A posteriori estimates for the stationary Stokes problem in exterior domains

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ A posteriori estimates for the stationary Stokes problem in exterior domains © 2020 American mathematical society Accepted version (Final draft) Pauly, D.; Repin, S. Pauly, D., & Repin, S. (2019). A posteriori estimates for the stationary Stokes problem in exterior domains. St. Petersburg Mathematical Journal, 31(3), Article 3. https://doi.org/10.1090/spmj/1613 2019 Algebra i analiz St. Petersburg Math. J. Tom. 31 (2019), }3 S 1061-0022(XX)0000-0 A POSTERIORI ESTIMATES FOR THE STATIONARY STOKES PROBLEM IN EXTERIOR DOMAINS D. PAULY AND S. REPIN Dedicated to the memory of S. G. Mikhlin Abstract. This paper is concerned with the analysis of the inf-sup condition arising in the stationary Stokes problem in exterior domains and applications to the derivation of computable bounds for the distance between the exact solution of the exterior Stokes problem and a certain approximation (which may be of a rather general form). In the first part, guaranteed bounds are deduced for the constant in the stability lemma associated with the exterior domain. These bounds depend only on known constants and the stability constant related to bounded domains that arise after suitable truncations of the unbounded domains. The lemma in question implies computable estimates of the distance to the set of divergence free fields defined in exterior domains. Such estimates are crucial for the derivation of computable majorants of the difference between the exact solution of the Stokes problem in exterior domains and an approximation from the admissible (energy) class of functions satisfying the Dirichlet boundary condition but not necessarily divergence free (solenoidal). Estimates of this type are often called a posteriori estimates of functional type. The constant in the stability lemma (or equivalently in the inf-sup or LBB condition) serves as a penalty factor at the term that controls violations of the divergence free condition. In the last part of the paper, similar estimates are deduced for the distance to the exact solution for nonconforming approximations, i.e., for those that may violate some continuity and boundary conditions. The case where the dimension of the domain equals 2 requires a special consideration because the corresponding weighted spaces differ from those natural for the dimension 3 (or larger). This special case is briefly discussed at the end of the paper where similar estimates are deduced for the distance to the exact solution of the exterior Stokes problem. §1. Introduction 1.1. Notation and nomenclature. Throughout the paper we consider domains in Rd, d≥2, with Lipschitz boundaries. The symbol ωis used for bounded domains and the boundary of such a domain is denoted by γ(typically, the latter is composed of two open and disjoint parts γDand γNassociated with the Dirichlet and Neumann parts). Exterior domains (i.e., those having the form Rd\ω) are denoted by Ω. By the letter D, we denote domains which may be bounded or unbounded depending on the context (if this property is not necessary to outline). For Lebesgue and Sobolev spaces of functions (scalar, vector, or tensor valued) with generalised square integrable derivatives of the first order we use the standard notation L2(ω) and H1(ω) (or L2(Ω) and H1(Ω)), respectively. The standard inner product, norm, and orthogonality in L2(ω) will be denoted by h·,·i0,ω,k · k0,ω, and ⊥0,ω. If γD6=∅, then the homogeneous Dirichlet boundary conditions are encoded in the space H1 γD(D), 2010 Mathematics Subject Classification. 35J57,65N15,76D07. Key words and phrases. Stationary Stokes problem, exterior domains, inf-sup condition, a posteriori estimates. c 2004 American Mathematical Society 1 2 D. PAULY AND S. REPIN which is defined as the closure of compactly supported smooth functions vanishing on γDin the norm of H1. Also, for bounded domains we use spaces with vanishing mean values1 L2 ⊥(ω) := L2(ω)∩R⊥0,ω =nφ∈L2(ω) : hφ, 1i0,ω = 0o=nφ∈L2(ω) : Zω φ= 0o, H1 ⊥(ω) := H1(ω)∩L2 ⊥(ω) = nφ∈H1(ω) : Zω φ= 0o. To handle the special case of γD=∅using a unified notation, we introduce the space L2 γD(ω) := (L2(ω) if γD6=γ, L2 ⊥(ω) if γD=γ, and for the case where γD=∅redefine H1 γD(ω) by setting H1 γD(ω) = H1 ⊥(ω). To further unify our definitions and extend them to exterior domains, we consider a domain (an open and connected set) D ⊂ Rd,d≥2. This domain may be bounded or unbounded. It has a Lipschitz boundary B, which consists of two relatively open and disjoint parts BD,BN⊂ B (such that B=BD∪ BN) associated with Dirichlet and Neumann boundary conditions. As before, we denote the standard Lebesgue and Sobolev spaces by L2(D) and H1(D), respectively. If BD6=∅, we introduce homogeneous Dirichlet boundary conditions in H1 BD(D) defined as the closure of C∞ BD(D) := nφ|D:u∈C∞(Rd),supp φis compact, dist(supp φ, BD)>0o in H1(D). As above we utilise the notations L2 B(D) = L2 ⊥(D), L2 BD(D), and H1 ∅(D) = H1 ⊥(D) provided that Dis bounded. Next, we introduce polynomially weighted spaces L2 ±1(D) := nφ∈L2 loc(D) : ρ±1φ∈L2(D)o, H1 −1(D) := nφ∈L2 −1(D) : ∇φ∈L2(D)o, where the weight function ρis defined by ρ(r) := (1 + r2)1/2, and r(x) := |x|. The inner product, norm, and orthogonality in L2 ±1(D) are denoted by h·,·i±1,D:= ρ±2·,·0,D,k · k±1,D,and ⊥±1,D, respectively. In the case of a bounded domain, there is no difference between the unweighted and weighted spaces (if we mean that the spaces coincide as sets and possess different inner products). However, in analysis of problems in exterior domains a proper selection of weights is important (in §4.6 devoted to the case of d= 2 we define the weighted spaces differently). As before, if BD6=∅, then the homogeneous Dirichlet boundary conditions are encoded in H1 −1,BD(D), the closure of C∞ BD(D) in H1 −1(D). Finally, for the Stokes equations, we introduce spaces of solenoidal fields S(D) := nϕ∈H1(D) : div ϕ= 0o,SBD(D) := H1 BD(D)∩S(D), S−1(D) := nϕ∈H1 −1(D) : div ϕ= 0o,S−1,BD(D) := H1 −1,BD(D)∩S−1(D). 1Throughout this paper, we do not express the respective measure in the notation of integrals, so that, e.g., we often use the notation like this: Zω f=Zω f dλ =Zω f dx, Zγ f=Zγ f do =Zγ f ds. THE STATIONARY STOKES PROBLEM IN EXTERIOR DOMAINS 3 1.2. Stability lemma and the Stokes problem in bounded domains. The classical stationary Stokes problem consists of finding a vector field u(velocity) and a scalar valued function p(pressure) that solve the system −ν∆u+∇p=fin ω,(1) div u= 0 in ω,(2) u=uDon γD,(3) σn = 0 on γN,(4) where σ:= ν∇u−pI,ν(viscosity) is a positive constant or a positive function taking values in [ν⊖, ν⊕], ν⊖, ν⊕>0, and f∈L2(ω). The boundary conditions are defined by the vector valued function uD. Henceforth, we assume that uDis given by a solenoidal vector field uD, i.e., the Dirichlet boundary condition is defined by uD∈S(ω) in the sense that u=uDon γD, i.e., u−uD∈H1 γD(ω). If γ=γD, then we additionally assume that Zγ n·uD=Zω div uD=huD,1i0,ω = 0.(5) The existence of the corresponding generalised solution follows from the well-known solution theory for uniformly elliptic linear equations and the stability lemma, which plays an important role in the theory of incompressible flows. Lemma 1.1 (stability lemma, [22, 1, 3, 13, 14]).There exists c > 0such that for any g∈L2 γD(ω)there is a vector field ug∈H1 γD(ω)with div ug=gand k∇ugk0,ω ≤ckgk0,ω.(6) Henceforth, the best constants in (6) and similar inequalities for unbounded domains are denoted by the letter κ, i.e., κ(ω, γD) is the smallest cin (6). For u∈H1 γD(ω) we also have the Friedrichs/Poincar´e inequality kuk0,ω ≤ck∇uk0,ω, and cFP (ω, γD) denotes the best constant c. Hence from Lemma 1.1, we conclude that ugsatisfies the inequalities 1 cFP (ω, γD)kugk0,ω ≤ k∇ugk0,ω ≤κ(ω, γD)kdiv ugk0,ω. We notice that in the theory of electrodynamics the function ugis called a regular potential as it admits (for Maxwell’s equations) an unphysical (high) regularity and boundary condition, which is much stronger than the usual normal boundary condition related to the divergence operator. Lemma 1.1 yields several important corollaries. First, it guarantees the solvability of the stationary Stokes problem (in the velocity-pressure posing). By setting g= div ug, Lemma 1.1 immediately yields the well-known inf-sup (or LBB) condition: (7) inf g∈L2 γD(ω)sup u∈H1 γD(ω) hg, div ui0,ω kgk0,ωk∇uk0,ω ≥1 κ(ω, γD)=: cLBB . Another direct corollary to Lemma 1.1 is an estimate of the distance between a vector field u∈H1 γD(ω) and the set SγD(ω) (see [35, 36]), dist u, SγD(ω):= inf v∈SγD(ω)∇(u−v)0,ω. 4 D. PAULY AND S. REPIN Corollary 1.2. For any u∈H1 γD(ω)there exists u0∈SγD(ω)such that dist u, SγD(ω)≤∇(u−u0)0,ω ≤κ(ω, γD)kdiv uk0,ω. Proof. For u∈H1 γD(ω), solve the equation div eu= div u∈L2 γD(ω) with eu∈H1 γD(ω) and the stability estimate k∇euk0,ω ≤κ(ω, γD)kdiv uk0,ω by Lemma 1.1. Note that for γD=γwe have Zγ n·u=Zω div u=hu, 1i0,ω = 0.(8) Then u0:= u−eu∈SγD(ω) and ∇(u−u0)0,ω =∇eu0,ω ≤κ(ω, γD)kdiv uk0,ω. In [38, 39, 40], this result was extended to vector fields satisfying nonhomogeneous Dirichlet boundary conditions (and also for vector fields in W1,q(Ω) for q∈(1,∞)) provided that such a vector field usatisfies div u∈L2 γD(Ω), i.e., the mean value condition (8), if γD=γ. Moreover, it was shown that if the mean value conditions hold true for a collection of subdomains whose union of closures coincides with the closure of ω, then estimates of the distance can be based on local constants associated with subdomains. In the case of nonhomogeneous boundary conditions, a modified version of Corollary 1.2 reads as follows. Corollary 1.3. For any u∈H1(ω)with div u∈L2 γD(ω)there exists a solenoidal u0∈ S(ω)such that u0−u∈H1 γD(ω), i.e., u0|γD=u|γD, and ∇(u0−u)0,ω ≤κ(ω, γD)kdiv uk0,ω. It should be noted that Corollary 1.3 can be also viewed as a lifting lemma, because a boundary datum u|γDis lifted to the domain ω. In this case lifting is performed with the help of a solenoidal representative. Estimates of the constant κ(ω, γD) have been studied in [10, 25, 29, 44, 6] and some other publications. It is not difficult to see that the constant cLBB in (7) is nonnegative and cannot exceed 1 so that κ(ω, γD)≥1. It is known that cLBB >0 for any bounded Lipschitz domain (e.g., cLBB = 1/√dfor a ball in Rd). However, the exact values of this constant are unknown except some for very special cases (for example, we do not know the constant even for a cube!). In [6], simply computable and sufficiently accurate estimates of the constant were obtained for domains in R2that are included in a ball of radius Rand are star-shaped with respect to a concentric ball of radius ρ. It was shown that (9) κ(ω, γ)≤√2 ζ1 + p1−ζ21/2, where ζ=ρ/R. For d= 3, estimates of cLBB are known only for domains with sufficiently regular boundaries (see [29]). A systematic numerical analysis of constants in the infsup condition (7) was performed in [11], where approximate values of the constants were computed for a wide collection of bounded domains. Computational approaches to the evaluation of the distance to the set of divergence free fields based on domain decomposition were suggested in [38, 39, 40]. In our subsequent analysis, we assume that, using the results and methods mentioned above, we are able to find a majorant of THE STATIONARY STOKES PROBLEM IN EXTERIOR DOMAINS 5 the constant κ(ω, γD) for bounded domains ωthat arise as truncations of an unbounded domain Ω. 1.3. A posteriori estimates. Estimates of the distance to SγD(ω) are not merely of theoretical value. They are important for the quantitative analysis of boundary value problems generated by incompressible media models (e.g., in the theory of viscous incompressible fluids). First, estimates of this type are necessary for getting computable bounds for the difference between the exact solution of a boundary value problem and an approximation obtained by some computational procedure. The term “computable” means that the corresponding estimates do not involve unknown functions and constants and can indeed be computed by means of an approximate solution only. Estimates of this type are often called a posteriori error estimates and nowadays are widely used in the quantitative analysis of mathematical problems. Unlike the a priori (asymptotic) convergence estimates, a posteriori estimates provide an explicit verification of the accuracy of a particular numerical solution. First methods of a posteriori error control for PDEs originates from the works of W. Prager and J. L. Synge [33] and S. G. Mikhlin [19]. The latter monograph contains a pioneering idea of a new approach to error estimation, which differs principally from asymptotic rate convergence estimates dominated at that time and several decades subsequently. For variational problems generated by quadratic type functionals (10) J(v) = 1 2a(v, v)−hf, vi, f ∈V, where Vis a Hilbert space and a:V×V→Ris a V-elliptic bilinear form, S. Mikhlin deduced the principal relation (11) 1 2a(u−v, u −v) = J(v)−J(u). Here uis the minimizer that satisfies J(u) = min w∈VJ(w) and v∈V is any function compared with u. Since the exact infimum is unknown, it is impossible to use (11) directly. In [19], it was suggested to estimate J(u) from below using a dual variational problem and further apply the orthogonal projection method of H. Weyl [48]. Certainly, these first estimates were derived for a rather limited set of problems and suffered from serious restrictions imposed on the set of functions that are admissible in the dual setting. For these reasons, they were rarely used in computational practice. Moreover, the methods developed in 1970–1980 for measuring errors of finite element approximations (such as the “gradient averaging” and “residual” methods, see, e.g., [47] and the references therein) were based on different grounds. These methods strongly exploit properties of a particular approximation computed on a particular mesh. In essence, they provide certain error indicators (for mesh adaptive procedures) rather than guaranteed error bounds. Subsequent studies focused on the problem of guaranteed error control (performed in the 1990s) confirmed the idea (encompassed in (11)) that the corresponding methods should be justified on the functional level by means of the same mathematical tools that are used in analysis of PDEs without attracting specific features of approximations and numerical methods. If we have a general (universal) estimate of the distance between a function and the exact solution of a boundary value problem, then it can be used with any approximation and requires no changes if one approximation (mesh) is replaced by another. In the last two decades computable bounds of this type has been derived and tested for a wide spectrum of problems (see [34, 23, 37, 17] and many other publications cited in these monographs). For clear reasons, they are often called 6 D. PAULY AND S. REPIN a posteriori estimates of functional type. They differ from others due to two important properties: the estimates (a) do not contain constants associated with a particular finite dimensional subspace (mesh) and a method used to solve the problem and (b) are valid for any approximation in the energy space and do not use special conditions required for the exact solution (e.g., extra regularity) or its approximation (e.g., Galerkin orthogonality, quasi–uniformity of meshes). A posteriori estimates of functional type involve only global constants generated by functional inequalities such as various embedding estimates, trace inequalities, Poincar´e, Maxwell, Korn inequalities, etc. It should be noted that although constants of this type do not appear in (11) (and in the estimates derived in [33]), the importance of studying them was already understood by Mikhlin (see [20]). The reader can find an overview of the history of a posteriori error estimation methods and a subsequent exposition of the functional approach to the problem in [37]. First estimates of the distance between a function in the energy space and the exact solution of the stationary Stokes problem in a bounded domain were derived in [35] (by means of the variational duality method) and in [36] (by transformations of the integral identity that defines the corresponding weak solution). It is worth starting a short overview of these results with the error identity (12) ν∇(v−u)2+ν−1kτf−σk2= 2(I(v)−I∗(τf)), which can be viewed as an analog of (11) for the stationary Stokes problem. Here v∈Sγ(ω) and τf∈L2 f(ω) := nτ∈L2(ω) : hτ, ∇wi0,ω =hf, wi0,ω ∀w∈Sγ(ω)o are regarded as approximations of the exact velocity field uand exact stress fiels σ, respectively. Identity (12) is fulfilled for any v∈Sγ(ω) and any τf∈L2 f(ω). However, it is not very useful for practice for the same reasons as (11), namely, the functions in Sγ(ω) and L2 f(ω) are subject to differential relations. In [35] (see also [37]), a way was shown to overcome these difficulties by using computable estimates of distances to the sets Sγ(ω) and L2 f(ω). As a result, the following estimates for the velocity and pressure fields were derived: ν∇(u−eu)0,ω ≤ kτ+epI−ν∇euk0,ω +cFP(ω, γ)kdiv τ+fk0,ω + 2νκ(ω, γ)kdiv euk0,ω, (13) 1 2κ(ω, γ)kp−epk0,ω ≤ kτ+epI−ν∇euk0,ω +cFP(ω, γ)kdiv τ+fk0,ω +νκ(ω, γ)kdiv euk0,ω. (14) Here νis a positive constant and eu∈H1(ω) is a vector-valued function satisfying the Dirichlet boundary conditions. The function euis regarded as an approximation of the exact velocity u. Similarly, epis a square integrable function (with zero mean value if the Dirichlet conditions are imposed on the whole boundary γ) viewed as an approximation of pand τ∈L2(ω) is an approximation of the exact stress field σ. The right-hand sides of (13) and (14) have a clear meaning: they contain three nonnegative terms that vanish if the approximations coincide with the exact velocity, pressure, and stress, respectively. In other cases, the terms can be viewed as penalties for possible violations of the three basic relations that form (1) and (2). It is easy to see that the constant κ(ω, γ) plays an important role in (13) and (14) and, therefore, it is indeed necessary to have guaranteed majorants of this constant. THE STATIONARY STOKES PROBLEM IN EXTERIOR DOMAINS 7 These constants arise in the a posteriori analysis of a numerical solution if it satisfies the divergence free condition only approximately. If the constant κ(ω, γD) is known, then by using Corollary 1.2 we can deduce guaranteed and fully computable error estimates. For problems in bounded Lipschitz domains the respective results were presented in [35, 36, 37, 39] and other publications cited therein. Also, it should be mentioned that explicit bounds of the constant κ(ω, γ) are required not only for Stokes type problems. They arise in other continuous media problems, which are not related to viscous fluids (e.g., see [42]). Subsequently analogous estimates for the velocity and pressure fields were obtained for the Stokes problem in the velocity-vorticity-pressure formulation [18] and for the generalised Stokes problem [41]. In [7], such estimates were derived for a class of stationary problems associated with nonlinear viscous fluids and in [24] for the evolutionary Stokes problem. We note that the approach used in these publications and in the present paper differs essentially from the so-called residual method often used in the finite element community for getting indicators of approximation errors (see, e.g., [46]). 1.4. Outline of the paper. In the first part of the paper, we recall some known results related to the analysis of boundary value problems in exterior domains, paying a special attention to the stability Lemma 2.3 and the corresponding corollaries. §3 is devoted to computable bounds κ⊕(Ω,ΓD) for the stability constant κ(Ω,ΓD). In Lemma 3.2, we obtain a desired estimate for κ⊕(Ω,ΓD), which involves known constants and the constant κ(ω, γD) associated with a bounded domain ω(which is a suitable truncation of Ω). Estimates of the last constant have been derived in several publications cited above, so that we view this problem as solvable (at least in the sense that a certain guaranteed bound of κ(ω, γD) can be derived). As a result, we obtain estimates of the distance between a vector field in Ω and the respective set of solenoidal fields defined in Ω and satisfying the same boundary conditions (Lemma 3.4). These estimates are used in §4, we derive a posteriori error estimates of functional type, which are valid for a wide class of approximate solutions to the stationary Stokes problem in exterior domains. Estimates for the velocity are obtained in three different forms. The first (and the simplest) form is valid for approximations in S(Ω), i.e., for solenoidal vector-valued functions that satisfy the Dirichlet boundary condition exactly (Theorem 4.1). These estimates do not contain the constant κ(Ω,ΓD). Estimates of the second type are valid for approximations in H1(Ω) still satisfying the boundary condition exactly but admit possibly nonsolenoidal functions (Theorem 4.2). They involve a term that penalises possible violation of the solenoidality condition and has the constant κ(Ω,ΓD) as a penalty factor. Finally, the most general form of the estimate is applicable for nonconforming approximations, which even may not belong to the energy class H1(Ω) (Theorem 4.6). It involves one more term that can be viewed as a measure of the distance to the energy class natural for the velocity function. Also, we deduce estimates for approximations of the pressure (Theorem 4.4 and Theorem 4.6) and the stress field (Subsection 4.4). In §4.5 we consider lower bounds and in §4.6 we adapt our results to the special case of space dimension d= 2. §2. Preliminaries 2.1. Exterior domain and main functional inequalities. We consider an exterior domain Ω ⊂Rd, where d≥3 (the special case of d= 2 is studied in Subsection 4.6), with a (strong) Lipschitz boundary Γ, which is composed of two open and disjoint parts ΓD,ΓN⊂Γ (Dirichlet and Neumann parts) with Γ = ΓD∪ΓN. Moreover, we assume 8 D. PAULY AND S. REPIN that there exist 0 < r1< r2such that Rd\Ω⊂Br1and denote (see Figure 1) (15) ω:= Ωr2:= Ω ∩Br2, γ = Γ ∪Sr2, γD:= ΓD∪Sr2, where Brand Srdenote the open ball and the sphere of radius rcentered at the origin in Rd. By ηwe denote a Lipschitz continuous cut-off function, which vanishes in the ball2Br1, equals 1 in Rd\Br2, and takes values in [0,1]. ω Ω η= 1 η= 0 ∇η6= 0 Rd\Ω Γ ΓΓD ΓD Sr2 Sr1 Figure 1. Rd\Ω (gray) surrounded by the boundary Γ (thin black lines), the boundary part ΓD(thick black lines), and the artificial boundary spheres (dashed lines). The two main ingredients for our proofs are Lemma 1.1 and a few elementary results from the theory of ∇-curl-div–systems in exterior domains and especially Rd(see, e.g., [16, 43] or [12, 26] and in particular [28] as well as references therein), which can be summarised in the two subsequent lemmas as follows. Lemma 2.1 (Friedrichs/Poincar´e lemma for exterior domains).The following weighted Friedrichs/Poincar´e estimates hold true. (i) There exists c > 0such that for all v∈H1 −1,ΓD(Ω) we have kvk−1,Ω≤ck∇vk0,Ω. The best constant cis called the Friedrichs/Poincar´e constant and we denote it by cFP(Ω,ΓD). (ii) If ΓD= Γ, then cFP (Ω,Γ) = cF(Ω) (the Friedrichs constant)and cF(Ω) ≤cd:= 2 d−2. Hence for all v∈H1 −1,Γ(Ω) the Friedrichs estimate kvk−1,Ω≤cdk∇vk0,Ωholds true. If ΓD=∅, then cFP (Ω,∅)is replaced by the Poincar´e constant cP(Ω). In this case, the Poincar´e estimate kvk−1,Ω≤cP(Ω)k∇vk0,Ω is fulfilled for all v∈H1 −1(Ω). 2For the sake of simplicity we henceforth operate with the balls Br1and Br2. However, if necessary Br1can be replaced by a Lipschitz domain containing R2\Ω and Br2by another Lipschitz domain containing Br1. THE STATIONARY STOKES PROBLEM IN EXTERIOR DOMAINS 15 i.e., euis possibly nonsolenoidal but satisfies the boundary condition exactly. Corollary 3.5 guarantees the existence of u0∈S−1(Ω) such that u0−eu∈H1 −1,ΓD(Ω) and k∇(u0−eu)k0,Ω≤κ(Ω,ΓD)kdiv euk0,Ω.(36) Hence u0=eu+u0−eu∈uD+H1 −1,ΓD(Ω), i.e., u0∈uD+S−1,ΓD(Ω), and by Theorem 4.1 we have ν1/2∇(u−eu)0,Ω≤ν1/2∇(u−u0)0,Ω+ν1/2∇(u0−eu)0,Ω ≤ν−1/2 ⊖cFP (Ω,ΓD)kdiv τ+fk1,Ω +ν−1/2(τ+qI−ν∇u0)0,Ω+ν1/2∇(u0−eu)0,Ω ≤ν−1/2 ⊖cFP (Ω,ΓD)kdiv τ+fk1,Ω +ν−1/2(τ+qI−ν∇eu)0,Ω+ 2ν1/2 ⊕∇(u0−eu)0,Ω. (37) In view of (36), we obtain the following result. Theorem 4.2. Let eu∈uD+H1 −1,ΓD(Ω). Then for all τ∈DΓN(Ω) and all q∈L2(Ω) we have ν1/2∇(u−eu)0,Ω≤ν−1/2 ⊖cFP(Ω,ΓD)kdiv τ+fk1,Ω +ν−1/2(τ+qI−ν∇eu)0,Ω+ 2ν1/2 ⊕κ(Ω,ΓD)kdiv euk0,Ω. If the approximation euis solenoidal, we recover Theorem 4.1 and, again, the upper bound coincides with the norm of the error on the left-hand side if τ=σ,q=p. If the approximation euis solenoidal only in, e.g., Rd\Br2, then we trivially get an estimate by Theorem 4.2, replacing the term kdiv euk0,Ωby kdiv euk0,ω. But with a moderate additional assumption on the decay of the approximation we can even do better in this case, replacing the constant κ(Ω,ΓD) by a stability constant κ(ω, γD) of the bounded domain ω. For this let eu=uD+w∈uD+H1 −1,ΓD(Ω) with div eu= div w= 0 in Rd\Br2and if γD=γ(i.e., ΓD= Γ), then eusatisfies div eu∈L2 ⊥(ω). To meet the last condition, we additionally assume for the case of ΓD= Γ that |w| ≤ c r−m, m > d −1(38) as r→ ∞ with some c > 0 independent of r(notice that r−m∈L2 −1(Rd\B1) if m > d/2−1). Indeed, it is easy to see that Zω div eu=Zω div w=ZSr n·w≤c rd−1−mr→∞ −−−→ 0 for r > r2. Now we consider the Ansatz u0:= eu+(uωin ω, 0 in Rd\Br2, where uω∈H1 γD(ω). Utilising Lemma 1.1, we find uω∈H1 γD(ω) such that div uω=−div euin ω 16 D. PAULY AND S. REPIN together with the stability estimate k∇uωk0,ω ≤κ(ω, γD)kdiv euk0,ω. The function u0so constructed satisfies the boundary condition on ΓDand it is a solenoidal field, i.e., u0∈uD+S−1,ΓD(Ω). Using u0as an admissible vector field in (37) yields the estimate ν1/2∇(u−eu)0,Ω≤ν−1/2 ⊖cFP(Ω,ΓD)kdiv τ+fk1,Ω +ν−1/2(τ+qI−ν∇eu)0,Ω+ 2ν1/2 ⊕∇(u0−eu)0,Ω | {z } =k∇uωk0,ω . Hence we have the following improved estimate for the case of a partially solenoidal approximation. Corollary 4.3. Let eu∈uD+H1 −1,ΓD(Ω) and let div eu= 0 in Rd\Br2. If ΓD= Γ, then we additionally impose condition (38). Then for all τ∈DΓN(Ω) and all q∈L2(Ω) we have ν1/2∇(u−eu)0,Ω≤ν−1/2 ⊖cFP(Ω,ΓD)kdiv τ+fk1,Ω +ν−1/2(τ+qI−ν∇eu)0,Ω+ 2ν1/2 ⊕κ(ω, γD)kdiv euk0,ω. Here the last term on the right-hand side is a penalty for a possible violation of the solenoidal condition in ω. 4.2. Estimates for the pressure. By Lemma 2.3 there exists a vector field ψ∈ H1 −1,ΓD(Ω) such that div ψ=p−epand k∇ψk0,Ω≤κ(Ω,ΓD)kp−epk0,Ω.(39) For all eu∈uD+H1 −1,ΓD(Ω) and all τ∈DΓN(Ω) we have kp−epk2 0,Ω=hp−ep, div ψi0,Ω =ν∇(u−eu),∇ψ0,Ω−hdiv τ+f, ψi0,Ω+hν∇eu−epI−τ, ∇ψi0,Ω ≤ν∇(u−eu)0,Ω+cFP(Ω,ΓD)kdiv τ+fk1,Ω +kν∇eu−epI−τk0,Ωk∇ψk0,Ω, where we have used Lemma 2.1 for ψand the relation div ψ=I:∇ψ. By (39) we obtain kp−epk0,Ω≤κ(Ω,ΓD)ν1/2 ⊕ν1/2∇(u−eu)0,Ω +cFP (Ω,ΓD)kdiv τ+fk1,Ω+ν1/2 ⊕ν−1/2(τ+epI−ν∇eu)0,Ω. In order to estimate the first term on the right-hand side, we use Theorem 4.2 with q=ep and arrive at the desired estimate for the pressure field. Theorem 4.4. Let ep∈L2(Ω). Then for all τ∈DΓN(Ω) and all eu∈uD+H1 −1,ΓD(Ω) THE STATIONARY STOKES PROBLEM IN EXTERIOR DOMAINS 17 we have kp−epk0,Ω≤κ(Ω,ΓD)(ν−1/2 ⊖ν1/2 ⊕+ 1)cFP(Ω,ΓD)kdiv τ+fk1,Ω + 2ν1/2 ⊕ν−1/2(τ+epI−ν∇eu)0,Ω+ 2ν⊕κ(Ω,ΓD)kdiv euk0,Ω. (40) Remark 4.5.The upper bound (40) consists of the same terms as the upper bound of Theorem 4.2 and vanishes if eu=u,τ=σ,ep=p. However, in this case, the estimate stronger depends on the stability constant κ(Ω,ΓD). A similar effect occurs in the estimates related to bounded domains (see [35, 36]). 4.3. Estimates for nonconforming approximations. The term nonconforming is usually applied to approximations that belong to a functional class wider than the natural energy class of the problem in question. For example, nonconformity of approximations may arise due to violation of continuity conditions or main boundary conditions. Nowadays such type approximations are widely used in computational practice (e.g., mortar, finite volume, and discontinuous Galerkin approximations) because they offer more freedom for various mesh adaptive procedures. Application of functional type a posteriori estimates to nonconforming approximations of elliptic problems was studied earlier in [4, 15, 37, 45]. In this section, we briefly discuss this question in the context of the exterior Stokes problem. Let us now assume that we have a nonconforming approximation e Υ∈L2(Ω) of the exact strain tensor field Υ := ∇u, u =uD+bu∈uD+S−1,ΓD(Ω) ⊂S−1(Ω). For example, e Υ as a “broken gradient” tensor field, the output of some discontinuous Galerkin method. By the triangle inequality, we estimate the difference between these tensor fields using a certain conforming approximation eu∈uD+H1 −1,ΓD(Ω) and obtain ν1/2(Υ −e Υ)0,Ω≤ν1/2∇(u−eu)0,Ω+ν1/2(∇eu−e Υ)0,Ω. Theorem 4.2 implies the estimate ν1/2(Υ −e Υ)0,Ω≤ν−1/2 ⊖cFP(Ω,ΓD)kdiv τ+fk1,Ω+ν−1/2(τ+qI−ν∇eu)0,Ω + 2ν1/2 ⊕κ(Ω,ΓD)kdiv euk0,Ω+ν1/2(∇eu−e Υ)0,Ω. Using the triangle inequality and Theorem 4.4, we obtain a posteriori error estimates for nonconforming approximations. Theorem 4.6. Let e Υ∈L2(Ω) and let ep∈L2(Ω). Then for all τ∈DΓN(Ω),q∈L2(Ω), and eu∈uD+H1 −1,ΓD(Ω) we have ν1/2(Υ −e Υ)0,Ω≤ν−1/2 ⊖cFP(Ω,ΓD)kdiv τ+fk1,Ω+ν−1/2(τ+qI−νe Υ)0,Ω + 2ν1/2 ⊕κ(Ω,ΓD)kdiv euk0,Ω+ 2ν1/2(∇eu−e Υ)0,Ω and kp−epk0,Ω≤κ(Ω,ΓD)(ν−1/2 ⊖ν1/2 ⊕+ 1)cFP (Ω,ΓD)kdiv τ+fk1,Ω + 2ν1/2 ⊕ν−1/2(τ+epI−νe Υ)0,Ω+ 2ν⊕κ(Ω,ΓD)kdiv euk0,Ω + 2ν1/2 ⊕ν1/2(∇eu−e Υ)0,Ω. 18 D. PAULY AND S. REPIN It is easy to see that in the case where e Υ = ∇euis generated by the conforming approximation eu∈uD+H1 −1,ΓD(Ω), the last term vanishes and we recover Theorem 4.2 and Theorem 4.4. 4.4. Estimates for the stress field. Error estimates for the stress tensor field follow directly from the estimates derived above for the velocity vector field and the pressure function. Indeed, let eσ∈L2(Ω) be an approximation of the exact stress tensor σ=ν∇u−pI=νΥ−pI. Moreover, let e Υ∈L2(Ω) and ep∈L2(Ω). Then, the respective error estimate follows from the triangle inequality keσ−σk0,Ω≤ keσ−νe Υ + epIk0,Ω+ν1/2 ⊕ν1/2(Υ −e Υ)0,Ω+d1/2kp−epk0,Ω. In particular, we can set e Υ = ∇eu, where eu∈uD+H1 −1,ΓD(Ω). The first term on the righthand side involves only known tensor fields and the second and third ones are estimated by, e.g., Theorem 4.2, Theorem 4.4, and Theorem 4.6. 4.5. Lower bounds of the error. Let eu∈uD+H1 −1,ΓD(Ω), i.e., u−eu∈H1 −1,ΓD(Ω). Obviously (since the subsequent max-property is true for any Hilbert3space), by (24) we have ν1/2∇(u−eu)2 0,Ω= max ϕ∈H1 −1,ΓD(Ω) 2ν∇(u−eu),∇ϕ0,Ω−kν1/2∇ϕk2 0,Ω ≥2hν∇u, ∇ϕi0,Ω−2hν∇eu, ∇ϕi0,Ω−kν1/2∇ϕk2 0,Ω = 2hf, ϕi0,Ω+ 2hq, div ϕi0,Ω−ν∇(2eu+ϕ),∇ϕ0,Ω+ 2hp−q, div ϕi0,Ω and the maximum is attained at ϕ=u−eu∈H1 −1,ΓD(Ω). The last term can be simply (but rather coarsly) estimated by Theorem 4.4 (ep=q), what yields the estimate presented below. Theorem 4.7. For all eu, v ∈uD+H1 −1,ΓD(Ω) and all ϕ∈H1 −1,ΓD(Ω),τ∈DΓN(Ω), and q∈L2(Ω) we have ν1/2∇(u−eu)2 0,Ω≥2hf, ϕi0,Ω+ 2hq, div ϕi0,Ω−ν∇(2eu+ϕ),∇ϕ0,Ω −2κ(Ω,ΓD)kdiv ϕk0,Ω(ν−1/2 ⊖ν1/2 ⊕+ 1)cFP(Ω,ΓD)kdiv τ+fk1,Ω + 2ν1/2 ⊕ν−1/2(τ+qI−ν∇v)0,Ω+ 2ν⊕κ(Ω,ΓD)kdiv vk0,Ω. In particular, v=euis possible. If ϕ∈S−1,ΓD(Ω), then the simple lower bound ν1/2∇(u−eu)2 0,Ω≥2hf, ϕi0,Ω−ν∇(2eu+ϕ),∇ϕ0,Ω is true. Moreover, equality occurs for ϕ=u−eu, provided that the approximation euis also solenoidal, i.e., eu∈uD+S−1,ΓD(Ω). To handle a nonconforming approximation e Υ∈L2(Ω), we can simply utilise the triangle inequality and apply Theorem 4.7. 3In any Hilbert space Hit is true that kxk2= maxy∈H2hx, yi − kyk2, in our case we can set H=L2(Ω). THE STATIONARY STOKES PROBLEM IN EXTERIOR DOMAINS 19 Theorem 4.8. Let e Υ∈L2(Ω). Then for all ϕ∈H1 −1,ΓD(Ω),τ∈DΓN(Ω),q∈L2(Ω), and eu, v ∈uD+H1 −1,ΓD(Ω), we have ν1/2(Υ −e Υ)0,Ω≥ν1/2∇(u−eu)0,Ω−ν1/2(∇eu−e Υ)0,Ω, ν1/2∇(u−eu)2 0,Ω≥2hf, ϕi0,Ω+ 2hq, div ϕi0,Ω−ν∇(2eu+ϕ),∇ϕ0,Ω −2κ(Ω,ΓD)kdiv ϕk0,Ω(ν−1/2 ⊖ν1/2 ⊕+ 1)cFP(Ω,ΓD)kdiv τ+fk1,Ω + 2ν1/2 ⊕ν−1/2(τ+qI−ν∇v)0,Ω+ 2ν⊕κ(Ω,ΓD)kdiv vk0,Ω. 4.6. Special case: 2D exterior domains. For a Lipschitz domain D ⊂ R2we introduce somewhat different weighted spaces by using logarithmic weights, namely, L2 ±1,ln(D) := nφ∈L2 loc(D) : ρln(e+ρ)±1φ∈L2(D)o, H1 −1,ln(D) := nφ∈L2 −1,ln(D) : ∇φ∈L2(D)o, where eis the Euler number, see, e.g., [16, 43, 9]. We notice that at infinity ρln(e+ρ)±1 behaves like (rln r)±1. The inner product in L2 ±1,ln(D) is defined by the relation hφ , ψ i±1,ln,D:= ρln(e+ρ)±2φ , ψ 0,D. All other weighted spaces and norms are modified and defined in a similar way. The sets Ω ⊂R2and ω⊂R2are defined as in §2, i.e., Ω ⊂R2is an exterior Lipschitz domain and ωis a certain truncation of Ω. The situation is now different from the case of d≥3 because the constants4will be integrable in our weighted spaces, i.e., 1 ∈L2 −1,ln(Ω). Introducing additionally H1 −1,ln,∅(Ω) := H1 −1,ln(Ω) ∩R⊥−1,ln,Ω we have the following Friedrichs/Poincar´e estimate for exterior domains. Lemma 4.9. There exists c > 0such that kvk−1,ln,Ω≤ck∇vk0,Ω for all v∈H1 −1,ln,ΓD(Ω). The best constant cis denoted by cFP(Ω,ΓD). In the special case where Be⊂Rd\Ωand ΓD= Γ, we have cFP (Ω,Γ) ≤2. This theorem follows from [27, Appendix 4.2, Lemma 4.1, Corollary 4.2, Remark 4.3], see also [43, Lemma 4.1] and [28, 16]. It should be noted that in this case we need boundary or mean value conditions as in the case of a bounded domain. Now, all results from the sections for d≥3 follow with obvious modifications. In particular, the stability Lemma 3.2 reads as follows. Lemma 4.10. There exists c > 0such that for any h∈L2(Ω) there exists uh∈ H1 −1,ln,ΓD(Ω) with div uh=hand k∇uhk0,Ω≤ckhk0,Ω. The best constant is denoted by κ(Ω,ΓD)and satisfies the estimate κ(Ω,ΓD)≤κ⊕(Ω,ΓD) := (1 + κ)1 + cP(R2)α ρ(r2) ln(e+ρ(r2)) with α=α(r1, r2, η) = max x∈Br2\Br1∇η(x)and κ = min κ(ω, γD), κ(ω, γ). 4Specifically, (rln r)−1∈L2(Bǫ), (rln r)−1/∈L2(B1+ǫ\B1−ǫ), (rln r)−1∈L2(R2\B1+ǫ) for 0< ǫ < 1. 20 D. PAULY AND S. REPIN If hhas compact support in ωand (if ΓD= Γ) additionally RΩh=Rωh= 0, then uhcan be chosen with a compact support in ωas well, in particular uh∈H1 γD(ω)⊂H1 ΓD(Ω).In this case, κ(Ω,ΓD)≤κ(ω, γD). Estimates of the distance to the set of solenoidal fields are derived quite similarly. Corollary 4.11. For any u∈H1 −1,ln,ΓD(Ω) there exists a solenoidal u0∈S−1,ln,ΓD(Ω) such that dist u, S−1,ln,ΓD(Ω)≤∇(u−u0)0,Ω≤κ(Ω,ΓD)kdiv uk0,Ω. For any u∈H1 −1,ln(Ω) there exists a solenoidal u0∈S−1,ln(Ω) such that u−u0∈H1 −1,ln,ΓD(Ω), i.e., u0|ΓD=u|ΓD, and ∇(u−u0)0,Ω≤κ(Ω,ΓD)kdiv uk0,Ω. Another obvious corollary is the inf-sup lemma for 2D exterior domains. Corollary 4.12. We have inf h∈L2(Ω) sup u∈H1 −1,ln,ΓD(Ω) hh, div ui0,Ω khk0,Ωk∇uk0,Ω≥1 κ(Ω,ΓD). As in the case of d≥3, the solvability of the Stokes problem and respective energy estimates follow. Below we recall these results. Let L2 1,ln,ΓN(Ω) := (L2 1,ln(Ω) if ΓD6=∅, L2 1,ln,⊥(Ω) if ΓD=∅, and L2 1,ln,⊥(Ω) := L2 1,ln(Ω) ∩(R2)⊥0,Ω=nφ∈L2 1,ln(Ω) : ZΩ φi= 0o. Corollary 4.13. For ν,f∈L2 1,ln,ΓN(Ω), and uD∈S−1,ln(Ω), the 2DStokes system is uniquely solvable with a solenoidal vector field u=uD+bu∈uD+S−1,ln,ΓD(Ω) ⊂S−1,ln(Ω) and p∈L2(Ω). Moreover, νk∇buk0,Ω≤cFP(Ω,ΓD)kfk1,ln,Ω+νk∇uDk0,Ω, νk∇uk0,Ω≤cFP(Ω,ΓD)kfk1,ln,Ω+ 2νk∇uDk0,Ω, kpk0,Ω≤2κ(Ω,ΓD)cFP(Ω,ΓD)kfk1,ln,Ω+νk∇uDk0,Ω. Now we address the subject of a posteriori error estimation and introduce the sets D(Ω) := nτ∈L2(Ω) : div τ∈L2 1,ln(Ω)o and DΓN(Ω) as the closure of C∞ ΓN(Ω)-tensor fields in the norm of D(Ω). For errors encompassed in approximation of the velocity field we have the following results, which repeat (with some modifications) those derived for d≥3. First, we present an analog of Theorem 4.2. Theorem 4.14. Let eu∈uD+H1 −1,ln,ΓD(Ω). Then for all τ∈DΓN(Ω) and q∈L2(Ω) we have ν1/2∇(u−eu)0,Ω≤ν−1/2 ⊖cFP(Ω,ΓD)kdiv τ+fk1,ln,Ω +ν−1/2(τ+qI−ν∇eu)0,Ω+ 2ν1/2 ⊕κ(Ω,ΓD)kdiv euk0,Ω. THE STATIONARY STOKES PROBLEM IN EXTERIOR DOMAINS 21 As in the case of d≥3, the estimate is simplified if div ev= 0 in R2\Br2and additionally (if ΓD= Γ) condition (38) is satisfied. Then, in Theorem 4.14 we can replace the constant κ(Ω,ΓD) by κ(ω, γD). If the approximation euis solenoidal in Ω, then the last term vanishes and we arrive at estimates similar to Theorem 4.1. They possess the same property: the upper bound coincides with the norm of the error on the left-hand side if τ=σand p=q(i.e., the estimate is sharp in the sense that there is no “gap” between its leftand right-hand sides). For the approximation of the pressure function, Theorem 4.4 is modified as follows. Theorem 4.15. Let ep∈L2(Ω). Then for all τ∈DΓN(Ω) and all eu∈uD+H1 −1,ln,ΓD(Ω) we have kp−epk0,Ω≤κ(Ω,ΓD)(ν−1/2 ⊖ν1/2 ⊕+ 1)cFP(Ω,ΓD)kdiv τ+fk1,ln,Ω + 2ν1/2 ⊕ν−1/2(τ+epI−ν∇eu)0,Ω+ 2ν⊕κ(Ω,ΓD)kdiv euk0,Ω. Finally, we consider a nonconforming approximation euand obtain analogs of the theorems exposed in Subsection 4.3. Theorem 4.16. Suppose that e Υ∈L2(Ω) and ep∈L2(Ω). Then for all eu∈uD+ H1 −1,ln,ΓD(Ω),τ∈DΓN(Ω), and q∈L2(Ω) we have ν1/2(Υ −e Υ)0,Ω≤ν−1/2 ⊖cFP (Ω,ΓD)kdiv τ+fk1,ln,Ω+ν−1/2(τ+qI−ν∇eu)0,Ω + 2ν1/2 ⊕κ(Ω,ΓD)kdiv euk0,Ω+ν1/2(∇eu−e Υ)0,Ω ≤ν−1/2 ⊖cFP (Ω,ΓD)kdiv τ+fk1,ln,Ω+ν−1/2(τ+qI−νe Υ)0,Ω + 2ν1/2 ⊕κ(Ω,ΓD)kdiv euk0,Ω+ 2ν1/2(∇eu−e Υ)0,Ω and kp−epk0,Ω≤κ(Ω,ΓD)(ν−1/2 ⊖ν1/2 ⊕+ 1)cFP(Ω,ΓD)kdiv τ+fk1,ln,Ω + 2ν1/2 ⊕ν−1/2(τ+epI−νe Υ)0,Ω+ 2ν⊕κ(Ω,ΓD)kdiv euk0,Ω + 2ν1/2 ⊕kν1/2(∇eu−e Υ)k0,Ω. For e Υ = ∇eu, where eu∈uD+H1 −1,ln,ΓD(Ω), we recover Theorem 4.14 and Theorem 4.15. As in Subsection 4.4, error estimates for the stress tensor field σfollow immediately by the triangle inequality. Finally, we briefly present lower bounds of the error derived in the spirit of Subsection 4.5. Theorem 4.17. For all eu, v ∈uD+H1 −1,ln,ΓD(Ω) and all ϕ∈H1 −1,ln,ΓD(Ω),τ∈DΓN(Ω), and q∈L2(Ω) we have ν1/2∇(u−eu)2 0,Ω≥2hf, ϕi0,Ω+ 2hq, div ϕi0,Ω−ν∇(2eu+ϕ),∇ϕ0,Ω −2κ(Ω,ΓD)kdiv ϕk0,Ω(ν−1/2 ⊖ν1/2 ⊕+ 1)cFP(Ω,ΓD)kdiv τ+fk1,ln,Ω + 2ν1/2 ⊕ν−1/2(τ+qI−ν∇v)0,Ω+ 2ν⊕κ(Ω,ΓD)kdiv vk0,Ω. In particular, v=euis possible. If ϕ∈S−1,ln,ΓD(Ω), the estimate simplifies to ν1/2∇(u−eu)2 0,Ω≥2hf, ϕi0,Ω−ν∇(2eu+ϕ),∇ϕ0,Ω and equality occurs for ϕ=u−eu, provided that eu∈uD+S−1,ln,ΓD(Ω). 22 D. PAULY AND S. REPIN Finally, to handle nonconforming approximations e Υ∈L2(Ω) we use again the triangle inequality to estimate ν1/2(Υ −e Υ)0,Ω≥ν1/2∇(u−eu)0,Ω−ν1/2(∇eu−e Υ)0,Ω, whenever eu∈uD+H1 −1,ln,ΓD(Ω). 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