INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES
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1 INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES Asymptotic preserving error estimates for numerical solutions of compressible Navier-Stokes equations in the low Mach number regime Eduard Feireisl Mária Lukáčová-Medviďová Šárka Nečasová Antonín Novotný Bangwei She Preprint No PRAHA 2016
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3 Asymptotic preserving error estimates for numerical solutions of compressible Navier-Stokes equations in the low Mach number regime Eduard Feireisl Mária Lukáčová-Medviďová Šárka Nečasová Antonín Novotný Bangwei She Institute of Mathematics of the Academy of Sciences of the Czech Republic Žitná 25, CZ Praha 1, Czech Republic Institute of Mathematics, Johannes Gutenberg-University Mainz Staudingerweg 6, Mainz, Germany Institut Mathématiques de Toulon, EA2134, University of Toulon, France BP 20132, La Garde, France Abstract We study the convergence of numerical solution of the compressible Navier-Stokes system to its incompressible limit. The numerical solution is obtained by the combined finite element-finite volume method based on the Crouzeix-Raviart finite elements for velocity and piecewise constant approximation for density. The convective terms are approximated using the upwinding. The distance between a numerical solution of the compressible problem and the strong solution of the incompressible Navier- Stokes equations is measured by means of the relative energy functional. For barotropic pressure exponent γ 3/2 and for well-prepared initial data we obtain the uniform convergence of order O { }, h a, ε, a = min 2γ 3 γ, 1. Extensive numerical simulations also confirm that the numerical solution of the compressible problem converges to the solution of the incompressible Navier-Stokes equations as the discretization parameters, h and the Mach number ε tend to zero. ey words: Navier-Stokes system, finite element numerical method, finite volume numerical method, error estimates, asymptotic preserving schemes, low Mach number AMS classification 35Q30, 65N12, 65N30, 76N10, 76N15, 76M10, 76M12 1 Introduction We consider the scaled compressible Navier-Stokes systems in the low Mach number regime: t ϱ + div x ϱu = 0, 1.1 The research of E.F. leading to these results has received funding from the European Research Council under the European Union s Seventh Framework Programme FP7/ / ERC Grant Agreement The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO: The research of M.L.-M. has been supported by the German Science Foundation under the grants LU 1470/2-3 and the Collaborative Research Centers TRR 146 and TRR 165. The research of Š. N. has been supported by the Czech Agency of the Czech Republic No S and by RVO The research of B.S. leading to these results has received funding from the European Research Council under the European Union s Seventh Framework Programme FP7/ / ERC Grant Agreement The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:
4 t ϱu + div x ϱu u + 1 ε 2 xpϱ = div x S x u, 1.2 where ϱ and u R 3 are the unknown density and velocity fields, while S and p stand for the viscous stress and pressure characterizing the fluid material properties via the constitutive relations S x u = µ x u + t xu 2 3 div xui + ηdivui, µ > 0, η 0, 1.3 p C 2 0, C 1 [0,, p0 = 0, p ϱ > 0 for all ϱ > 0, 1.4 p ϱ lim ϱ ϱ γ 1 = p > 0, p ϱ lim ϱ 0+ ϱ α = p 0 > 0 where γ 1 and α 1. The small number ε > 0 is the Mach number. We notice that assumptions 1.4 are compatible with the isentropic pressure law pϱ = ϱ γ provided 1 γ 2. To avoid the problems with physical boundaries, we suppose the fluid motion is space-periodic, specifically ϱ = ϱt, x, u = ut, x, t [0, T, x T 3, where T 3 = The problem is completed by imposing initial conditions [0, 1] {0,1} 3. ϱ0, = ϱ 0, u0, = u 0, ϱ 0 > 0 in T When ε 0, solutions of are expected to approach the incompressible Navier-Stokes system ϱ t V + V x V + x Π = µ V, div x V = 0, ϱ > for suitable initial data V 0 = V 0, 1.7 see e.g. the seminal work by leinerman and Majda [30]. Problem 1.6, 1.7 admits a regular solution in the class l tv C l [0, T ]; W k l,2 T 3 ; R 3, l = 0, 1, 2, j t Π Cj [0, T ]; W k 1 j T 3, j = 0, 1, k defined on some maximal time interval [0, T max provided V 0 W k,2 T 3 ; R 3, div x V 0 = 0, see e.g. [30]. The norm of V, Π in the function spaces 1.8 will be denoted by V, Π X k T,T 3. Our goal is to estimate the distance between a discrete solution of a convenient numerical approximation of the compressible problem and the exact solution V of the system 1.6, 1.7 in terms of the numerical mesh size h, the time step, the Mach number ε, in a convenient numerical discrete function space with a convenient projection to the discrete numerical space of the unique strong solution of problem and a suitable norm of the difference of the initial data ϱ 0, u 0 of the discrete compressible problem and ϱ, V 0 of the incompressible target problem. The multiplicative constant in this estimate must be independent of the specific numerical solution and of course of h, and ε; it may however depend on the norm of the strong solution V. In particular, we shall not require any other extra restrictions on the numerical solution than those imposed by the numerical scheme and in this sense our error analysis results are unconditional. Taking into account the low Mach number regime, this type of error estimates are in particular referred as asymptotic preserving. The concept of asymptotic preserving schemes has been introduced by Jin et al., see [25], [28] and the references therein. A numerical scheme is called asymptotic preserving, if it is uniformly consistent with respect to a singular parameter - the Mach number ε in our case. In 2
5 particular, the scheme reduces to a consistent approximation of the limit equation. In [2], [3], [4], [8], [27], [37] asymptotic preserving properties, in the sense of uniform consistency with respect to ε, has been studied for the Euler equations of gas dynamics and some related problems, see also [1], [17], [18], [31], [35], [36], [39] for some well-known low Mach number schemes. We would like to point out that these schemes are usually based on a suitable splitting between the stiff and non-stiff parts of the underlying PDE and the corresponding semi-implicit or implicit-explicit time discretization. To the best of our knowledge, asymptotic error estimates uniform with respect to all three parameters ε,, h have not yet been obtained for any discrete scheme approximating fully nonlinear Euler or Navier-Stokes equations. On the other hand, in the case of a linear kinetic transport equation, such asymptotic error estimates in a diffusive scaling have been recently presented in [27], [34]. In spite of the importance of this property for applications, the mathematical literature on this subject is in a short supply, mostly due to the complexity of the problem. In this paper we present unconditional and uniform result for the compressible Navier-Stokes equations providing quantitatively an uniform convergence rate in terms of space-time discretization h, and Mach number ε. In particular, our scheme is asymptotic preserving in the sense of Jin, cf. [28]. The relative energy method introduced on the continuous level in [11], [16], [15] and its numerical counterpart developed in Gallouët et al. [22] seem to provide the convenient strategy to achieve this goal. It is worth to mention the recent work of Fischer [19], which can be seen as a counterpart of our approach. Specifically, he uses the relative energy functional to rigorously estimate the difference between an approximate numerical solution to the incompressible Navier-Stokes equations and any weak solution to the compressible Navier-Stokes system in the low Mach number regime. We shall apply the relative energy technique to the rather academic numerical scheme suggested in arper [29] with fully implicit time discretization, where the convergence to a weak solution was proved for γ > 3. Using the concept of the relative energy method the error estimates could be obtained for γ > 3/2 by Gallouët et al. [22]. The convergence result has been recently extended by Feireisl and Lukáčová [13] via dissipative measure valued solution for physically relevant range γ 1, 2. The application to other less academic numerical schemes, as mentioned above, is in the course of investigation. 2 Preliminaries Before formulating our main result, it seems convenient to introduce the basic notation and knows facts concerning the numerical apparatus used in the text. 2.1 Physical domain, mesh approximation We assume that T 3 = Th, 2.1 where T h is a family of closed tetrahedra called triangulation of T 3 having the following properties: If L, L, then L is either a common face, or a common edge, or a common vertex. By E, we denote the set of the faces of the element T h. The set of all faces of the mesh is denoted by E. For each face of the mesh = L, n, stands for the normal vector of, oriented from to L so that n, = n,l. We denote by and the 3 and 2 dimensional Lebesgue measure of the tetrahedron and of the face respectively, and by h and h the diameter of and respectively. We measure the regularity of the mesh by the parameter θ defined by { } ξ θ = inf, T h 2.2 h where ξ stands for the diameter of the largest ball included in. We denote by h the maximal size of the mesh, h = max T h
6 The triangulation T h is supposed to be regular, meaning θ θ 0 > In the sequel, we frequently omit the subscript h and write simply T instead of T h. 2.2 Spaces of discrete functions The symbol Q h T 3 denotes the space of piecewise constant functions: for v in CT 3, we set v = 1 Q h T 3 = {q L 2 T 3 T, q R}; 2.5 v dx for T and Π Q h vx = T where 1 is the characteristic function of. We introduce the Crouzeix-Raviart finite element spaces: V h T 3 = {v L 2 T 3, T, v P 1, E, = L, v 1 x, x T 3, 2.6 v ds = v L ds}, 2.7 where P 1 denotes the set of affine functions on, along with the associated projection Π V h : CT 3 V h T 3, Π V h [φ] ds x = φ ds x for all E. 2.8 For a Sobolev function v W 1,1 T 3 having well defined traces on the faces, we set v = 1 vds for E. 2.9 We recall that each function v V h T 3 can be written in the form vx = E v ϕ x, x T 3, 2.10 where the set {ϕ } E V h T 3 is the Crouzeix-Raviart basis determined by, E 2 1, ϕ ds = δ, Next, we recall the standard properties of the projection Π V h summarized in Lemmas below; the relevant proofs can be found in the Appendix of [22]. The reader may consult also the monograph Brezzi, Fortin [5], the Crouzeix and Raviart paper [7], or Gallouët, Herbin, Latché [21]. Lemma 2.1. The following estimates hold true: for all T and φ C; and φ Π V h [φ] L p for all T and φ C s. Π V h [φ] L φ L, 2.12 h s s φ L p ;R 3s, s = 1, 2, 1 p, 2.13 φ Π V h [φ] L p ;R d ch s 1 s φ L p ;R 3s, s = 1, 2, 1 p,
7 Remark 2.1. Here and hereafter expression a b means inequality a cb, where the positive number c may depend on the Sobolev and Lebesgue exponents involved in a and b here s, p, diameter of T 3, the Lebesgue measure T 3 of T 3 and θ 0. In particular, it is independent of θ, h and T h. Lemma 2.2. Let 1 p. Then with any v V h T 3. h E v p v p L p T 3, 2.15 We shall need the following version of [12, Lemma 2.2] see also [14, Theorem 10.7]. Lemma 2.3. Let r 0 be such that 0 r dx = a, T 3 r γ T 3 dx b. Then the following Poincaré Sobolev type inequality holds true: v 2 L 6 T 3 2 x v 2 dx + r v dx, 2.16 T T 3 for any v V h T 3, where the constant c depends on a and b but not on h. Lemma 2.4. There holds: T q div Π V h [v] dx = q div v dx, T for all v C 1 T 3, R d and all q Q h T 3. In particular, if divv = 0, then div Π V h [v] = 0 on each triangle T. Lemma 2.5 Jumps over faces in the Crouzeix-Raviart space. For all v V h T 3 there holds 1 h E [v] 2,n ds x v 2 dx, 2.18 T where [v],n is a jump of v with respect to a normal n to the face, { v x v x = L E, [v],n x = L x if n = n, v L x v x if n = n,l. We will frequently use the Poincaré, Sobolev and interpolation inequalities on tetrahedra reported in the following lemma. Lemma We have, v v L p v v L p v W 1,p, where 1 p. 2 There holds v v L p v v L p v W 1,p, 1 p 3, where p = 3p 3 p. h v L p ;R 3 for all T 2.19 h v L p ;R 3 for all E 2.20 v L p ;R 3 for all T 2.21 v L p ;R 3 for all E
8 3 We have, v v L q v v L q v W 1,p, 1 p 3, where 1 q = β p + 1 β p, p q p. h β v L p ;R 3 for all T 2.23 h β v L p ;R 3 for all E 2.24 The following estimates are easy to obtain by means of scaling arguments: v L q from which we readily deduce that 1 h 1/q v L q + h x v L q ;R 3, 1 q for any v C 1, 2.25 w L q 1 h 1/q w L q for any 1 q, w P m, 2.26 where P m denotes the space of polynomials of order m. In a similar way, we obtain h 3 1 p 1 q w L q 1 q p, w P m, 2.27 w L p and making use of the algebraic inequality 2.31 we deduce the global version w L p T 3 h 3 1 p 1 q w L q T 3 1 q p, for any w P m, T For future use, we record a version of 2.27 and 2.28 for the functions of the time variable t 0, T, where the discretization is of order, specifically, 1 p 1 q w L q 1 q p, 2.29 and w L p w L p 0,T 1 p 1 q w L q 0,T 1 q p We finish the section of preliminaries by recalling two usefull algebraic inequalities 1 the imbedding inequality L a i p 1/p L a i q 1/q, 2.31 i=1 for all a = a 1,..., a L R L, 1 q p and the discrete Hölder inequality for all a = a 1,..., a L R L, b = b 1,..., b L R L, 1 q + 1 p = 1. 3 Main results We will systematically use the following abbreviated notation: i=1 L L a i b i a i q 1/q L a i p 1/p, 2.32 i=1 i=1 i=1 ˆφ = Π Q h [φ], φ h = Π V h [φ], 3.1 where the projections Π Q h, ΠV h are defined in 2.6 and 2.8. For a function v C[0, T ], L1 T 3 we set v n x = vt n, x, 3.2 6
9 where t 0 = 0 t 1... t n 1 t n t n+1... t N = T is a partition of the interval [0, T ]. Finally, for a function v V h T 3 we denote h vx = T x vx1 x, div h vx = T div x vx1 x. 3.3 To ensure positivity of the approximate densities, we shall use the upwinding technique to discretize the convective term in the mass equation. For q Q h T 3 and u V h T 3 ; R 3, the upwinding of q with respect to u is defined, for = L E by q up = { q if u n, > 0 q L if u n, Numerical scheme Solutions ϱ, u of the scaled system will be approximated by ϱ n, u n = ϱ n,,h,ε, u n,,h,ε satisfying the following system of algebraic equations numerical scheme: For given initial data ϱ 0, u 0 we define T ϱn ϱn 1 φ + T ϱ n Q h T 3, ϱ n > 0, u n V h T 3 ; R 3, n = 0, 1,..., N, 3.5 E ϱ n,up u n n, φ = 0 for any φ Q h T 3 and n = 1,..., N, 3.6 T ϱ n ûn ϱ n 1 1 ε 2 T µ η T pϱ n ûn 1 v + T E E v n, + µ T ϱ n,up û n,up [u n n, ] v 3.7 u n : v dx divu n divv dx = 0, for any v V h T 3 ; R 3 and n = 1,..., N. Although the numerical solutions depend on the size h of the space discretization, the time step, and the Mach number ε, we shall use a concise notation ϱ n, u n instead of ϱ n,,h,ε, u n,,h,ε. The numerical method was proposed in [29, Definition 3.1]; it is strongly nonlinear and implicit. As shown by arper [29, Proposition 3.3], problem admits a solution ϱ n h, un h, ϱ n h Q h T 3, u n h V h T 3 ; R 3, n = 0, 1,..., N, for any fixed h > 0, > 0, and, moreover, ϱ n h > 0, n = 1,..., N, provided ϱ0 h > 0. The proof uses the topological degree theory in the spirit of [20]. 3.2 Error estimates We introduce the relative energy functional E ε ϱ, u z, v = T 3 ϱ u v ε 2 Eϱ z dx, 3.8 7
10 where ϱ Eϱ z = Hϱ H zϱ z Hz, Hϱ = ϱ 1 ps ds. 3.9 s2 Note that under the assumption p ϱ > 0, the function ϱ Hϱ is strictly convex in 0, ; whence Eϱ z 0 and Eϱ z = 0 ϱ = z. We are ready to state the main result of the paper. Theorem 3.1. Let the pressure satisfy 1.4 with γ 3/2. Let {ϱ n, u n } 0 n N be a family of numerical solutions resulting from the scheme , where the mesh satisfies assumptions and initial data ϱ 0, u 0 obey E ε ϱ 0 ε, u 0 ε ϱ, V 0 E 0, M 0 /2 ϱ 0 εdx 2M 0, M 0 = ϱ T T 3 with M 0 > 0, E 0 > 0 and ϱ > 0 independent of ε, h,. Moreover, suppose that [Π, V ] is a classical solution to the initial-boundary value problem in [0, T ] T 3 in the regularity class 1.8, emanating from the initial data V 0 W k,2 T 3 ; R 3, div x V 0 = 0, k 4. Then there exists a positive number c = c M 0, E 0, ϱ, p C 1 [ϱ/2,2ϱ], V, Π X k, 3.11 T,T 3 such that where sup E ε ϱ n, û n ϱ, V tn, + h u n x V t n, 2 1 n N 1 n N T 3 dx 3.12 c + h a + ε + E ε ϱ 0 ε, û 0 ε ϱ, V 0, { } 2γ 3 a = min, γ Here and hereafter, we denote by c a generic positive constant that may depend on the parameters from Remark 2.1 and also on T. Its possible dependence on other parameters is always explicitly indicated in the argument of c as in Moreover, c is always independent of ε, h, θ and T h. Remark 3.1. If the initial data are ill prepared, specifically Eϱ 0 ε ϱ ε 2, ϱ 0 ε û 0 T 3 ε V 0 2 dx 1, we obtain just an upper bound for the error. On the other hand, if the initial data are well prepared, meaning Eϱ 0 ε ϱ ε 2+ξ, ϱ 0 ε û 0 T 3 ε V 0 2 dx ε ξ, ξ > 0, Theorem 3.1 gives uniform convergence as h,, ε 0 of the numerical solution to the strong solution of the incompressible Navier-Stokes equations, including the rate of convergence on the 8
11 time interval on which the strong solution is known to exist. norm induced by the relative energy functional, namely The distance is measured in the E ε ϱ n, û n ϱ, V n ϱ n û n V t n 2 dx + ϱ n ϱ 2 T 3 ε, q = min{2, γ}. L q T 3 Theorem 3.1 holds also in the 2D case for any 0 a 2γ 2 γ if γ 1, 2] and a = 1 if γ > 2. Note that in this case the limit system 1.6, 1.7 admits global-in-time smooth solutions as long as the initial data are regular. The remaining part of the paper is devoted to the proof of Theorem 3.1. The main tool is a discrete version of the relative energy inequality derived in [22]. To simplify presentation, we drop the index ε in the notation used for initial data. 4 Uniform estimates If we take φ = 1 in formula 3.6 we get immediately the conservation of mass: n = 1,...N, ϱ n dx = T 3 ϱ 0 dx. T It is absolutely crucial for the subsequent discussion that the numerical scheme gives rise to a discrete version of the energy inequality stated below. The reader can consult Section 4.1 in Gallouët et al. [20, Lemma 4.1] for its laborious but rather straightforward proof. Lemma 4.1. Let ϱ n, u n be a solution of the discrete problem with the pressure p satisfying 1.4. Then there exist such that T ϱ n [minϱ n, ϱ n L, maxϱ n, ϱ n L], = L E, n = 1,..., N, ϱ n 1,n [minϱ n 1, ϱn, maxϱ n 1, ϱn ], T, n = 1,..., N, 1 2 ϱm u m ε 2 Eϱm ϱ 1 2 ϱ0 u ε 2 Eϱ0 ϱ T m + µ x u n 2 dx + µ 3 + η divu n 2 dx T for all m = 1,..., N, with the dissipation defect [D m, u time ] = m T [D m, ϱ time ] = 1 ε 2 m [D m, u space ] = [Dspace m, ϱ ] = ε 2 + [D m, u time ϱ n 1 T m = L E m = L E ] + [D m, ϱ time u n un 1 2 H ϱ n 1,n 2 ] + [Dspace m, u ] + [Dspace m, ϱ ] = 0, 4.2, 4.3a 2 ϱn ϱn 1 2 ϱ n,up u n un L 2 2 H ϱ n ϱn ϱn L 2 2, 4.3b u n n,, 4.3c u n n,. 4.3d 9
12 Lemma 4.1 yields the following corollary. Corollary 4.1. Under the assumptions of Lemma 4.1, and provided 3.10 holds, we have: There exists c = cm 0, E 0 > 0 independent of n, h and such that where = = L E N x u n 2 dx c, 4.4 N u n 2 L 6 T 3 ;R 3 c, 4.5 sup n=0,...n ϱ n û n 2 L 1 T 3 c. 4.6 sup n=0,...n ϱ n L γ T 3 c, 4.7 sup n=0,...n Eϱ n ϱ cε tor N Dϱ n, u n cε 2, 4.9 Dϱ n, u n ϱ n ϱn L 2 [maxϱ n, 1 {ϱ n ϱn 1} u n n, + L ]2 γ+ = L E ϱ n ϱ n L 2 1 {ϱ n 1} u n n, Proof If condition 3.10 is satisfied, then the right hand side of the energy inequality 4.2 is uniformly bounded. Estimates 4.4, thus follow directly from this inequality. Estimate 4.5 can be deduced from 4.1, 4.4, 4.6, 4.7, the discrete Sobolev inequality 2.16, and the discrete Poincaré inequality established in Lemma 2.3. Finally, estimate 4.9 follows from the bound 4.3d for the upwind dissipation by standard elementary reasoning employing assumption Discrete relative energy inequality The starting point of our error analysis is the discrete relative energy inequality for the numerical scheme derived in Gallouët et al. [22, Theorem 5.1], slightly modified in order to accommodate the presence of the small scaling parameter ε. Lemma 5.1. Let ϱ n, u n be a solution of the discrete problem with the pressure p satisfying 1.4. Then there holds for all m = 1,..., N, T 1 2 ϱ m u m U m 2 ϱ 0 u 0 U ε 2 + m T T Eϱ m r m Eϱ 0 r 0 µ x u n U n 2 dx + µ 3 + η divu n U n 2 dx 6 T i, i=
13 for any 0 r n Q h T 3, U n V h T 3 ; R 3, n = 1,..., N, where T 1 = T 2 = T 3 = T 4 = ε 2 T 5 = ε 2 T 6 = ε 2 m T m T m T m T m T m T µ x U n : x U n u n dx + µ 3 + η divu n divu n u n dx, ϱ n 1 E = L E = L U n U n 1 U ϱ n,up n + U n L 2 rn ϱ n E = L U n 1 + U n 2 pϱ n [U n n, ], H r n H r, n 1 ϱ n,up H r n 1 [un n, ]. u n 1, û n,up U n [u n n, ], 5.2 Remark 5.1. It is worth-noting that the relative energy inequality in the form 5.1 holds for any pair of test functions r n, U n that can be chosen conveniently according to the expected asymptotic limit. 6 Approximate discrete relative energy inequality The next step is to take the discrete relative energy inequality 5.1 with test function ϱ, V n h, where V belongs to the same regularity class 1.8 as the velocity field of the target system, div x V = 0, and ϱ is a positive constant. Since ϱ is a constant, terms T 5 and T 6 vanish trivially. Term T 4 vanishes as well, by virtue of Lemma 2.4. We keep term T 1 in its present form and transform terms T 2 and T 3 conveniently. This will be done in several steps. Transforming term T 2 We have T 2 = T 2,1 + R 2,1 + R 2,2, with T 2,1 = m T ϱ n 1 V n h, V n 1 h, V n h, u n, 6.1 and where and m m R 2,1 = R n, 2,1, R 2,2 = R2,2, n T R n, 2,1 = V n 2 ϱn 1 h, V n 1 h, 2 R n 2,2 = T = 2 ϱn 1 [V n V n 1 ] h, 2, ϱ n 1 V n h, V n 1 h, u n 1 un. As the function t V t, x is continuously differentiable, we get [V n V n 1 ] h, = 1 [ 1 [ t n ] ] t V z, xdz dx t n 1 h 11
14 = 1 [ 1 tn ] [ t V z t n 1 ]h xdz dx [ t V ] h,0 L 0,T T 3 ;R 3 t V, L 0,T T 3 ;R 3 where we have used Therefore, thanks to the mass conservation 4.1, we get R n, 2,1 M 0 2 tv 2 L 0,T T 3 ;R To treat term R n 2,2 we use the discrete Hölder inequality and identity 4.1 in order to get +cm 1/2 0 T R n 2,2 cm 0 t V 2 L 0,T ;W 1, T 3 ;R 3 ϱ n 1 un 1 un 2 1/2 t V L 0,T T 3 ;R 3 ; whence, by virtue of estimate 4.2 for the upwind dissipation term 4.3a, one obtains R 2,2 cm 0, E 0, t V L 0,T T 3 ;R Transforming term T 3. Employing the definition 3.4 of upwind quantities, we easily establish that T 3 = T 3,1 + R 3,1, m with T 3,1 = ϱ n,up û n,up T E n,up ˆV h, V n h,u n n,, R 3,1 = m E R n, 3,1, and R n, 3,1 = V n ϱn h, V n h,l 2 [u n n, ] + + ϱ n V n h,l V n h, 2 L [u n n,l ] +, = L E. 2 2 Writing V n h, V n h,l = V n h, V n h + V n h V n h, + V n h, V n h + V n h V n h,l, = L E, and employing estimates 2.19 then 2.14 s=1 and 2.20 after 2.14 s=1 to evaluate the L -norm of the first and second terms, and performing the same for the last two terms, we get consequently V n h, V n h,l L L;R 3 ch V L L;R 9 ; 6.4 R n, 3,1 h2 c V 2 L 0,T T 3 ;R 9 ϱn + ϱ n L u n, = L E, whence R 3,1 h c V 2 L 0,T T 3 ;R 9 h ϱ n + ϱ n L 6/5 5/6 T = L E [ m h u n 6 1/3] 1/2 h cm0, E 0, V L Q T ;R 9, T E 6.5 provided γ 6/5, thanks to the discrete Hölder inequality, the equivalence of norms 2.15 and energy bounds listed in Corollary 4.1. Clearly, for each face = L E, u n n, + u n n,l = 0; whence, finally T 3,1 = m T E ϱ n,up û n,up n,up ˆV h, V n h, V n h, u n n,
15 Before the next transformation of term T 3,1, we write V n h, V n h, = V n h, V n h + V n h V n h,; whence by virtue of and 2.14 s=1, similarly to 6.4, V n h, V n h, L ;R 3 ch x V L 0,T T 3 ;R 3 3,. 6.7 Let us now decompose the term T 3,1 as m T 3,1 = T 3,2 + R 3,2, with R 3,2 = R3,2, n m T 3,2 = ϱ n,up n,up ˆV h, T E R3,2 n = ϱ n,up n,up ˆV h, T E û n,up V n h, V n h, û n,up n,, and û n,up V n h, V n h, By virtue of discrete Hölder s inequality and estimate 6.7, we get R n 3,2 c x V L Q T ;R 9 T E h ϱ n,up γ 0 h ϱ n,up T E 1/2γ0 û n,up T E u n û n,up ˆV n,up h, 2 1/2 n,. h u n û n,up q 1/q, where γ q = 1, γ 0 = min{γ, 3} and γ 3/2. For the sum in the last term of the above product, we have h u n û n,up q c h u n u n q T c T E T E E u n u n q L q ;R 3 + u n u n q L q ;R 3 ch 2γ 0 3 q 2γ 0 T T x u n 2 L 2 ;R 9 q/2, where we have used the definition 3.4, the discrete Minkowski inequality, interpolation inequalities and the discrete imbedding inequality Now we can go back to the estimate of R3,2 n, and, taking into account the upper bounds 4.4, , we may use Young s inequality to get m R 3,2 h a cm 0, E 0, x V L Q T ;R 9 + E ε ϱ n, û n ϱ, ˆV n h, 6.8 provided γ 3/2, where a is given in Finally, we rewrite term T 3,2 as whence m T 3,2 = T 3,3 + R 3,3, with R 3,3 = R3,3, n m T 3,3 = ϱ n,up n,up ˆV h, T E R3,3 n = ϱ n,up n,up ˆV h, T E û n,up V n h, V n n,up h, ˆV h, n,, and û n,up V n h, V n h, û n,up n,up ˆV h, n, ; 6.9 m R 3,3 c x V L Q T,R 9 E ε ϱ n, û n ˆr n, ˆV n h,
16 We have proved the following lemma: Lemma 6.1 Approximate relative energy inequality. Let ϱ n, u n be a solution of the discrete problem emanating from the initial data 3.10, where the pressure satisfies 1.4 with γ 3/2. Then there exists c = c M 0, E 0, V W 1, 0,T T 3 ;R 3, such that for all m = 1,..., N, we have: ϱ m û m ˆV m h ε 2 Eϱm ϱ dx T 3 + m T T 3 ϱ 0 û 0 ˆV 0 h, ε 2 Eϱ0 ϱ dx µ x u n V n h 2 dx + µ 3 + η divu n V n h 2 dx for any pair of functions ϱ, V, where ϱ = const. > 0 and Here and S 1 = S 2 = S 3 = m T m T m T E ε V C 1 [0, T ] T 3 ; R 3, div x V = 0. µ x V n h : x V n h u n dx ϱ n 1 V n h, V n 1 h, V n h, u n, ϱ n,up n,up ˆV h, 3 S i + Rh, m +Gm, i= û n,up V n h, V n n,up h, ˆV h, n,, 6.12 m G m c E ε ϱ n, û n n ϱ, ˆV, R m h, c + h a, 6.13 with the power a given by 3.13 and with the relative energy functional E ε introduced in 3.8. Here, in accordance with our notation convention 2.9, , we have set V n h = ΠV h [V t n], V n h, = [V n h ], V n h, = [V n h ], ˆr n = Π Q h [rt n], where projections Π Q, Π V are defined in 2.6 and Consistency error This section is devoted to the proof of a discrete identity satisfied by any strong solution of problem in the class 1.8. Lemma 7.1 A discrete identity for strong solutions. Let ϱ n, u n be a solution of the discrete problem emanating from the initial data 3.10, with the pressure satisfying 1.4, where γ 3/2. Let Π, V belonging to the regularity class 1.8 with k 4 be a strong solution of the incompressible Navier-Stokes system Then there exists c = c M 0, E 0, V, Π X k > 0, T,T 3 such that 3 S i + R m h, = 0, for all m = 1,..., N, 7.1 i=1 14
17 where and S 1 = S 2 = S 3 = m T m T m T E µ x V n h : x V n h u n dx ϱ V n h, V n 1 h, n,up ϱ ˆV h, V n h, u n, û n,up V n h, V n n,up h, ˆV h, n, { } 5γ 6 R m h, h c b + + ε, b = min 2γ, 1. The remaining part of this section is devoted to the proof of Lemma Essential and residual sets We start with an auxiliary algebraic inequality whose straightforward proof is left to the reader. Lemma 7.2. Let p satisfies assumptions 1.4. Let ϱ > 0. Then there exists c = cϱ > 0 such that for all ϱ [0, and there holds ϱ Eϱ ϱ c 1 R+ \[ϱ/2,2ϱ] + ϱ γ 1 R+ \[ϱ/2,2ϱ] + ϱ ϱ 2 1 [ϱ/2,2ϱ], 7.2 where E is defined in 3.9. Now, for fixed number ϱ and fixed functions ϱ n, n = 0,..., N, we introduce the residual and essential subsets of T 3 relative to ϱ n as follows: and we set N n ess = {x T ϱ ϱn x 2ϱ}, N n res = T 3 \ N n ess, 7.3 [g] ess x = gx1 N n ess x, [g] res x = gx1 N n res x, x T 3, g L 1 T 3. Integrating inequality 7.2 we deduce cϱ [ ε 2 1 [ϱ ]res + n γ] [ + ϱ n ϱ res ] 2 ess dx E ε ϱ n, u n ϱ, V n, 7.4 for any pair ϱ, V with V belonging to the class 1.8 and any ϱ n Q h T 3, ϱ n 0, u n V h T 3 ; R 3. We are now ready to prove Lemma 7.1. Since Π, V satisfies on 0, T T 3 and belongs to the class 1.8, equation 1.6 can be rewritten in the form ϱ t V + ϱv V + Π µ V = 0 in 0, T T 3. From this fact, we deduce the identity where T 1 = T 3 = m m T 3 µ V n V n u n dx, T 3 ϱv n V n V n u n dx, In the steps below, we handle each term T i. 5 T i = 0, 7.5 i=1 15 T 2 = T 4 = m m T 3 ϱ[ t V ] n V n u n dx, T 3 Π n u n dx.
18 7.2 Viscous term Integrating by parts, we get: T 1 = T 1,1 + R 1,1, m with T 1,1 = µ V n h : V n h u n dx, and R 1,1 = I 1 + I 2, with T m I 1 = µ V n V n h : V n h u n dx, T m I 2 = µn, V n V n h u n ds T E m [ = µn V n V n h u n] ds, E,n where in the last line n is the unit normal to the face and [ ],n is the jump over sigma with respect to n defined in Lemma 2.5. To estimate I 1 we use the Cauchy-Schwartz inequality, employ estimates 2.14 s=2 to evaluate the norms involving V n V n h, and use 2.13 s=1, 4.4, the Minkowski inequality to estimate the norms involving V n h un. We get I 1 h cm 0, E 0, V, 2 V L 0,T ;L T 3 ;R 36. Since the integral over any face E of the jump of a function from V h T 3 is zero, we may write m [ ] I 2 = µn x V n x V n u n V n h ds;,n E whence by means of the mean value formula applied x V n x to evaluate the differences x V n x V n and Hölder s inequality, Therefore, I 2 h c 2 V L Q T ;R 27 h c 2 V L Q T ;R 27 m E m E 1 ] h h [u n V n L h,n 2 ;R 3 h + 1 h [ u n V n 2 h ],n L 2 ;R R 1,1 h cm 0, E 0, V L 0,T ;W 2, T 3 ;R 3, 7.7 where we have employed Lemma 2.5, together with 4.4 and Term with the time derivative Let us now decompose the term T 2. Step 1: T 2 = T 2,1 + R 2,1, m with T 2,1 = ϱ V n V n 1 m V n h u n dx, R 2,1 = T T and R n, 2,1 = ϱ[ t V ] n V n V n h dx + ϱ [ t V ] n V n V n 1 R n, 2,1, V n h u n dx. 16
19 The remainder R n, 2,1 can be rewritten as follows R n, 2,1 = ϱ[ t V ] n V n V n h dx + 1 whence, by the Hölder inequality, [ t n tn ] ϱ t 2 V z, dzds V n h u n dx; t n 1 s R n, 2,1 + hϱ [ t V L Q T ;R 3 5/6 V n L 6 + V n h,0 L 6 ] + t 2 V n L 6/5 T 3 ;R 3 un L 6 + V n h,0 L 6. Consequently, R 2,1 + h c M 0, E 0, ϱ, V, t V, V L Q T ;R 15, t 2 V L 2 0,T ;L 6/5 T 3 ;R 3, 7.8 where we have used the discrete Hölder and Young inequalities, the estimates 2.12 and the energy bound 4.4 from Corollary 4.1. Step 2: Term T 2,1. We decompose the term T 2,1 as T 2,1 = T 2,2 + R 2,2, m with T 2,2 = ϱ V n h, V n 1 m h, V n h u n dx, R 2,2 = T T and R n, V n 2,2 = V n 1 [ V n V n 1 ] ϱ V n h,0 u n dx h [ V n V n 1 [ V ϱ ]h n V n 1 ] V n h u n dx= I1 + I2. h, Next, we calculate I2 = 1 [ ϱ t n [ t V zdz t n 1 ]h tn ] t V zdz u n V n h dx t n 1 h, ϱ h tn ] x [ t V z h L 6/5 ;R 3 un V n h,0 L 6 ;R 3, t n 1 R n, 2,2, where we have used the Fubini theorem, Hölder s inequality and 2.19, 2.14 s=1. Further, employing the Sobolev inequality on the Crouzeix-Raviart space V h T 3, 2.16, the Hölder inequality and estimate 2.14 s=1, we get T I 2 h ϱ un V n h L 6 T 3 ;R 3 tn t n 1 x t V z dz. L 6/5 T 3 ;R 3 We reserve the similar treatment to the term I1. Resuming these calculations, and summing over n from 1 to m we get by using Corollary 4.1, 2.12, R 2,2 h cm 0, E 0, ϱ, V, x V, t V L Q T ;R 16, t V L 2 0,T ;L 6/5 T 3 ;R Step 3: Term T 2,3. We rewrite this term in the form T 2,3 = T 2,4 + R 2,4, R 2,4 = with T 2,4 = and R n, 2,4 = m T m T r n 1 V n h, V n 1 h, R n, 2,4, r n 1 V n h, V n 1 h, u n V n h, dx, u n u n V n h V n h, dx
20 We estimate the L norm of V n n 1 h, V h, as in 6.2 and get, similarly to the above, R 2,4 h c M 0, E 0, ϱ, V W 1, 0,T T 3 ;R Convective term Step 1: Term T 3. Let us first decompose T 3 as T 3 = T 3,1 + R 3,1, m m with T 3,1 = ϱv n h, V n V n h, u n dx, R 3,1 = T T and R n, 3,1 = ϱv n V n h V n V n h u n dx + ϱv n h V n h, V n V n h u n dx + ϱv n h, V n V n h V n h, u n u n dx. We have, by virtue of 2.13 s=1, V n V n h L ;R 3 and, by virtue of 2.19, 2.13 s= s=1, and, finally, by virtue of 2.19, V n h V n h, L ;R 3 u n u n L 2 ;R 3 h V n L ;R 9, h V n L ;R 9 h x u n L 2 ;R 9. R n, 3,1, Consequently, employing Hölder inequality for the integrals over and the discrete Hölder inequality for the sums over T, and using estimate 4.4, we arrive at R 3,1 h cm 0, E 0, ϱ, V, x V L Q T ;R Now we shall deal with term T 3,1. Integrating by parts, we get: ϱv n h, V n V n h, u n dx = E due to the the fact that E V n h, n,ds = 0. Step 2: T erm T 3,1 Next we write ϱ[v n h, n, ]V n V n h, V n h, u n, T 3,1 = T 3,2 + R 3,2, m R 3,2 = R n 3,2, 18
21 m n,up T 3,2 = ϱ[ ˆV h, n, ]V n V n n,up h, ˆV h, û n,up, 7.13 T E and R n 3,2 + T T E E ϱ [V n h, n,up ϱ[ ˆV h, n, ]V n V n h, n,up ˆV h, n, ]V n V n h, V n h, u n V n h, ˆV n,up h, u n û n,up h,. We use repeatedly the Taylor formula along with 2.13 s=1, 2.19, 2.14 s=1, to get the bound We obtain, by Hölder inequality, T E T 3 R n 3,2 h cϱ 1 + V W 1, Q T ;R 3 h u n T 2 +cϱ 1 + V W 1, Q T ;R 3 T h u n c T [ h u n u n c T E h u n 6 1/6 c u n u n 2 L 2 ;R 3 T h u n u n. x u n 2 L 2 ;R 9 1/2, 1/2 h c T x u n 2 L 2 ;R 9 1/2, where we have used 2.21 p=2, p=2. Consequently, we may use 4.4 to conclude R 3,2 h c M 0, E 0, ϱ, V, x V L Q T ;R Step 3: Term T 3,2 Finally, we replace in T 3,2 the term V n V n h, by V n h, V n h,. We get T 3,2 = T 3,3 + R 3,3, m R 3,3 = R n 3,3, T 3,3 = m T E n,up ϱ[ ˆV h, n, ]V n h, V n n,up h, ˆV h, û n,up, 7.15 and R n 3,3 = T E ϱv n h, n, [V n V h ] n [V n h V n n,up h] ˆV h, û n,up, committing the error R n 3,3 h c M 0, E 0, ϱ, V, x V L Q T ;R
22 7.5 Pressure term Step 1: A useful formula for upwind discretization We will need the following formula proved in [12, Formula 2.16, Lemma 8.1]. ru x φdx = r T 3 up u n, F 7.17 T + T E F φ [ ] r [u n, ] ds+,n, T E E E φru u n, + T F φrdivudx for any r, F Q h T 3, u V h T 3, φ C 1 R 3. We also report the straightforward formula: r up u n, F = ] r up u n [F 7.18,n T E for any r, F Q h T 3, u V h T 3. The strategy is to use formula 7.17 with r = ϱ n, u = u n, φ = Π n, F = ˆΠ n in combination with the discrete continuity equation 3.6 to show that the term T 4 is proportional to ε. Step 2: Discrete time derivative in the continuity equation 3.6 We get by direct calculation Therefore, as in 6.2, N T3 ϱ n ϱ n 1 N J ˆΠ n dx = T 3 ϱ n ϱ ϱ n 1 ϱ ˆΠn dx N = ϱ n ϱπ n ϱ n 1 ϱπ n 1 N dx + ϱ n 1 ϱˆπ n 1 ˆΠ n dx T 3 T 3 N T3 = ϱ n ϱπ N ϱ 0 ϱπ 0 + ϱ n 1 ϱ ˆΠ n 1 ˆΠ n dx. T 3 T 3 N J = T ϱ n ϱn 1 p = max{2, γ }, where we have used 4.8 and Lemma 7.2. Step 3: Error of upwind discretization We write where T 4 = ϱ N R 4,1 = Π n dx ε1 + cm 0, E 0, Π L Q T, t Π L 1 0,T ;L p T 3, 7.19 ϱ n u n x Π n dx + T 3 ϱ ϱ N T 3 ϱ ϱ n u n x Π n dx = T 4,1 R 4,1, 7.20 N ϱ ϱ n u n x Π n ε cm 0, E 0, ϱ, x Π L 0,T T 3. T 3 Using the formula 7.17 with r = ϱ n, u = u n, F = Π Q h [Πn ] we check without difficulty that R 4,2 ϱ n u n x Π n dx ϱ n,up T 3 u n n, Π n T 20 E
23 T E = T E Π n Π n [ ϱ n],n, Π n ϱ n u n u n n, ds x + T We shall now estimate conveniently terms I n 1 In 3 Step 3a: Term I 1. First, we have by using Hölder s inequality, 1 E T E h x Π L 0,T T 3 = L E T E maxϱ n, ϱ n L γ ds x δ 2γ [u n n] ds x Π n Π n ϱ n div x u n dx. = I n 1 + I n 2 + I n in three steps. Π n Π n [ ϱ n],n, [ϱ n] 2 [u n n, ] ds x,n 1, [u n E maxϱ n,, n] 1/2 dsx ϱn L δ T E u n n, γ γ δ 2γ γ δ ds x with any 0 δ γ and any E T 3, where we have used estimate Therefore employing and estimates 4.5, 4.7, 4.9, we obtain Step 3b: Term I 2. We observe that N I1 n εh 1/2 cm 0, E 0, x Π L 0,T T I n 2 = Π n Π n ϱ n u n u n n, ds x T E γ 0 L γ 0 1 ;R 3 E Π n Π n L ϱn L γ 0 u n h u n h, x Π L 0,T T 3 ϱ n L γ 0 u n u n T γ 0 L γ 0 1 ;R 3 h 5γ 0 6 2γ 0 ϱ n 1/2 x L γ 0 T 3 x u n 2 L 2 ;R 3 3 Π L 0,T T 3. T with γ 0 = min{2, γ}, where we have used Hölder s inequality, trace estimates , and interpolation estimates 2.24 together with estimate 4.4. Consequently, N I n 2 t h 5γ 0 6 2γ 0 cm 0, E 0, x Π L T Step 3c:Term I 3. We have I3 n = Π n Π n ϱ n div x u n dx h ϱ n L γ div x u n L 2 x Π L 0,T T 3 T T 21
24 whence Step 4: Conclusion According to 7.20, 7.21 and 3.6, N I2 n t hcm 0, E 0, x Π L 0,T T T 4 = ϱ N R n 4,2 + R 4,1 1 ϱ J ; whence, taking into account 7.19, 7.22, 7.23, 7.24, we infer T 4 ε + h 5γ 0 6 2γ 0 cm 0, E 0, ϱ, Π, x Π L 0,T T 3 ;R 4 t Π L 1 0,T ;L 3 T 3, γ 0 = min{2, γ} Now it suffices to put together formulas 7.6, 7.10, 7.15 together with 7.7, 7.8, 7.9, 7.11, 7.12, 7.14, 7.16 and 7.25 in order to get the statement of Lemma A Gronwall type inequality In this section we put together the relative energy inequality 6.11 and the identity 7.1 derived in the previous section. The final inequality resulting from this manipulation is formulated in the following lemma. Lemma 8.1. Let ϱ n, u n be a solution of the discrete problem emanating from initial data 3.10 with the pressure satisfying 1.4, where γ 3/2. Then there exists a positive number such that for all m = 1,..., N, there holds: c = c M 0, E 0, ϱ, p C 1 [ϱ/2,2ϱ], V, Π X k T,T 3 E ε ϱ m, u m ϱ, ˆV m h + µ 2 m T [ c h a + + ε + E ε ϱ 0, u 0 ϱ, ˆV ] h 0 with any couple Π, V belonging to the class defined in 3.13 and E ε is given in 3.8. Proof. Gathering the formulas 6.11 and 7.1, one gets x u n V n h 2 dx m + c E ε ϱ n, u n ϱ, ˆV n h, 1.8 k 4 satisfying on [0, T T 3, where a is E ε ϱ m, u m m ϱ, ˆV h E ε ϱ 0, u 0 m ϱ, ˆV h 0 + µ u n V n h 2 P L T 2 ;R P 2 + Q, 8.1 where P 1 = P 2 = m ϱ n 1 T m T = L E Q = R m h, + Rm h, + Gm. ϱv n h, V n 1 h, ϱ n,up n,up ϱ ˆV h, V n h, u n, û n,up V n h, V n h, V n,up h, n,, 22
25 Now, we estimate conveniently the terms P 1 and P 2 in two steps. Step 1: Term P 1. We estimate the L norm of V n n 1 h, V h, by L norm of t V in the same manner as in 6.2. According to Lemma 7.2, ϱ r γ 1 R+ \[ϱ/2,2ϱ]ϱ cpe p ϱ r, with any p 1; in particular, ϱ r 6/5 1 R+ \[ϱ/2,2ϱ]ϱ ceϱ r. 8.2 We get by using the Hölder inequality, [ T ϱ n 1 T ϱv n h, V n 1 h, V n h, u n c t V L Q T ;R 3 1/2 ϱ n 1 5/6 ] ϱ 2 1 [ϱ/2,2ϱ] ϱ + ϱ n 1 ϱ 6/5 1 R+ \[ϱ/2,2ϱ]ϱ T T V n h, u n 6 1/6 c t V L Q T ;R 3 εeε 1/2 ϱ n 1, û n 1 ϱ, +ε 5/3 Eε 5/6 ϱ n 1, û n 1 n 1 1/6, ϱ, ˆV h V n h, u n 6 L 6 ;R 3 T ˆV n 1 h where we have used 8.2 and estimate 4.8 to obtain the last line. Now, we write V n h, un = [V n h un ] V n h un + V n h un and use the Minkowski inequality together with formulas 2.21, 2.16 to get 1/6 V n h, u n 6 1/2. L 6 ;R 3 V n h u n 2 L 2 ;R 3 T T Finally, employing Young s inequality, and estimate 4.8, we arrive at P 1 ε cδ, M 0, E 0, ϱ, V, x V, t V L Q T,R 15 E ε ϱ 0, û 0 ˆr 0, ˆV 0 m h + E ε ϱ n, û n ˆr n, ˆV n m h + δ V n h u n 2 L 2 ;R 3, 8.3 T with any δ > 0, provided ε 0, 1. Step 2: Term P 2. We rewrite V n h, V n h, = V n h, V n h, + [V n h V n h ] + [V n h V n h ] and estimate the L norm of this expression by h x V L Q T ;R 9 by virtue of , 2.14 s=1. Now we write P 2 = m P2 n where Lemma 7.2 and the Hölder inequality yield, similarly as in the previous step, P2 n cϱ, V L Q T ;R 9 h E 1/2 ϱ n,up ϱ + E 2/3 ϱ n,up n,up n,up ϱ ˆV h,0, ˆV h,0, û n,up T E [ cϱ, V, V L Q T ;R 12 + T E heϱ n,up ϱ T E 2/3 ] 1/2 h Eϱ n,up ϱ T E h ˆV n,up h, û n,up 6 1/6, provided γ 3/2. Next, we observe that the contribution of the face = L to the sums T E heϱn,up ϱ and n,up T E h ˆV h, û n,up 6 is less or equal than 2 heϱ n ϱ + 23
26 Eϱ n L ϱ, and than 2 h V n h, un 6 + V n h,l un L 6, respectively. Consequently,we get by the same reasoning as in the previous step, under assumption γ 3/2, m P 2 cδ, M 0, E 0, ϱ, V, V L Q T ;R 12 E ε ϱ n, û n ϱ, ˆV n m h+δ V n h u n 2 L 2 ;R 3. T 8.4 Gathering the formulas 8.1 and with δ sufficiently small with respect to µ, we conclude the proof of Lemma 8.1 and a fortiori also proof of Theorem 3.1 by applying Gronwall inequality to Lemmma Numerical experiments To illustrate the behavior of the present scheme, we perform two numerical tests in the simplified 2D geometry. We use the equation of state pϱ = ϱ γ with the Poisson adiabatic constant of dry air γ = 1.4. In order to guarantee convergence of the fixed point iterations of the discrete nonlinear system, we impose the time step via the standard CFL-condition = CFL h u +c/ε, c = γp/ρ, with the choice of CFL = Example 1 Following Haack et al. [25] we consider the flow of a vortex in the box Ω = [ 1, 1] 2, supplemented with the no-slip boundary conditions for the velocity. The initial data of the compressible Navier-Stokes system are u 1 x, y, 0 = sin 2 πx sin2πy u 2 x, y, 0 = sin2πx sin 2 πy 9.1 ρx, y, 0 = 1 ε2 2 tanhy 0.5. Time evolution of the flow field at t = 0, 0.1, 0.2 is shown in Figure 1. As time evolves the amplitude of solution decreases due to physical diffusion. Table 1 presents the experimental order of convergence for the scheme for different values of the Mach number ε ranging from 0.8 to The error is evaluated by means of the reference solution computed at a fine mesh h = 1/256. The simulations are performed on consecutively refined meshes with h = 1/2 N, N = 3,..., 7, T = We can clearly recognize the expected first order convergence rate in the momentum q 1, q 2, which is uniform with respect to ε. Numerical convergence rate for velocity tends to the second order, as will be expected. The velocity gradients as well as density converge linearly. The first order convergence rate for density for low Mach number ε = would be recovered using finer grid resolution. Relative energy error between the compressible and the incompressible Navier-Stokes equations in the low Mach number limit In order to highlight the convergence result of Theorem 3.1, we evaluate the difference between the compressible and incompressible numerical solutions. The solution of the compressible Navier-Stokes system is computed by means of the scheme for different meshes. The solution of the limit incompressible Navier-Stokes equations is obtained by the pressure stabilized Lagrange-Galerkin method of Notsu and Tabata [38] at a fine mesh h = 1/512. Let ϱ, p = ϱ γ, u be the numerical solution of the compressible Navier-Stokes systen, and let π, V be the solution of incompressible Navier-Stokes equations; we set z = 1 + ε 2 Π 1/γ. 24
27 The error is measured by e E = sup E ε ρ n, û n zt n,, V t n,, e xu = x u V L 2 0,T ;L 2 Ω, 1 n N e u = u V L 2 0,T ;L 2 Ω, e ρ = ρ z L 2 0,T ;L 2 Ω, e p = p z γ L 2 0,T ;L 2 Ω. Table 2 presents the results for two different viscosity constants, µ = 0.01 and µ = 1. We set ε = h. Numerical simulations show the second order convergence rate for the relative entropy with respect to h and ε and indicate that our theoretical convergence rates of first order for γ 3 are suboptimal. Furthermore, numerical experiments show that pressure and density of the compressible Navier-Stokes system converge to their incompressible limit with the second order rate. 9.2 Example 2 In this numerical experiment we choose a particular solution of the incompressible Navier-Stokes equations, the so-called unsteady Taylor vortex flow with periodic boundary conditions in x and y directions V 1 x, y, t = sin2πx cos2πy e 8π2 µt V 2 x, y, t = cos2πx sin2πy e 8π2 µt Πx, y, t = 1 4 cos4πx + cos4πy e 16π2 µt. For the compressible Navier-Stokes system the corresponding initial data are 9.2 ρx, y, 0 = 1 + ε 2 Πx, y, 0 u 1 x, y, 0 = V 1 x, y, 0 u 2 x, y, 0 = V 2 x, y, Time evolution of the flow field at t = 0, 0.1, 0.2 is shown in Figure 2. Analogously as in the previous experiment we present in Table 3 experimental orders of convergence for different Mach numbers varying from 0.8 to We obtain the first order convergence rates for the momentum, velocity gradients and density uniformly with respect to the Mach number. The L 2 -norm of velocity converges with the second order. To study the relative energy error between the compressible solution and its incompressible limit we use the exact solution of the incompressible Taylor vortex flow 9.2 and apply the numerical scheme to solve numerically the compressible Navier-Stokes system In Figure 3 the error of velocity component u 1 for different Mach numbers and mesh sizes is presented. As expected, the error between the compressible and incompressible solution decreases, when either the mesh size h or the Mach number ε tend to 0. Table 4 and 5 demonstrate the same results. Table 4 presents the convergence rates for physically relevant adiabatic exponent γ = 1.4, which is not covered by our theoretical analysis. On the other hand, Table 5 shows the convergence rates, for the case, when Theorem 3.1 gives the first order convergence of the relative energy. In both cases we obtained for different viscosities µ = 0.01 and µ = 1 the second order convergence of the relative energy, density and pressure and the first order rates for the velocity and its gradient. 10 Conclusion In the present paper we have studied both theoretically and numerically convergence of the solution of the compressible Navier-Stokes esystem to its incompressible limit The distance between a discrete solution of a convenient numerical approximation of the compressible problem and the exact solution of the incompressible Navier-Stokes equations is measured by means of the relative energy functional E, cf For the numerical scheme we use the combined finite element-finite volume method based on the Crouzeix-Raviart finite elements for velocity and piecewise constant approximation for density. The convective terms are approximated by means of upwinding. 25
28 Our main result is presented in Theorem 3.1. We have shown that in the case that the data are wellprepared, cf. Remark 3.1, we obtain the uniform convergence of order O { }, h a, ε, a = min 2γ 3 γ, 1, γ 3/2. First of all we have shown the uniform apriori estimates in suitable norms, see Section 4. The crucial step of the proof is to derive the discrete relative energy inequality. Consistency analysis presented in Section 7 shows suitable discrete identities and estimates for the discrete relative energy, that are obtained in the case when the strong solution of the incompressible Navier-Stokes equations is a test function in the discrete relative energy. A Gronwall-type argument concludes the proof and yields the corresponding convergence rates. As far as we know this is the first result for the error estimates between the numerical solution of the compressible Navier-Stokes equations and the solution of the limiting incompressible Navier-Stokes equations. We would also like to point out that these estimates are uniform with respect to all three parameters, h and the Mach number ε. Thus, our results demonstrate asymptotic preserving property of the proposed numerical scheme. Our extensive numerical experiments show second order convergence rate of the relative energy. Consequently, theoretical results presented in Theorem 3.1 seem to be suboptimal. References [1] H. Bijl, P. Wesseling. A unified method for computing incompressible and compressible flows in boundary-fitted coordinates. J. Comput. Phys., 1412: , 1998 [2] G. Bispen. IMEX finite volume schemes for the shallow water equations. PhD-thesis, University of Mainz, [3] G. Bispen,. R. Arun, M. Lukáčová-Medviďová, S. Noelle. IMEX large time step finite volume methods for low Froude number shallow water flows. Comm. Comput. Phys., 16: , 2014 [4] G. Bispen, M. Lukáčová-Medviďová, L. Yelash. Asymptotic preserving IMEX finite volume schemes for low Mach number Euler equations with gravitation. Preprint JGU Mainz, 2016 [5] F. Brezzi, M. Fortin. Mixed and hybrid finite elements methods. Springer series in computational mathematics. [6] C. Cancès, H. Mathis, N. Seguin. Relative entropy for the finite volume approximation of strong solutions to systems of conservation laws. HAL : hal , [7] M. Crouzeix, P. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Série Rouge, 7:33 75, 1973 [8] P. Degond, M. Tang. All speed scheme for the low Mach number limit of the isentropic Euler equations. Commun. Comput. Phys., 101:1 31, [9] R. Eymard, T. Gallouët, M. Ghilani, R. Herbin. Error estimates for the approximate solutions of a non-linear hyperbolic equation given by a finite volume scheme. IMA J. Numer. Anal. 184: , 1998 [10] L. C. Evans, R. F. Gariepy. Measure theory and fine properties of functions. CRC Press, 1992 [11] E. Feireisl, B. J. Jin, A. Novotný. Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech., 144: , 2012 [12] E. Feireisl, T.. arper, A. Novotný. A convergent numerical scheme for the full Navier-Stokes- Fourier system. IMA J. Numer. Anal., on line first
29 [13] E. Feireisl, M. Lukáčová-Medviďová. Convergence of a mixed finite element-finite volume scheme for the isentropic Navier-Stokes system via dissipative measure-valued solutions. Preprint Czech Academy of Sciences IM , arxiv: , 2016 [14] E. Feireisl, A. Novotný. Singular limits in thermodynamics of viscous fluids. Birkhauser, 2009 [15] E. Feireisl, A. Novotný. Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal., 2042: , 2012 [16] E. Feireisl, A. Novotný, Y. Sun. Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids. Indiana Univ. Math. J., 602: , 2011 [17] M. Feistauer, V. Dolejší, V. učera. On the discontinuous Galerkin method for the simulation of compressible flow with wide range of Mach numbers. Computing and Visualization in Science, 10:17 27, 2007 [18] M. Feistauer, V. učera. On a robust discontinuous Galerkin technique for the solution of compressible flow. J. Comput. Phys.,224 : , 2007 [19] J. Fischer. A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation. SIAM J. Numer. Anal., 53 : , 2015 [20] T. Gallouët, L. Gastaldo, R. Herbin, J.-C. Latché. An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations. M2AN Math. Model. Numer. Anal., 422: , 2008 [21] T. Gallouët, R. Herbin, J.-C. Latché. A convergent finite element-finite volume scheme for the compressible Stokes problem. I. The isothermal case. Math. Comp., 78267: , 2009 [22] T. Gallouët, R. Herbin, D. Maltese, A. Novotný. Error estimate for a numerical approximation to the compressible barotropic navier-stokes equations. IMA J. Numer. Anal., 2015 published on-line [23] L. Gastaldo, R. Herbin, W. heriji, C. Lapuerta, J.-C. Latché. Staggered discretizations, pressure correction schemes and all speed barotropic flows, Finite volumes for complex applications. VI. Problems and perspectives. Volume 1, 2, Springer Proc. Math., 4: , Springer, Heidelberg, 2011 [24] L. Gastaldo, R. Herbin, J.-C. Latché, N. Therme. Consistency result of an explicit staggered scheme for the Euler equations. Preprint, 2014 [25] J. Haack, S. Jin, J.-G. Liu. An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equations. Commun. Comput. Phys., 12: , 2012 [26] R. Herbin, W. heriji, J.-C. Latché. On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations, ESAIM Math. Model. Numer. Anal., 486: , 2014 [27] J. Jang, F. Li, J.-M. Qiu, T. Xiong. Analysis of asymptotic preserving DG-IMEX schemes for linear kinetic transport equations in a diffusive scaling. SIAM J. Numer. Anal., 524: , 2014 [28] S. Jin. Asymptotic preserving AP schemes for multiscale kinetic and hyperbolic equations: a review. Riv. Mat. Univ. Parma, 3: , 2012 [29] T.. arper. A convergent FEM-DG method for the compressible Navier-Stokes equations. Numer. Math., 1253: , 2013 [30] S. lainerman, A. Majda. Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math., 34: ,
30 [31] R. lein. Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I. One-dimensional flow. J. Comput. Phys., 1212: , 1995 [32] B. Liu. The analysis of a finite element method with streamline diffusion for the compressible Navier-Stokes equations. SIAM J. Numer. Anal., 381:1 16 electronic, 2000 [33] B. Liu. On a finite element method for three-dimensional unsteady compressible viscous flows. Numer. Methods Partial Differential Equations, 203: , 2004 [34] J. Liu, L. Mieussens. Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit. SIAM J. Num. Anal., 48: , 2010 [35] M. Lukáčová Medviďová, A. Müller, V. Wirth, L. Yelash. Adaptive discontinuous evolution Galerkin method for dry atmospheric flow. J. Comput. Phys., 268: , 2014 [36] C.-D. Munz, S. Roller, R. lein,.-j. Geratz. The extension of incompressible flow solvers to the weakly compressible regime. Computers & Fluids, 322: , 2003 [37] S. Noelle, G. Bispen,. R. Arun, M. Lukáčová-Medviďová, C.-D. Munz. A weakly asymptotic preserving low Mach number scheme for the Euler equations of gas dynamics. SIAM J. Sci. Comput., 366: , 2014 [38] H. Notsu, M. Tabata. Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations. M2AN, 502: , 2016 [39] J. H. Park, C.-D. Munz. Multiple pressure variables methods for fluid flow at all Mach numbers, Internat. J. Numer. Methods Fluids, 498: ,
31 a streamline b u 1 c u 2 Figure 1: Example 1: Time evolution of the streamline and velocity components with µ = 0.01, h = 1/64, ε = at t = 0, 0.1, 0.2 from left to right. 29
32 Table 1: Example 1: Experimental order of convergence, µ = a ε = 8e 1 h ρ L 2 EOC q 1 L 2 EOC q 2 L 2 EOC u L 2 EOC u L 2 EOC 1/8 6.61e e e e e-01 1/ e e e e e / e e e e e / e e e e e / e e e e e b ε = 1e 1 h ρ L 2 EOC q 1 L 2 EOC q 2 L 2 EOC u L 2 EOC u L 2 EOC 1/8 3.49e e e e e-01 1/ e e e e e / e e e e e / e e e e e / e e e e e c ε = 1e 2 h ρ L 2 EOC q 1 L 2 EOC q 2 L 2 EOC u L 2 EOC u L 2 EOC 1/8 2.09e e e e e+00 1/ e e e e e / e e e e e / e e e e e / e e e e e d ε = 1e 3 h ρ L 2 EOC q 1 L 2 EOC q 2 L 2 EOC u L 2 EOC u L 2 EOC 1/8 1.17e e e e e-01 1/ e e e e e / e e e e e / e e e e e / e e e e e
33 Table 2: Example 1: Convergence of the compressible solution to the incompressible solution for different viscosities µ, t = a ε = h, µ = 0.01 h e E EOC e xu EOC e u EOC e ρ EOC e p EOC 1/8 1.12e e e e e-04 1/ 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 b ε = h, µ = 1 h e E EOC e xu EOC e u EOC e ρ EOC e p EOC 1/8 5.42e e e e e-03 1/ 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
34 a streamline b u 1 c u 2 Figure 2: Example 2: Time evolution of the streamline and velocity components with µ = 0.01, h = 1/64, ε = at t = 0, 0.1, 0.2 from left to right. 32
35 Table 3: Example 2: Experimental order of convergence, µ = a ε = 8e 1 h ρ L 2 EOC q 1 L 2 EOC q 2 L 2 EOC u L 2 EOC u L 2 EOC 1/8 3.06e e e e e+00 1/ e e e e e / e e e e e / e e e e e / e e e e e b ε = 1e 1 h ρ L 2 EOC q 1 L 2 EOC q 2 L 2 EOC u L 2 EOC u L 2 EOC 1/8 1.53e e e e e+00 1/ e e e e e / e e e e e / e e e e e / e e e e e c ε = 1e 2 h ρ L 2 EOC q 1 L 2 EOC q 2 L 2 EOC u L 2 EOC u L 2 EOC 1/8 1.02e e e e e+00 1/ e e e e e / e e e e e / e e e e e / e e e e e d ε = 1e 3 h ρ L 2 EOC q 1 L 2 EOC q 2 L 2 EOC u L 2 EOC u L 2 EOC 1/8 1.00e e e e e+00 1/ e e e e e / e e e e e / e e e e e / e e e e e
36 a ε = 1e 1 b ε = 1e 3 Figure 3: Example 2: Error of velocity component u 1 for different mesh sizes and Mach numbers; h = 1 16, 1 64, left to right, ε = 0.1, top to bottom. 34
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