applying (2.3), (2.26) with r= 2, (4.18), and (4.10), the bound (4.12) follows. Finally, since standard spectral theory of positive self-adjoint operators shows that A−1/2 h ... uh2 0≤A−3/2 h ... uh0A1/2 h ... uh0, by applying H¨older’s inequality the bound (4.11) follows from (4.9) and (4.12). 5. Error estimates. In this section we obtain error estimates for the temporal errors enof the two BDF described in section 3.2, the backward Euler method and the two-step formula (3.21)–(3.22), for which, an equivalent formulation is (5.1) dtU(n) h=−AhU(n) h−BhU(n) h,U(n) h+Π hf(tn),l 0≤n≤N. We remark that although higher regularity was required in section 2.3, in what follows it is only required that Ω is of class C2and that (2.11)–(2.13) hold for l=2. A simple calculation shows that for a sequence (yn)N n=0 in Vh, (5.2) n j=l (yj,Dy j)=1 2yn2 0−1 2yl−12 0+1 2 n j=lDyj2 0,1≤l≤n≤N, and, for 2 ≤l≤n≤N, (see, e.g., [10, (2.4b)]) n j=lyj,D+1 2D2yj=1 4yn2 0+1 4yn+Dyn2 0+1 4 n j=lD2yj2 −1 4yl−12 0−1 4yl−1+Dyl−12 0. (5.3) Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 612 J.DEFRUTOS,B.GARC ´ IA-ARCHILLA, AND J. NOVO As mentioned in section 3.2, we shall assume that e0= 0 although in some of the previous lemmas this condition will not be required. It must be noticed that e0=0 is not a serious restriction, since, on the one hand, it is usually satisfied in practice, and, on the other hand, were it not satisfied, there are standard ways to show that the effect of e0= 0 decays exponentially with time. The finite-element approximation uhto the velocity satisfies (5.4) dtuh(tn)+Ahuh(tn)+Bh(uh(tn),u h(tn)) −Πhf(tn)=τn, where (5.5) τn=dtuh(tn)−˙uh(tn)= 1 ktn tn−1 (t−tn−1)¨uh(t)dt, n =1,2,...,N, for the backward Euler method, and, for the two-step BDF, (5.6) τn=1 ktn tn−22(t−tn−1)+−1 2(t−tn−2)¨uh(t)dt, where for x∈R,x+=max(x, 0), and, also, (5.7) τn=1 2ktn tn−22(t−tn−1)+2−1 2(t−tn−2)2d3 dt3uh(t)dt. Subtracting (5.1) from (5.4), we obtain that the temporal error ensatisfies (5.8) dten+Ahen+Bh(en,u h(tn)) + BU(n) h,e n=τn,n=2,3,...,N. We shall now prove a result valid for both the backward Euler method and the two-step BDF. Lemma 5.1. Fix T>0and M>0. Then, there exist positive constants k0 and C, such that for any for k≤k0with Nk =T, and any four sequences (Yn)N n=0, (Vn)N n=0,(Wn)N n=0,and(gn)N n=0 in Vhsatisfying (5.9) max (AhVn,AhWn)≤M, n =0,1,...,N, and (5.10) dtYi+AhYi+Bh(Yi,V i)+Bh(Wi,Y i)=gi,i=l0,...,N, where l0is the value defined in (3.23), the following bound holds for n=l0,...,N, and j=−2,−1,0,1,2. Aj/2 hYn 2 0+k n i=l0 A(j+1)/2 hYi 2 0 ≤C2 Aj/2 hY0 2 0+(l0−1) Aj/2 hY1 2 0+k n i=l0 A(j−1)/2 hgi 2 0. (5.11) When j=0, condition (5.9) can be relaxed to AhVn≤M,forn=0,1...,N. Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. POSTPROCESSING FINITE-ELEMENT METHODS 613 Proof. Take inner product with AjYiin (5.10) so that we have (5.12) dtAj/2 hYi,A j/2 hYi+ A(j+1)/2 hYi 2 0≤Zi,A j hYi+A(j−1)/2 hgi,A (j+1)/2 hYi, where (5.13) Zi=B(Yi,V i)+B(Wi,Y i). Applying H¨older’s inequality to the last term on the right-hand side of (5.12) and rearranging terms we have (5.14) dtAj/2 hYi,A j/2 hYi+1 2 A(j+1)/2 hYi 2 0≤Zi,A j hYi+1 2 A(j−1)/2 hgi 2 0. For j>−2, we write (Zi,A j hYi)=(A(j−1)/2 hZi,A (j+1)/2 hYi), so that applying H¨older’s inequality and Lemma 4.2 with Viand Witaking the role of vhand whin (4.5), and recalling (5.9) we have Zi,A j hYi≤ A(j−1)/2 hZi 2 0+1 4 A(j+1)/2 hYi 2 0 ≤C2M2 Aj/2 hYi 2 0+1 4 A(j+1)/2 hYi 2 0. (5.15) Notice that when j= 0, due to the skew-symmetry property (2.1) of the trilinear form bwe have |(Zi,Y i)|=|b(Yi,V i,Y i)|, so that only AhVi0≤Mis necessary for (5.15) to hold; that is, no condition on AhWiis required. For j= 2, on the other hand, we write (Zi,A j hYi)=(Aj/2 hZi,A j/2 hYi), so that arguing similarly we have Zi,A −2 hYi≤ A−1 hZi 0 A−1 hYi 0≤CM A−1/2 hYi 0 A−1 hYi 0 ≤C2M2 A−1 hYi 2 0+1 4 A−1/2 hYi 2 0. (5.16) In all cases, then, from (5.14) it follows that for an appropriate constant C0>0 dtAj/2 hYi,A j/2 hYi+1 4 A(j+1)/2 hYi 2 0≤C2 0M2 Aj/2 hYi 2 0+1 2 A(j−1)/2 hgi 2 0, so that multiplying this inequality by kand summing from l0to n, and recalling (5.2– 5.3), after some rearrangements we can write 1 2l0 Aj/2 hYn 2 0+k 4 n i=l0 A(j+1)/2 hYi 2 0 ≤1 2l0 Aj/2 hYl0−1 2 0+(l0−1) Aj/2 h(Yl0−1+DYl0−1) 2 0 +C2 0M2k n i=l0 Aj/2 hYi 2 0+k 2 n i=l0 A(j−1)/2 hgi 2 0. (5.17) A simple calculation shows that Aj/2 hY1 2 0+ Aj/2 h(Y1+DY1) 2 0≤7 Aj/2 hY1 2 0+3 Aj/2 hY0 2 0, Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 614 J.DEFRUTOS,B.GARC ´ IA-ARCHILLA, AND J. NOVO so that multiplying by 2l0in (5.17) and taking into account that 2l0/4≤1, for an appropriate constant Cwe may write Aj/2 hYn 2 0+k n i=l0 A(j+1)/2 hYi 2 0≤2l0C2 0M2k n i=l0 Aj/2 hYi 2 0 +C Aj/2 hY0 2 0+(l0−1) Aj/2 hY1 2 0+k n i=l0 A(j−1)/2 hgi 2 0. Now, for ksufficiently small so that 2l0C2 0M2k<1/2, applying a standard discrete Gronwall lemma (e.g., [34, Lemma 5.1]) we have that (5.11) holds, with C2being Cexp(4l0C2 0M2T). Lemma 5.2. Let (2.11) hold for l=2. Then, there exist positive constants k0 and c1such that the errors ensatisfy the following bound for 1≤n≤N,k≤k0: (5.18) En≡en2 0+k n i=l0 A1/2 hei 2 0≤c2 1e02 0+(l0−1)e12 0+k2I2,−1(tn). Proof. We apply Lemma 5.1 to (5.8) in the case where j=0andYi=ei, Vi=uh(ti), and Wi=U(i) h. Observe that since we are in the case j= 0, only one of the two sequences (Ahuh(ti))N i=0,(AhU(i) h)N i=0 has to be bounded, and, in the present case, the first one is bounded according to (2.23). Thus, we have (5.19) en2 0+k n i=l0 A1/2 hei 2 0≤C2e02 0+(l0−1)e12 0+k n i=1 A−1/2 hτj 2 0. Now, applying H¨older’s inequality to the right-hand side of (5.5) we have k A−1/2 hτj 2 0≤1 ktj tj−1 (t−tj−1)2dttj tj−1 A−1/2 h¨uh(t) 2 0dt =k2 3tj tj−1 A−1/2 h¨uh(t) 2 0dt. Similarly, for the two-step BDF, applying H¨older’s inequality to the right-hand side of (5.6) we have k A−1/2 hτn 2 0≤1 4ktn tn−2 (4(t−tn−1)+−(t−tn−2))2dttn tn−2 A−1/2 h¨uh(t) 2 0dt ≤5 3k2tn tn−2 A−1/2 h¨uh(t) 2 0dt. Thus, the statement of the lemma follows from (5.19). The previous result allows us to deduce a bound for Aheh0in the following lemma. The values of I2,−1,I2,0are those of Proposition 2.1. Lemma 5.3. Under the conditions of Lemma 5.2, there exist positive constants ˜ k0 and ˜c0such that if e0=0and, in the case of the two-step BDF, also U(1) his given by the backward Euler method, the following bound holds for k≤˜ k0and n=1,2,...,N: (5.20) Ahen0≤˜c0Jn, where Jn=(I2,−1(tn)+I2,0(tn))1/2. Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. POSTPROCESSING FINITE-ELEMENT METHODS 615 Proof.Ife0= 0, from (5.18) for the Euler method (for which l0= 1) it follows that (5.21) en0≤ck(I2,−1(tn))1/2≤ckJn,n=1,2...,N for c=c1. In the case of the two-step BDF, if U(1) his obtained by the backward Euler method, if we allow for a larger value of c, it is clear that (5.21) also holds. Furthermore, it is immediate to check that dten0≤2l0k−1max0≤i≤l0en0,so that, in view of (5.21), from (5.8) it follows that Ahen0≤cJn+ Bh(en,u h(tn)) + BU(n) h,e n 0+τn0, for some constant c >0. For the Euler method, recalling the expression of τnin (5.5) we can write τn2 0= 1 ktn tn−1 (t−tn−1)¨uhdt 2 0≤1 k2tn tn−1 (t−tn−1)2 tdttn tn−1 t¨uh2 0dt. A simple calculation shows that the first factor on the right-hand side above can be bounded by k/tn≤1forn=1,2,...,N. Furthermore, a similar bound can be also obtained in the case of the two-step BDF. Thus, we have Ahen0≤cJn+ Bh(en,u h(tn)) + BU(n) h,e n 0, for an appropriate constant c >0. Finally, Bh(en,u h(tn)) + BU(n) h,e n 0=Bh(en,u h(tn)) + B(uh(tn),e n)−Bh(en,e n)0. Applying (4.5) and (4.6) we get Ahen0≤cJn+C A1/2 hen 0Ahuh0+C A1/2 hen 3/2 0Ahen1/2 0, and, thus, 1 2Ahen0≤cJn+C A1/2 hen 0Ahuh0+1 2C2 A1/2 hen 3 0. Since Ahuh0is bounded (recall Proposition 2.1) and, arguing as in (5.21), we have A1/2 hen0≤c1(kI2,−1(tn))1/2, the bound (5.20) follows for ksufficiently small. Remark 5.1. Observe that from the previous lemma and Proposition 2.1 it follows that AhU(n) h0≤c˜ M3where c=1+˜c0√2. Thus, as long as e0= 0 and, in case of the two-step BDF, also U(1) his given by the Euler method, we may apply Lemma 5.1 for j=0,withVnand Wnreplaced by uh(tn)andU(n) h, respectively. We now study the errors Ahtnen. We deal first with the backward Euler method. Observe that D(tnen)=tnDen+ken−1, so that multiplying by tnin (5.8), after some rearrangements we get (5.22) dt(tnen)+Ah(tnen)+Bh(tnen,u h)+BhU(n) h,t nen=en−1+tnτn. We have the following result. Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 616 J.DEFRUTOS,B.GARC ´ IA-ARCHILLA, AND J. NOVO Theorem 5.4. Let (2.11) hold for l=2. Then, there exist positive constants k2 and c2such that for k≤k2,ife0=0the errors n=tnensatisfy (5.23) Ahn2 0+k n i=1 A3/2 hi 2 01/2 ≤c2k(I2,−1(tn)+I2,1(tn))1/2,1≤n≤N. Proof. Applying Lemma 5.1 with j= 2 to (5.22) we have Ahn2 0+k n i=1 A3/2 hi 2 0≤ck n−1 i=1 A1/2 hei 2 0+k n i=1 tiA1/2 hτi 2 0. The first term on the right-hand side above is bounded Lemma 5.2 by c2 1k2I2,−1(tn). For the second one we notice that k tiA1/2 hτi 2 0=k ti kti ti−1 t−ti−1 ttA1/2 h¨uh(t)dt 2 0 , so that, for i≥2sinceti/t ≤ti/ti−1≤2, if t∈(ti−1,t i), we may bound ktjA1/2 hτj2 0 by 4 kti ti−1 (t−ti−1)2dtti ti−1 t2 A1/2 h¨uh(t) 2 0dt≤4k2 3ti ti−1 t2 A1/2 h¨uh(t) 2 0dt, and, for i=1,sincet1/k =1,and(t−t0)/t = 1, we may bound ktiA1/2 hτi2 0by kti ti−1 dtti ti−1 t2 A1/2 h¨uh(t) 2 0dt≤k2ti ti−1 t2 A1/2 h¨uh(t) 2 0dt. Thus, (5.23) follows. Lemma 5.5. Under the conditions of Lemma 5.2, there exist positive constants k 0and c 1such that for k≤k 0,ife0=0, in the backward Euler method e1satisfies (5.24) A−1 he1 2 0+k A−1/2 he1 2 0≤c 1k4G1, where G1=max 0≤s≤kF2,−2(s). Proof. We take inner product with 2kA−2 he1in (5.8) for n= 1, and recalling (5.2) and taking into account that e0= 0, after some rearrangements we have (5.25) A−1 he1 2 0+2k A−1/2 he1 2 0≤2kZ1,A −2 he1+2kA−1 hτ1,A −1 he1, where Z1is as in (5.13) but with Y1,V1,andW1replaced by e1,uh(t1), and U(1) h, respectively. Thus, arguing as in (5.16) 1−2kC2˜ M2 3 A−1 he1 2 0+3 2k A−1/2 he1 2 0≤2kA−1 hτ1,A −1 he1, so that taking into account that 2kA−1 he1,A −1 hτ1≤ A−1 he1 2 0/2+k2 A−1 hτ1 2 0, Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. POSTPROCESSING FINITE-ELEMENT METHODS 617 from (5.25) it follows that (5.26) 1 2−2kC2˜ M2 3 A−1 he1 2+3 2k A−1/2 he1 2 0≤k2 A−1 hτ1 2 0. Recalling the expression of τnin (5.5) we can write A−1 hτ1 2 0≤max 0≤t≤t1 A−1 h¨uh(t) 2 01 kk 0 tdt2 =k2 4max 0≤t≤t1 A−1 h¨uh(t) 2 0, we then have that (5.24) follows from (5.26) provided that kis sufficiently small. Lemma 5.6. Under the conditions of Lemma 5.2, let e0=0and let U(1) hbe given by the backward Euler method. Then, there exist positive constants k0and c1such that the errors enof the two-step BDF satisfy (5.27) E n≡ A−1 hen 2 0+k n j=2 A−1/2 hej 2 0≤c1k4(G1+I3,−3(tn)),2≤n≤N, where G1is given after (5.24). Proof. In view of the comments in Remark 5.1, we can apply Lemma 5.1 with j=−2 to (5.8), so that, recalling that e0=0,wehave (5.28) A−1 hen 2+k n i=2 A−1/2 hei 2 0≤c⎛ ⎝ A−1 he1 2 0+k n j=2 A−3/2 hτj 2 0⎞ ⎠. Notice that, as we showed in Lemma 5.5, the first term on the right-hand side above is bounded by c 1k4G1. For the second term on the right-hand side of (5.28), in view of (5.7) a simple calculation shows that k A−3/2 hτj 2 0≤Ck4tn tn−2 A−3/2 h d3uh(s) ds3 2 0 ds. Thus, (5.27) follows. For any two sequences (yn)∞ n=0 and (zn)∞ n=0, it is easy to check that D(ynzn)= ynDzn+zn−1Dyn,forn=1,2,..., and, also, D2(ynzn)=ynD2zn+2DynDzn−1+zn−2D2yn. Thus, for the two-step BDF, multiplying (5.8) by tnand t2 nand rearranging terms, for j=2,3,...,N,wehave dt(tnen)+Ahtnen+Bh(uh(tn),t nen) −tnBU(n) h,t nen=tnτn+(en−1+Den−1), (5.29) and (5.30) dtt2 nen+Aht2 nen+Bht2 nen,u h(tn)−BU(n) h,t 2 nen=t2 nτj+σn−1, where (5.31) σn−1=(tn+tn−1)(en−1+Den−1)+ken−2. Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 618 J.DEFRUTOS,B.GARC ´ IA-ARCHILLA, AND J. NOVO Theorem 5.7. Under the conditions of Theorem 5.4, there exist positive constants k1and c2such that for k≤k1if e0=0and U(1) hbe given by the backward Euler method, the errors n=tnenand n=t2 nenof the two-step BDF satisfy the following bounds for 2≤n≤N: (5.32) En≡n2 0+k n i=2 A1/2 hi 2 01 2 ≤c2k2(H1+I(tn)) 1 2, (5.33) E n≡Ah n2 0+k n i=2 A3/2 h i 2 01 2 ≤c2k2H1+I(tn)+J2 1+I3,1(tn) 1/2, where J1is given after (5.20), H1=I2,−1(t1)+G1,whereG1is given after (5.24), and I(t)=I3,−1(t)+I3,−3(t). Proof. To prove (5.32) we apply Lemma 5.1 with j= 0 to (5.29), so that taking into account that e0=0wehave (5.34) n2 0+k n j=2 A1/2 hj 2 0≤c12 0+k n−1 i=1 A−1/2 h(ei+Dei) 2 0+k n i=2 tiA−1/2 hτi 2 0 . Notice that since ei+Dei=2ei−ei−1, the second term on the right-hand side above can be bounded by 7kn−1 i=1 A−1/2 hei2 0, a quantity that has already been bounded in Lemma 5.6. Also, by writing tiτi=ti 2kti ti−2 1 t2(t−ti−1)+2−1 2(t−ti−2)2td3 dt3uh(t)dt, and noticing that ti/ti−2≤3fori≥3, and t2/k ≤2, a straightforward calculation shows that (5.35) k A−1/2 htiτj 2 0≤Ck4ti ti−2 t2 A−1/2 h ... uh(t) 2 0dt, j =2,...,N. Finally, noticing that 12 0=k2e12 0and recalling Lemma 5.2, we have that (5.32) follows from (5.34) and (5.35). To prove (5.33) we apply Lemma 5.1 with j= 2 to (5.30) to get (5.36) Ah n2 0+k n j=2 A3/2 h j 2 0≤cAh 12 0+k n−1 i=1 A1/2 hσi 2 0+k n i=2 t2 iA1/2 hτi 2 0. For the first term on the right-hand side above, in view of Lemma 5.3 we can write (5.37) Ah 10=k2Ahe1≤k2˜c0J1. For the second term on the right-hand side of (5.36), we first recall the expression of σiin (5.31) and then we notice that for i≥1wehavethatk≤ti,ti+2/ti≤3and ti+1/ti≤2, so that, recalling that e0= 0, for an appropriate constant C>0wemay write (5.38) k n−1 i=1 A1/2 hσi 2 0≤Ck A1/2 h1 2 0+k n−1 i=2 A1/2 hi 2 0≤Ck3 A1/2 he1 2 0+E2 n, Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. POSTPROCESSING FINITE-ELEMENT METHODS 619 Enbeing the quantity in (5.32). Also, by writing t2 iτi=t2 i 2kti ti−2 1 t22(t−ti−1)+2−1 2(t−ti−2)2t2d3 dt3uh(t)dt, and noticing that t2 i/t2 i−2≤9fori≥3, and t2 2/k ≤4k,weget (5.39) k A1/2 ht2 iτi 2 0≤Ck4ti ti−2 t4 A1/2 h ... uh(t) 2 0dt, j =2,...,N. Thus, (5.33) follows from (5.36), (5.37), (5.38), (5.32), (5.18), and (5.39). Although not strictly necessary for the analysis of the postprocessed approximation, for the sake of completeness we include an error bound for the pressure. We first notice that as a consequence of the LBB condition (2.7) we have that the error πn=ph(tn)−P(n) hsatisfies πnL2(Ω)/R≤1 βsup φh∈Xh,r |(πn,∇·φh)| φh1 . Furthermore subtracting (3.21) from (2.18), we have (πn,∇·φh)=˙uh−dtU(n) h,φ h+(∇en,∇φh)+b(en,u h,φ h)+bU(n) h,e n,φ h for all φh∈Xh,r. Using standard bounds for the trilinear form b(e.g., [34, (3.7)]) we can write (5.40) πnL2(Ω)/R≤Cen1+ ˙uh−dtU(n) h −1. Recalling the expression of d∗ tU(n) hin (3.4), we see that ˙uh−dtU(n) h=˙uh−d∗ tU(n) h, so that applying Lemma 3.2 and taking into account the equivalence (2.15) between en1and A1/2 hen0,wehaveπnL2(Ω)/R≤CA1/2 hen0. Since using standard spectral theory of positive self-adjoint operators it is straightforward to show that A1/2 hen0≤Cen1/2 0Ahen1/2 0, applying Lemma 5.2 and Theorem 5.4 in the case of the backward Euler method, and Theorem 5.7 in the case of the two-step BDF, we conclude the following result. Theorem 5.8. Under the conditions of Theorem 5.4, there exist positive constants k3and c4such that if e0=0and U(1) his obtained by the backward Euler method, the following bound holds for k<k 3and for n=l0,...,N: For the backward Euler method, ph(tn)−P(n) h L2(Ω)/R≤c4C1/2 1 k t1/2 n , where C1=(I2,−1(I2,−1+I2,1))1/2, and, for the two-step BDF, ph(tn)−P(n) h L2(Ω)/R≤c4C1/2 2 k2 t3/2 n , where C2is the product of the quantities between parentheses on the right-hand sides of (5.32) and (5.33). Downloaded 04/17/17 to 150.214.182.208. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
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