Adaptive Uzawa algorithm for the Stokes equation

Transcription

1 ASC Report No. 34/2018 Adaptive Uzawa algorithm for the Stokes equation G. Di Fratta. T. Fu hrer, G. Gantner, and D. Praetorius Institute for Analysis and Scientific Computing Vienna University of Technology TU Wien ISBN

2 Most recent ASC Reports 33/2018 C.-M. Pfeiler, M. Ruggeri, B. Stiftner, L. Exl, M. Hochsteger, G. Hrkac, J. Schöberl, N.J. Mauser, and D. Praetorius Computational micromagnetics with Commics 32/2018 A. Gerstenmayer and A. Jüngel Comparison of a finite-element and finite-volume scheme for a degenerate crossdiffusion system for ion transport 31/2018 A. Bespalov, D. Praetorius, L. Rocchi, and M. Ruggeri Convergence of adaptive stochastic Galerkin FEM 30/2018 L. Chen, E.S. Daus, and A. Jüngel Rigorous mean-field limit and cross diffusion 29/2018 X. Huo, A. Jüngel, and A.E. Tzavaras High-friction limits of Euler flows for multicomponent systems 28/2018 A. Jüngel, U. Stefanelli, and L. Trussardi Two time discretizations for gradient flows exactly replicating energy dissipation 27/2018 A. Jüngel and M. Ptashnyk Homogenization of degenerate cross-diffusion systems 26/2018 F. Auer and E.B. Weinmüller Convergence of the collocation schemes for systems of nonlinear ODEs with a time singularity 25/2018 M. Karkulik, J.M. Melenk H-matrix approximability of inverses of discretizations of the fractional Laplacian 24/2018 M. Bernkopf and J.M. Melenk Analysis of the hp-version of a first order system least squares method for the Helmholtz equation Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstraße Wien, Austria admin@asc.tuwien.ac.at WWW: FAX: ISBN c Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors.

3 1 ADAPTIVE UZAWA ALGORITHM FOR THE STOKES EQUATION 2 GIOVANNI DI FRATTA, THOMAS FÜHRER, GREGOR GANTNER, AND DIRK PRAETORIUS Abstract. Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates, if the arising linear systems are solved iteratively, e.g., by PCG. Our analysis avoids the use of efficiency estimates for the residual error estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement Introduction The mathematical analysis of adaptive finite element methods AFEMs has significantly increased over the last years. Nowadays, AFEMs are recognized as a powerful and rigorous tool to efficiently solve partial differential equations arising in physics and engineering Model problem. In this paper, we focus on an adaptive algorithm for the solution of the steady-state Stokes equations, which after a suitable normalization read 1 u + p = f in Ω, u = 0 in Ω, u = 0 on Ω. In the literature, the first equation is referred to as momentum equation, the second as mass equation, and the third as no-slip boundary condition. Here, Ω R d with d {2, 3} is a bounded polygonal resp. polyhedral Lipschitz domain. Given the body force f, one seeks the velocity field u of an incompressible fluid and the associated pressure p. With { } 2 V := H0Ω 1 d, P := q L 2 Ω : q dx = 0, it is well-known that the Stokes problem admits a unique solution u, p V P, where p can be characterized as the unique null average solution of the elliptic Schur complement equation; see, e.g., [Bra03]. More precisely, the pressure solves the elliptic equation 3 Sp = 1 f with the Schur complement operator S := 1 : P P. Ω Date: December 20, Mathematics Subject Classification. 65N30, 65N50, 65N15, 41A25. Key words and phrases. adaptive finite element method; optimal convergence; Uzawa algorithm; Stokes equation. Acknowledgement. The authors thankfully acknowledge the support by the Austrian Science Fund FWF through grant P27005 DP, P29096 GG, DP, as well as grant F65 GDF, DP and by CON- ICYT through FONDECYT project P TF. Moreover, GG thanks Peter Binev for his explanations on [BD04, Bin15]. 1

4 The latter equation can be reformulated as a fixpoint problem for the operator 4 N α : P P, q I αsq + α 1 f. Note that S is self-adjoint. Since the norm of self-adjoint operators coincides with their spectral radius and S has positive spectrum, one has that I αs < 1 whenever 1 α S < 1. It follows that N α is a contraction for 0 < α < 2 S 1 ; see Appendix A. Moreover, elementary calculation proves that S 1. Hence, for all 0 < α < 2 and any initial guess p 0 P, the generalized Richardson iteration 5 p j+1 := N α p j = I αsp j + α 1 f converges to the exact pressure of the Stokes problem. It follows that u = lim j u[p j ] in V with u[p j ] := 1 f p j, so that, at the continuous level, the full iterative process can be expressed in the form u[p j ] = 1 f p j, 6 p j+1 = p j α u[p j ]. In the spirit of [KS08], the iterative scheme 6, usually referred to as Uzawa algorithm for the Stokes problem, is the starting point of our AFEM analysis State of the art. Although AFEMs for the analysis of mixed variational problems issuing from fluid dynamics have a long history in the engineering and physics literature, only in the last decade, [DDU02] introduced an adaptive wavelet method based on the Uzawa algorithm for solving the Stokes problem. In [BMN02], the adaptive wavelet method is replaced by an AFEM. Their numerical experiments suggested that the latter algorithm leads to optimal algebraic convergence rates. Indeed, by addition of a mesh-coarsening step to this method, [Kon06] proved optimal convergence rates. Later, in [KS08], the original algorithm of [BMN02] was modified by adding an additional loop, which separately controls the triangulations on which the pressure is discretized. We also note that for a standard conforming AFEM with Taylor Hood elements, the first proofs of convergence were presented in [MSV08, Sie10]. The work [Gan14] gives an optimality proof under the assumption that some general quasi-orthogonality is satisfied. This assumption has only recently been verified in [Fei17]. For adaptive nonconforming finite element methods, convergence and optimal rates have been investigated and proved in [BM11, HX13, CPR13] Adaptive Uzawa FEM. In this work, we further investigate the algorithm of [KS08], which is described in the following: Given a possibly non-conforming partition P i of Ω, we denote by p i P i the best approximation to p, with respect to the S-induced energy norm P, from the corresponding discrete space P i P of piecewise polynomials of degree m 1 with vanishing integral mean. With the corresponding velocity u i := u[p i ] defined analogously to 6 and the L 2 -orthogonal projection Π i : L 2 Ω P i, one can show that u i, p i is the unique solution of the reduced problem u i + p i = f in Ω, 7 Π i u i = 0 in Ω, u i = 0 on Ω. December 20,

5 In general, the velocity u i is not discrete, and hence this problem can still not be solved in practice. In an inner loop, the velocity u i is approximated by some FEM approximation U ijk V ijk via a standard adaptive algorithm of the form SOLVE ESTIMATE MARK REFINE for the vector-valued Poisson problem steered by a weighted-residual error estimator η ijk. Here, V ijk V denotes the space of all continuous piecewise polynomials on some conforming triangulation T ijk, which is a refinement of the possibly non-conforming P i. In the next loop, we apply a discretized version of the Uzawa algorithm 6 to obtain an approximation P ij P i of p i. Here, the update reads P ij+1 = P ij Π i U ijk. The last loop employs an adaptive tree approximation algorithm from [BD04] to obtain a better approximation p i+1 P i+1 of p on a refinement P i+1 of the partition P i such that ϑ U ijk Ω Π i+1 U ijk Ω for some bulk parameter 0 < ϑ < 1. We will see in Section 3.1 that U ijk Ω is related to p p i P and Π i+1 U ijk Ω to p i+1 p i P. In contrast to [KS08], in [BMN02] the latter loop was not present, since the same triangulation for the discretization of the pressure and the velocity, i.e., P i = T ijk was used. Under the assumption that the right-hand side f is a piecewise polynomial of degree m 1, [KS08] proved that the approximations U ijk and P ij converge with optimal algebraic rate to the exact solutions u and p. To generalize this result for arbitrary f, as in the seminal work [Ste07], which proves optimal convergence of a standard AFEM for the Poisson problem, [KS08] applies an additional outer loop to resolve the data oscillations appropriately. However, [KS08] only outlines the proof of this generalization. Moreover, as in the seminal work [Ste07], the analysis of [KS08] hinges on the following interior node property: Given marked elements M ijk of the current velocity triangulation T ijk, the next velocity triangulation T ijk+1 is the coarsest refinement via newest vertex bisection NVB such that all T M ijk and all T T ijk, which share a common n 1-dimensional hyperface, contain a vertex of T ijk+1 in their interior. In particular for n = 3, this property is highly demanding; see, e.g., the 3D refinement pattern in [EGP18] Contributions of present work. In the spirit of [CKNS08], which generalizes [Ste07], we prove that the algorithm of [KS08] without the data approximation loop leads to convergence of the combined error estimator η ijk + U ijk Ω which is equivalent to the error plus data oscillations at optimal algebraic rate with respect to the number of elements #T ijk if one uses standard newest vertex bisection without interior node property for the velocity triangulations. We also prove that the combined estimator sequence converges linearly in each step, i.e., it essentially contracts uniformly in each step. Moreover, our algorithm allows for the inexact solution of the arising linear systems for the discrete velocities by iterative solvers like PCG. On a conceptual level, our proofs show that even for general saddle point problems and adaptive strategies based on Richardson-type iterations, the analysis of rate optimal adaptivity can be conducted without exploiting efficiency estimates of the corresponding a posteriori error estimators Outline. The paper is organized as follows: Section 2 rewrites the Stokes problem in its variational form, introduces newest vertex bisection, and fixes some notation for the discrete ansatz spaces. In Section 3, we consider the reduced Stokes problem and the December 20,

6 corresponding Galerkin approximations, recall some well-known results on a posteriori error estimation, and introduce the tree approximation Algorithm 3.6 from [BD04] as well as our variant of the adaptive Uzawa Algorithm 3.6 from [KS08]. In Section 4, we state and prove linear convergence of the resulting combined error estimator in each step of the algorithm Theorem 4.1. To this end, we show that each increase of either i, j, or k essentially leads to a uniform contraction of the combined error estimator. Finally, Section 5 is dedicated to the main Theorem 5.3 on optimal convergence rates for the combined error estimator and its proof. As an auxiliary result of general interest, Lemma 5.1 proves that the two different definitions of approximation classes from the literature, which are either based on the accuracy ε > 0 see, e.g., [Ste08, KS08] or the number of elements N see, e.g., [CKNS08, CFPP14], are exactly the same. While all constants in statements of theorems, lemmas, etc. are explicitly given, we abbreviate the notation in proofs: For scalar terms A and B, we write A B to abbreviate A C B, where the generic constant C > 0 is clear from the context. Moreover, A B abbreviates A B A Preliminaries 2.1. Continuous Stokes problem. The vector-valued velocity fields v V are denoted in boldface, the scalar pressures q P in normal font. Let, Ω be the L 2 Ω scalar product with the corresponding L 2 Ω norm Ω. With the bilinear forms a : V V R and b : V P R defined by aw, v := w, v Ω and bv, q := v, q Ω, the mixed variational formulation of the Stokes problem 1 reads as follows: Given f L 2 Ω d, let u, p V P be the unique solution to 8 au, v + bv, p = f, v Ω for all v V, bu, q = 0 for all q P. On the velocity space V, we consider the a, -induced energy norm v V := av, v 1/2 = v Ω v H 1 Ω. We note that v P for all v V and 9 v Ω v Ω = v V for all v V, which follows from integration by parts; see Appendix B. Define the operators A : V V, B : V P, and B : P V by Av := av,, Bv := bv,, B q := b, q. Then, the Schur complement operator S := BA 1 B : P P P is bounded, symmetric, and elliptic; see [KS08, Lemma 2.2]. Thus, it holds that q P := Sq, q 1/2 Ω q Ω on P. More precisely, there exists a constant C div 1, which depends only on Ω, such that 10 C 1 div q Ω q P q Ω for all q P. Here, the upper bound with constant 1 follows from S 1, which itself follows from 9. December 20,

7 Partitions, triangulations, and newest vertex bisection NVB. Throughout, P is a finite possibly non-conforming partition of Ω into compact non-degenerate simplices, which is used to discretize P, while T is a finite conforming triangulation of Ω into compact non-degenerate simplices, which is used to discretize V. Throughout, we use NVB refinement; see, e.g., [Ste08, KPP13] for the precise mesh-refinement rules. We write P := bisectp, M for the partition obtained by one bisection of all marked elements M P, i.e., M = P\P and #M = #P #P. We write P T nc P, if there exists J N 0 and partitions P j and M j P j for all j = 0,..., J, such that P = P 0, P j = bisectp j 1, M j 1 for all j = 1,..., J, and P = P J. We write T := refinet, M for the coarsest triangulation such that at least all marked elements M T have been bisected, i.e., M T \T. We write T T c T, if there exists J N 0 and triangulations T j and M j T j for all j = 0,..., J, such that 11 T = T 0, T j = refinet j 1, M j 1 for all j = 1,..., J, and T = T J. Let T init be a given initial conforming triangulation of Ω. We define the sets T nc := T nc T init and T c := T c T init of all non-conforming and conforming NVB refinements of T init. Clearly, T c T nc. We write T := closep if P T nc is a partition and T T c is the coarsest conforming refinement of P. Existence and uniqueness of T follow from the fact that NVB is a binary refinement rule, and the order of the bisections does not matter. In particular, this also implies that refinet, M = closebisectt, M for all T T c and M T. It follows from elementary geometric observations that NVB refinement leads only to finitely many shapes of simplices T ; see, e.g., [Ste08]. Hence, all NVB refinements are uniformly γ-shape regular, i.e., 12 diamt γ := sup max <. P T nc T P T 1/d Finally, we recall the following properties of NVB, where C son, C cls > 0 are constants, which depend only on T init and the space dimension d 2: M1 overlay estimate: For all P, P T nc, there exists a unique coarsest common refinement P P T nc P T nc P. It holds that #P P #P+#P #T init. If P, P T c are conforming, it also holds that P P T c. M2 finite number of sons: For all T T c, M T, and T := refinet, M, it holds that {T T : T T } = T and #{T T : T T } C son for all T T. M3 mesh-closure estimate: For all sequences T j T c such that T 0 = T init and T j = refinet j 1, M j 1 with M j 1 T j 1 for all j N, it holds that 13 J 1 #T J #T init C cls #M j for all J N 0. M4 conformity estimate: For all partitions P T nc, it holds that 14 j=0 #closep #T init C cls #P #T init. December 20,

8 The overlay estimate M1 is first proved in [Ste07] for d = 2, but the proof transfers to arbitrary dimension d 2; see [CKNS08]. For d = 2, M2 obviously holds with C son = 4, while it is proved in [GSS14] for general dimension d 2. The closure estimate M3 is first proved in [BDD04] for d = 2. For general d 2, it is proved in [Ste08]. While the proofs of [BDD04, Ste08] require an admissibility condition on T init, the work [KPP13] proves M3 for d = 2, but arbitrary conforming triangulation T init. We refer to Appendix D for the fact that M3 implies M Discrete function spaces. Given a fixed polynomial degree m N as well as P T nc and T T c, we consider the discrete spaces 15 PP := {Q P P : T P Q P T is polynomial of degree m 1}, VT := {V T V : T T which consist of piecewise polynomials. V T T is polynomial of degree m}, 2.4. Auxiliary problems. Let P T nc. Then, p P PP denotes the best approximation of the exact pressure p with respect to P, i.e., 16 p p P P = min p Q P P. Q P PP By definition of the operator S from 3, there exists a unique u P V such that u P, p P V PP is the unique solution to the reduced Stokes problem 17 au P, v + bv, p P = f, v Ω for all v V, bu P, Q P = 0 for all Q P PP; see [KS08, Section 4]. Note that the second condition can equivalently be stated as Π P u P = 0 in Ω, where Π P : L 2 Ω PP is the orthogonal projection with respect to Ω. Thus, 17 is just the variational formulation of 7 with P i replaced by P. Even though, p P is a discrete function, it can hardly be computed since p is unknown. Given q P, let u[q] V be the unique solution to the vector-valued Poisson equation 18 au[q], v = f, v Ω bv, q for all v V. Note that u P = u[p P ]. Finally, let T T nc P T c be a conforming refinement of P. Then, U T [q] VT is the unique solution to the Galerkin discretization of au T [q], V T = f, V T Ω bv T, q for all V T VT. Note that U T [q] is the Galerkin approximation to u[q] in VT. Since V denotes the energy norm corresponding to a,, there holds the Céa lemma 20 u[q] U T [q] V = min u[q] V T V, V T VT Recall the operators A, B, B from Section 2.1. Note that u[q] u[r] = A 1 B r q for arbitrary q, r P, which yields that u[q] u[r] 2 V = B r q, A 1 B r q V V. By definition of the operator S = BA 1 B and the norm P, we thus see that 21 U T [q] U T [r] V u[q] u[r] V = q r P. December 20,

9 Notational conventions. Throughout this work, u, p V P denotes the exact solution of the continuous Stokes problem 8. All occurring functions u P, u[q], and U T [q] are approximations of u. All occurring functions p P and P P are approximations of p. We employ bold face symbols for velocity functions, e.g., v V or V T VT, and normal font for pressure functions, e.g., q P, Q P PP. Finally, small letters indicate functions, which are continuous or not computable, e.g., u, p, and p P, while computable discrete functions are written with capital letters, e.g., U T [Q P ]. The corresponding partitions P T nc resp. triangulations T T c are always indicated by indices Abbreviate notation for adaptive algorithm. The adaptive algorithm below generates nested partitions P i T nc and triangulations T ijk T c for certain indices i, j, k Q N 3 0 such that T ijk T nc P i T c. Furthermore, it provides approximations 22 p P ij P i := PP i as well as u U ijk V ijk := VT ijk. More precisely and with the notation from Section 2.4, it holds that 1 23 P ij p i := p Pi as well as U ijk U Tijk [P ij ] u[p ij ] =: u ij. Besides this notation, let 24 Π i := Π Pi : L 2 Ω PP i be the L 2 Ω-orthogonal projection with respect to Ω and let 25 η ijk := ηt ijk ; U ijk, P ij ηt ijk ; U Tijk [P ij ], P ij be the computable a posteriori error estimator from Section 3.1 below. 3. Adaptive Uzawa algorithm 3.1. A posteriori error estimation. Throughout this section, let P T nc be a partition of Ω R d and T T nc P T c be a conforming refinement. We recall the residual a posteriori error estimator: For T T, Q P PP, and V T VT, define 26 η T V T, Q P 2 := T 2/n f Q P + V T 2 T + T 1/n [Q P n V T n] 2 T Ω, where [ ] denotes the jump of its argument over T. Then, the error estimator reads ηm; V T, Q P 2 := T M η T V T, Q P 2 for all M T In the following, we recall some important properties of η from [CKNS08, KS08]. We start with the available reliability results. Lemma 3.1 reliability [KS08, Prop. 5.1, Prop. 5.5]. There exists a constant C rel > 0 such that, for all Q P PP, it holds that 28 u[q P ] U T [Q P ] V C rel ηt ; U T [Q P ], Q P. Moreover, it holds that 29 u P U T [Q P ] V + p P Q P P C rel ηt ; UT [Q P ], Q P + Π P U T [Q P ] Ω 1 Do not mistake the pressure p i with the iterates p j of the exact Uzawa algorithm 6. December 20,

10 as well as 30 u U T [Q P ] V + p Q P P C rel ηt ; UT [Q P ], Q P + U T [Q P ] Ω. The constant C rel depends only on γ-shape regularity. For some fixed discrete pressure Q P, we recall the localized upper bound in the current form of [CKNS08], which improves [KS08, Prop. 5.1]. Lemma 3.2 discrete reliability [CKNS08, Lemma 3.6]. Let T T c T. There exists a constant C drel > 0 such that, for all Q P PP, it holds that 31 U T [Q P ] U T [Q P ] V C drel ηt \ T ; U[Q P ], Q P. The constant C drel depends only on γ-shape regularity. Next, we note that the estimator depends Lipschitz continuously on the arguments. The result is slightly stronger than [KS08, Prop. 5.4], but the proof is standard [CKNS08]. Lemma 3.3 stability [CKNS08, Prop. 3.3]. Let T T c T. There exists a constant C stab > 0 such that, for all V T V T, W T VT, and Q P, R P PP, it holds that ηt T ; V T, Q P ηt T 32 ; W T, R P C stab V T W T V + Q P R P P. The constant C stab depends only on the polynomial degree m and γ-shape regularity. The following reduction property follows from the reduction of the mesh-size on refined elements. The proof is standard [CKNS08]. Lemma 3.4 reduction [CKNS08, Proof of Cor. 3.4]. Let T T c T. Let Q P PP. Then, with q red = 2 1/n+1, there holds the reduction property 33 η T \ T ; U T [Q P ], Q P q red ηt \ T ; U T [Q P ], Q P + C red U T [Q P ] U T [Q P ] V. The constant C red > 0 depends only on the polynomial degree m and γ-shape regularity. Finally, for the divergence contribution to the Stokes error estimator, we recall the following equivalence. The result is slightly stronger than [KS08, Prop. 5.7]. Lemma 3.5. Let C div 1 be the norm equivalence constant from 10. Let Π T : L 2 Ω PT be the L 2 Ω-orthogonal projection. If Q P PP, then it holds that 34 Π T u[q P ] Ω u T u[q P ] Ω p T Q P P C div Π T u[q P ] Ω. Moreover, it holds that u[q P ] Ω p Q P P C div u[q P ] Ω Proof. From the definition of the Schur complement operator, we have that 36 u T u[q P ] = Sp T Q P. December 20,

11 305 Taking into account 10, we obtain that 306 u T u[q P ] 2 Ω 36 = Sp T Q P, u T u[q P ] Ω = p T Q P, u T u[q P ] P p T Q P P u T u[q P ] P 10 p T Q P P u T u[q P ] Ω. Together with Π T u T = 0, this proves that Π T u[q P ] Ω u T u[q P ] Ω p T Q P P. On the other hand, note that T T nc P implies that Π T p T Q P = p T Q P. The norm equivalence 10 and the Cauchy-Schwarz inequality thus imply that C div p T Q P P Π T u[q P ] Ω 10 p T Q P Ω Π T u[q P ] Ω p T Q P, Π T u[q P ] Ω = p T Q P, Π T u T u[q P ] Ω = p T Q P, u T u[q P ] Ω 36 = Sp T Q P, p T Q P Ω = p T Q P 2 P and therefore p T Q P P C div Π T u[q P ] Ω. This concludes the proof of 34. The proof of 35 follows along the same lines with p = p T and hence 0 = u = u T Adaptive refinement of pressure triangulation. To refine the partitions P i, we apply the following algorithm from [Bin15, Section 2] which slightly differs from the well-known thresholding second algorithm of [BD04]: Algorithm 3.6. Input: Partition P := P T nc, triangulation T T nc P T c, function V T VT, adaptivity parameter 0 < ϑ 1. Loop: Iterate the following steps i iii until ϑ V T Ω Π P V T Ω : 326 i Compute et := inf{ V T Q 2 T : Q polynomial of degree m 1} for all 327 T P, for which et has not been already computed. 328 ii For all T P for which ẽt has not been already defined, define ẽt := et if T P and ẽt := et ẽ T /et + ẽ T otherwise, where T 329 denotes the 330 unique father element of T. 331 iii Choose one element T P with ẽt = max T P ẽt and employ newest vertex 332 bisection to generate P := bisectp, {T } Output: Triangulation P = bisectp, T, V T ; ϑ T nc P with T T nc P T c. According to [Bin15, Theorem 2.1], the output P is a quasi-optimal mesh in T nc P with ϑ V T Ω Π P V T Ω : This means that for all ϑ < ϑ < 1 and all P T nc P with ϑ V T Ω Π P V T Ω, it holds that #P #P C bin # P #P for some C bin > 1, which depends only on the ratio 1 ϑ 2 /1 ϑ 2. The same reference states that the effort of Algorithm 3.6 is O#T log#t if 0 < ϑ < 1. To obtain optimal algebraic convergence rates of the error estimator, one has to choose ϑ sufficiently small and ϑ sufficiently close to ϑ; see Theorem 5.3 below. December 20,

12 Adaptive Uzawa algorithm. We investigate the following adaptive Uzawa algorithm, which goes back to [KS08, Section 7]. Algorithm 3.7. Input: Conforming initial triangulation P 0 := T 000 := T init of Ω, initial approximation P 00 = 0, counters i = j = k = 0, adaptivity parameters 0 κ 1 < 1, 0 < κ 2 < 1, 0 < κ 3 < 1, 0 < ϑ 1, 0 < θ 1, and C mark 1. Loop: Iterate the following steps i iv: i Compute U ijk V ijk as well as all local contributions of the corresponding error estimator η ijk = ηt ijk ; U ijk, P ij such that the exact Galerkin approximation U Tijk [P ij ] V ijk of u ij satisfies that U Tijk [P ij ] U ijk V κ 1 η ijk. ii while η ijk + Π i U ijk Ω κ 2 ηijk + U ijk Ω do Determine P i+1 := bisectp i, T ijk, U ijk ; ϑ by Algorithm 3.6. Define M ijk :=, P i+10 := P ij, and T i+100 := T ijk. Update counters i, j, k i + 1, 0, 0. end while iii if η ijk κ 3 Π i U ijk Ω then Define M ijk :=, P ij+1 := P ij Π i U ijk P i, and T ij+10 := T ijk. Update counters i, j, k i, j + 1, 0. iv else Determine a set M ijk T ijk of up to the fixed factor C mark minimal cardinality, which satisfies the Dörfler marking criterion 37 θ ηijk 2 ηm ijk ; P ij, U ijk 2. Generate T ijk+1 := refinet ijk, M ijk. Update counters i, j, k i, j, k + 1. end if Remark 3.8. The actual implementation of Algorithm 3.7 will replace the triple indices i, j, k by one single index n N 0, which is increased in each step ii iv. However, the present statement of the algorithm makes the numerical analysis more accessible Lemma 3.9. Define the index set Q := {i, j, k N 3 0 : U ijk is defined by Algorithm 3.7}. Then, for i, j, k N 3 0, there hold the following assertions a c: a If i, j, k + 1 Q, then i, j, k Q. b If i, j+1, 0 Q, then i, j, 0 Q and ki, j := max{k N 0 : i, j, k Q} <. c If i+1, 0, 0 Q, then i, 0, 0 Q and ji := max{j N 0 : i, j, 0 Q} <. Throughout, we shall make the following conventions for the triple index: If we write η ijk etc. see, e.g., Lemma 4.5, then implicitly k = ki, j. If we write η ijk etc. see, e.g., Lemma 4.6, then implicitly j = ji and k = ki, j. 377 Proof. Each step ii iv of the algorithm increases either i or j or k by one Remark Unlike the algorithm from [KS08], our formulation of the adaptive Uzawa algorithm avoids any special treatment of the data oscillations i.e., to resolve f by a December 20,

13 piecewise polynomial in an additional outer loop. This is achieved by the fact that our analysis avoids to exploit any efficiency of the a posteriori estimator η. Remark We note that the choice U ijk := U Tijk [P ij ] i.e., κ 1 = 0 is admissible in step i of Algorithm 3.7. In the spirit of [FHPS18], one can also employ the PCG algorithm [GVL13, Algorithm ] with optimal preconditioner. With κ 1 and an additional index l N 0 for the PCG iteration and initially l := 0, repeat the following three steps, until U ijk := U ijkl+1 satisfies U ijkl+1 U ijkl V κ 1 η ijkl+1 : Do one PCG step to obtain U ijkl+1 V ijk from U ijkl V ijk. Compute all local contributions of the estimator η ijkl+1 := ηt ijk ; U ijkl+1, P ij. Update counters i, j, k, l i, j, k, l + 1. If the preconditioner is optimal, i.e., the preconditioned linear system has uniformly bounded condition number, then it follows that PCG is a uniform contraction [FHPS18, Section 2.6]: There exists 0 < q pcg < 1 such that U Tijk [P ij ] U ijkl+1 V q pcg U Tijk [P ij ] U ijkl V for all l N 0. Hence, the PCG loop terminates, and the triangle inequality proves that U Tijk [P ij ] U ijkl+1 V q pcg 1 q pcg U ijkl+1 U ijkl V q pcg 1 q pcg κ 1 η ijkl+1, i.e., the criterion of step i of Algorithm 3.7 is satisfied for κ 1 := κ 1q pcg /1 q pcg Convergence 4.1. Main theorem on linear convergence. To state linear convergence, we need an ordering of the set Q from Lemma 3.9: For i, j, k, i, j, k Q, write i, j, k < i, j, k if the index i, j, k appears earlier in Algorithm 3.7 than i, j, k. Define 38 i, j, k := #{i, j, k Q : i, j, k < i, j, k} N 0. Note that i, j, k coincides with the single index n from Remark 3.8. Then, we have the following theorem. The proof is given in Section 4.3. Theorem 4.1. Let 0 < κ 1 < θ 1/2 /C stab. Suppose that 0 < κ 2, κ 3 < 1 are sufficiently small as in Lemma 4.5 and Lemma 4.6 below. Let 0 < ϑ 1 and 0 < θ 1. Then, there exist constants C lin > 0 and 0 < q lin < 1 such that η ijk + U ijk Ω C lin q i,j,k i,j,k 39 lin ηi j k + U i j k Ω for all i, j, k, i, j, k Q with i, j, k < i, j, k. The constants C lin and q lin depend only on the domain Ω, γ-shape regularity, the polynomial degree m, and the parameters κ 1, κ 2, κ 3, ϑ, and θ. Remark 4.2. The adaptive Uzawa algorithm from [BMN02] employs only one triangulation for both, the pressure and the velocity. Similarly, we can additionally update P i := T ijk+1 in step iv of Algorithm 3.7. Since 0 < κ 2 < 1 and Π i U ijk = U ijk, December 20,

14 then the condition in ii will always fail. We note that the convergence analysis of Section 4.2 and in particular, linear convergence Theorem 4.1 clearly remain valid for this modified algorithm, while our proof of optimal convergence rates Theorem 5.3 fails Auxiliary results. The first lemma provides links between the exact Galerkin solutions U Tijk [P ij ] and its approximations U ijk Lemma 4.3. Let i, j, k Q. For all S T ijk, it holds that 40 ηs; U Tijk [P ij ], P ij ηs; U ijk, P ij κ 1 C stab η ijk, where C stab > 0 is the constant from Lemma 3.3. This particularly yields the equivalence 41 1 κ 1 C stab η ijk ηt ijk ; U Tijk [P ij ], P ij 1 + κ 1 C stab η ijk. as well as the reliability estimates 42 u ij U ijk V C relκ 1 η ijk, u i U ijk V + p i P ij P C relκ η ijk + Π i U ijk Ω, u U ijk V + p P ij P C relκ 1 44 η ijk + U ijk Ω, where C rel κ 1 := 1 + κ 1 C stab C rel + κ 1 C rel + 1 C rel with C rel > 0 from Lemma 3.1. Proof. To shorten notation, we set ηijk := ηt ijk; U Tijk [P ij ], P ij. The stability 40 follows from Lemma 3.3 and U Tijk [P ij ] U ijk V κ 1 η ijk, which is guaranteed by step i of Algorithm 3.7. Taking S = T ijk, 41 is an immediate consequence. To see 42, we use reliability 28, step i of Algorithm 3.7, and 41 to see that u ij U ijk V 28 C rel η ijk + U Tijk [P ij ] U ijk V κ 1 C stab C rel + κ 1 η ijk. To prove 43, we apply 29 u i U ijk V + p i P ij P 29 C rel η ijk + Π i U Tijk [P ij ] Ω + UTijk [P ij ] U ijk V Similarly, 44 follows from κ 1 C stab C rel + κ 1 η ijk + C rel Π i U ijk Ω. The following three lemmas prove that Algorithm 3.7 leads to contraction if either i, j, or k is increased. Throughout, let 0 < ϑ 1, 0 < θ 1, and, if not stated otherwise, 0 κ 1 < 1, 0 < κ 2, κ 3 < Lemma 4.4. Let i, j, 0 Q and define k := max{k N 0 : i, j, k Q} N 0 { }. If 0 κ 1 < θ 1/2 /C stab, then, there exist constants 0 < q 1 < 1 and C 1 > 0, which depend only on γ-shape regularity, the polynomial degree m, κ 1, and θ, such that 45 Moreover, it holds that 46 η ijk+n C 1 q n 1 η ijk for all k, n N 0 with k k + n k. η ijk η ijk + U ijk Ω 1 κ κ 3 η ijk for all 0 k < k. If k =, this yields that u U ijk V + p P ij P 0 as k with p = p i = P ij. December 20,

15 Proof. We split the proof into three steps. Step 1. If U ijk = U Tijk [P ij ] for all i, j, k Q, step iv of Algorithm 3.7 is the usual adaptive step in an adaptive algorithm for, e.g., the vector-valued Poisson model problem. Hence, 45 follows from reliability 28, stability 32 and reduction 33; see, e.g., [CFPP14, Theorem 4.1 i]. For general U ijk, 45 follows from [CFPP14, Theorem 7.2] under the constraint 0 κ 1 < θ 1/2 /C stab. Step 2. If k < k, the structure of Algorithm 3.7 implies that the conditions in step ii and iii are false, i.e., η ijk + Π i U ijk Ω > κ 2 ηijk + U ijk Ω and η ijk > κ 3 Π i U ijk Ω. Hence, η ijk η ijk + U ijk Ω < 1 κ 2 ηijk + Π i U ijk Ω < 1 κ κ 3 η ijk which proves 46. Step 3. For k =, the estimates imply that u U ijk V + p P ij P 44 η ijk + U ijk Ω 46 η ijk k 0. Note that k = also implies that neither i nor j are increased, i.e., P ij remains constant as k. Hence, p = P ij P i and therefore also p = p i Lemma 4.5. Let i, 0, 0 Q and define j := max{j N 0 : i, j, 0 Q} N 0 { }. If 0 < κ 3 1 is sufficiently small see 55 in the proof below, then there exist constants 0 < q 2 < 1 and C 2 > 0 such that 47 p i P ij+n P q n 2 p i P ij P for all j, n N 0 with j j + n j. Moreover, it holds that 48 C 1 2 p i P ij P η ijk + U ijk Ω C 2 p i P ij P for all 0 j < j. If j =, this yields convergence u U ijk V + p P ij P 0 as j. While q 2 depends only on the domain Ω, γ-shape regularity, κ 1, and κ 3, the constant C 2 depends additionally on κ 2. Proof. We split the proof into three steps. Step 1. If j < ji and k = ki, j, then necessarily ki, j <. The structure of Algorithm 3.7 implies that the condition in step ii is false, while the condition in step iii is true, i.e., 49 η ijk + Π i U ijk Ω > κ 2 η ijk + U ijk Ω and η ijk κ 3 Π i U ijk Ω. First, this proves that κ 2 η ijk + U ijk Ω < η ijk + Π i U ijk Ω 1 + κ 3 Π i U ijk Ω 1 + κ 3 U ijk Ω 1 + κ 3 η ijk + U ijk Ω. December 20,

16 Second, reliability 42 gives that 51 Π i u ij U ijk Ω u ij U ijk V 42 C relκ 1 η ijk 49 κ 3 C relκ 1 Π i U ijk Ω. The triangle inequality yields that 52 1 κ 3 C relκ 1 Π i U ijk Ω 51 Π i u ij Ω κ 3 C relκ 1 Π i U ijk Ω. This leads us to 53 C 1 div 1 κ 3 C rel κ κ 3 C rel κ 1 p 34 i P ij P 1 κ 3C rel κ κ 3 C rel κ 1 Π i u ij Ω 52 1 κ 3 C relκ 1 Π i U ijk Ω 52 Π i u ij Ω 34 p i P ij P. If κ 3 C rel κ 1 < 1, the combination of 53 and 50 proves 48. Step 2. Starting from P ij, one step of the exact Uzawa iteration for the reduced Stokes problem leading to the auxiliary quantity p ij+1 guarantees the existence of some 0 < q Uzawa < 1 such that the following contraction holds see [KS08, Eq. 4.3]: 54 p i p ij+1 P q Uzawa p i P ij P with p ij+1 = P ij Π i u ij. The contraction constant q Uzawa is the norm of the operator from 4 with α = 1. Indeed, the proof of 54 works exactly as in Appendix A if S : P P is replaced by the operator Π i S : P i P i. In particular, q Uzawa does neither depend on i nor on j. Since P ij+1 = P ij Π i U ijk, we are thus led to p i P ij+1 P p i p ij+1 P + p ij+1 P ij+1 P q Uzawa p i P ij P + Π i u ij U ijk P 51 q Uzawa p i P ij P + κ 3 C relκ 1 Π i U ijk Ω q Uzawa + κ 3C rel κ 1 1 κ 3 C rel κ p i P ij P =: q 2 p i P ij P Let 0 < κ 3 1 be sufficiently small, i.e., 55 0 < κ 3 C relκ 1 < 1 and 0 < q 2 := q Uzawa + κ 3C rel κ 1 1 κ 3 C rel κ 1 < 1. Then, induction proves that p i P ij+n P q2 n p i P ij P for every j, n N 0 with j j + n j. This proves 47. Step 3. For j =, the estimates imply that j u U ijk V + p P ij P η ijk + U ijk Ω p i P ij P 0. This concludes the proof. Note that i := max{i N 0 : i, 0, 0 Q} < in Algorithm 3.7 implies that either j := ji = or ki, j =. According to Lemma 4.4 for k = and Lemma 4.5 for j =, it only remains to analyze the case i =. December 20,

17 Lemma 4.6. Let i := max{i N 0 : i, 0, 0 Q} N 0 { }. If 0 < κ 2 1 is sufficiently small see 61 in the proof below, then there exist constants 0 < q 3 < 1 and C 3 > 0 such that 56 p P i+nj P q n 3 p P ij P for all i, n N 0 with i i + n i. Moreover, it holds that 57 C 1 3 p P ij P η ijk + U ijk Ω C 3 p P ij P for all 0 i < i. While C 3 depends only on the domain Ω, γ-shape regularity, κ 1 and κ 2, the contraction constant q 3 depends additionally on 0 < ϑ 1. If i =, this yields convergence u U ijk V + p P ij P 0 as i. Proof. We split the proof into five steps. Step 1. According to Algorithm 3.7, it holds that 58 For 0 < κ 2 < 1, this implies that Recall that η ijk + Π i U ijk Ω κ 2 ηijk + U ijk Ω. η ijk + Π i U ijk Ω κ 2 1 κ 2 U ijk Ω. U ijk Ω u ij Ω + u ij U ijk Ω 42 u ij Ω + C relκ 1 η ijk We abbreviate Cκ 1, κ 2 := C rel κ 1 κ 2 /1 κ 2. For sufficiently small 0 < κ 2 1 with 0 < Cκ 1, κ 2 < 1, the combination of the last two estimates implies that U ijk Ω 1 Cκ 1, κ 2 1 u ij Ω. With we are hence led to 59 Conversely, C κ 1, κ 2 := Cκ 1, κ 2 1 Cκ 1, κ 2, u ij U ijk V 42 C relκ 1 η ijk + Π i U ijk Ω Cκ1, κ 2 U ijk Ω C κ 1, κ 2 u ij Ω 35 C κ 1, κ 2 p P ij P. p P ij P 35 C div u ij Ω C div Uijk Ω + u ij U ijk Ω In particular, this proves 57. Step 2. Recall from Step 1 that max{1, C relκ 1 } C div Uijk Ω + η ijk. u ij U ijk Ω + Π i U ijk Ω 42 max{1, C relκ 1 } η ijk + Π i U ijk Ω 59 max{1, C relκ 1 } C κ 1, κ 2 p P ij P. December 20,

18 We hence observe that p i P ij P 34 C div Π i u ij Ω C div Πi u ij U ijk Ω + Π i U ijk Ω 60 C div max{1, C relκ 1 } C κ 1, κ 2 p P ij P. Step 3. From Algorithm 3.6, we obtain that According to 59, it holds that as well as ϑ U ijk Ω Π i+1 U ijk Ω. u ij Ω U ijk Ω + u ij U ijk Ω Cκ 1, κ 2 U ijk Ω, Π i+1 u ij U ijk Ω u ij U ijk V 59 C κ 1, κ 2 u ij Ω. Combining the last three estimates, we see that Π i+1 u ij Ω Π i+1 U ijk Ω Π i+1 u ij U ijk Ω ϑ 1 + Cκ 1, κ 2 C κ 1, κ 2 u ij Ω. Recall the constant C div > 0 from Lemma 3.5. If 0 < κ 2 1 is sufficiently small, it holds that C κ 2 := ϑ 1+Cκ 1,κ 2 C κ 1, κ 2 /C div > 0. This implies that 34 ϑ p i+1 P ij P Π i+1 u ij Ω 1 + Cκ 1, κ 2 C κ 1, κ 2 u ij Ω 35 C κ 2 p P ij P. Together with the Pythagoras theorem, we are hence led to p p i+1 2 P = p P ij 2 P p i+1 P ij 2 P 1 C κ 2 2 p P ij 2 P. Step 4. Combining Step 2 and Step 3, we obtain that p P i+1j 2 P = p p i+1 2 P + p i+1 P i+1j 2 P 1 C κ 2 2 p P ij 2 P + C 2 div max{1, C relκ 1 2 } C κ 1, κ 2 2 p P i+1j 2 P. For sufficiently small 0 < κ 2 1, i.e., 61 Cκ 1, κ 2 = C rel κ 1κ 2 1 κ 2 < 1, 0 < C κ 2 = 0 < q 2 3 := we hence see that 1 C κ C 2 div max{1, C rel κ 1 2 } C κ 1, κ 2 2 < 1, By induction, we conclude 56. p P i+1j 2 P q 2 3 p P ij 2 P. ϑ 1 + Cκ 1, κ 2 Cκ 1, κ 2 C 1 div 1 Cκ 1, κ 2, December 20,

19 Step 5. For i =, the estimates imply that i u U ijk V + p P ij P η ijk + U ijk Ω p P ij P 0. This concludes the proof Proof of Theorem 4.1. To prove Theorem 4.1, we need the following two lemmas. A slightly weaker version of the first lemma is already proved in [CFPP14, Lemma 4.9]. The proof, however, immediately extends to the following generalization and is therefore omitted. Lemma 4.7. Let a l l N0 be a sequence with a l 0 for all l N 0. With the convention 0 1/s :=, the following three statements are pairwise equivalent: a There exist a constant C > 0 such that n=l a n Ca l for all l N 0. b For all s > 0, there exists C > 0 such that l n=0 a 1/s n Ca 1/s l for all l N 0. c There exist 0 < q < 1 and C > 0 such that a l+n Cq n a l for all n, l N 0. Here, in each statement, the constants C > 0 may differ Lemma 4.8. Let 0 < κ 1 < θ 1/2 /C stab. Suppose that κ 2, κ 3 are sufficiently small as in Lemma 4.5 and Lemma 4.6. Let i, j, 0 Q. Then, there hold the assertions a d: 622 a If i 1, then η i00 + U i00 Ω C mon ηi 1jk + U i 1jk Ω. 623 b If j 1, then η ij0 + U ij0 Ω C mon ηij 1k + U ij 1k Ω. c η ijk + U ijk Ω C mon ηijk + U ijk Ω for all 0 k 624 k ki, j. d η ijk + U ijk Ω C mon ηij k + U ij k Ω for all 0 j 625 j < ji The constant C mon > 0 depends only on Ω, C stab, C rel, C 1, and C 2. Proof. To shorten notation, we set ηijk := ηt ijk; U Tijk [P ij ], P ij and Uijk := U T ijk [P ij ]. To prove a, recall from step ii of Algorithm 3.7 that T i00 = T i 1jk as well as P i0 = P i 1j. Hence, Ui00 = Ui 1jk and consequently η i00 = ηi 1jk as well as U i00 Ω = Ui 1jk Ω. Since κ 1 < θ 1/2 C 1 stab 630 C 1 stab, we can apply the equivalence 41 in both 631 directions. With step i of Algorithm 3.7, we see that η i00 + U i00 Ω 41 η i00 + U i00 Ω + U i00 U i00 V η i00 + U i00 Ω + η i00 41 η i00 + U i00 Ω = η i 1jk + U i 1jk Ω 41 η i 1jk + U i 1jk Ω + U i 1jk U i 1jk V η i 1jk + U i 1jk Ω. To prove b, recall from step iii of Algorithm 3.7 that T ij0 = T ij 1k and P ij = P ij 1 Π i U ij 1k. According to the discrete variational form 19, it holds that au ij0 U ij 1k, V ij0 = bv ij0, Π i U ij 1k for all V ij0 VT ij0 = VT i 1jk. December 20,

20 This proves that Uij0 Uij 1k V Π i U ij 1k Ω U ij 1k Ω. First, it follows that U ij0 Ω U ij 1k Ω + U ij0 U ij 1k V U ij 1k Ω + U ij0 U ij 1k V + U ij0 U ij0 V + U ij 1k U ij 1k V U ij 1k Ω + κ 1 η ij0 + κ 1 η ij 1k. Second, stability of the error estimator Lemma 3.3, T ij0 = T ij 1k and the previous estimate prove that η ij0 32 η ij 1k + C stab Uij0 U ij 1k V + Π i U ij 1k Ω 1 + κ 1 C stab η ij 1k + C stab U ij 1k Ω + κ 1 C stab η ij0. Recall that κ 1 C stab < θ 1/2 1. Thus, combining the last two estimates, we conclude the proof of b. To prove c, note that Lemma 4.4 implies that 62 η ijk 45 C 1 η ijk for all 0 k < k k := ki, j. Moreover, the Pythagoras theorem, reliability 28, and the equivalence 41 prove that U ijk Ω U ijk Ω + U ijk U ijk V + U ijk U ijk V + U ijk U ijk V U ijk Ω + u ij U ijk V + κ 1 η ijk + κ 1 η ijk U ijk Ω + η ijk + η ijk 41 U ijk Ω + η ijk. To prove d, note that Lemma 4.5 implies that η ijk + U ijk Ω 48 p i P ij P 47 p i P ij P 48 η ij k + U ij k Ω. This concludes the proof Proof of Theorem 4.1. For all 0 i i i, define ji N 0 by { 0 if i < i, ji := j if i = i. For all 0 i i i and all ji j ji, define ki, j N 0 by { 0 if i < i or j < j, ki, j := k if i = i and j = j. As for j and k, we write j = ji and k = ki, j if i and j are clear from the context. Further, we abbreviate µ ijk := η ijk + U ijk Ω. December 20,

21 675 With this notation and according to Lemma 4.7, 39 is equivalent to i,j,k Q i,j,k i,j,k µ ijk = i ji ki,j i=i j=ji k=ki,j µ ijk µ i j k for all i, j, k Q. We prove 63 in the following three steps. Step 1. For ki, j < ki, j <, Lemma 4.8 c proves that µ ijk µ ijk Hence, Lemma 4.4 in combination with the geometric series allows to estimate the sum over k 64 i ji ki,j i=i j=ji k=ki,j i ji i=i j=ji µ ijk c µ ijk = i ji ki,j 1 i=i j=ji k=ki,j ji j=ji µ i jk + 46 µ ijk i ji i=i +1 j=ji i ji ki,j 1 i=i j=ji k=ki,j µ ijk = ji j=j µ i jk + 45 η ijk i i i=i +1 j=0 ji i=i j=ji ji µ ij0. Step 2. In this step, we bound the first summand of 64 by µ i j k. It holds that ji j=j µ i jk = µ i j k + ji j=j +1 ji µ i jk = µ i j k + j=j +1 µ i j0. Lemma 4.8 b and Lemma 4.5 in combination with the geometric series show that ji j=j +1 µ i j0 b ji j=j +1 µ i j 1k = ji 1 j=j µ i jk 48 ji 1 j=j p i P i j P η ijk 47 p i P i j P 29 µ i j k. 689 Step 3. In this step, we bound the second summand of 64 by µ i j k. First, we 690 consider only the terms where j > 0. As in Step 2, Lemma 4.8 b and Lemma 4.5 in 691 combination with the geometric series show that i ji i=i +1 j=1 µ ij0 b Hence, it holds that i i ji i=i +1 j=1 ji i=i +1 j=0 µ ij 1k = µ ij0 = i i=i +1 i i=i +1 µ i00 + ji 1 j=0 i i=i +1 j=1 Lem.4.5 µ ijk ji µ ij0 i i=i +1 i i=i +1 µ i0k c µ i00. i i=i +1 µ i00. Lemma 4.8 a and Lemma 4.6 in combination with the geometric series show that i a i 1 i 1 i µ i00 µ i 1jk = µ ijk p P ij P p P i j P µ i jk. i=i +1 i=i +1 i=i i=i If j = ji, then Lemma 4.8 c yields that µ i jk = µ i j k µ i j k. Otherwise, if j < ji, then Lemma 4.8 b d yield that µ i jk December 20, c b d c µ i j0 µ i j 1k µ i j k µ i j k.

22 704 Altogether, we have derived 63, which concludes the proof Convergence rates 5.1. Main theorem on optimal convergence rates. The first lemma relates two different characterizations of approximation classes from the literature, which are either based on the accuracy ε > 0 see, e.g., [Ste08, KS08] or the number of elements N see, e.g., [CKNS08, CFPP14] Lemma 5.1. Recall that T c = T c T init. Let ϱ : T c R 0 satisfy that inf T T c ϱt = 0. Let s > 0 and define 65 A c sϱ := sup N + 1 s min ϱt, where T c N N 0 T T c N := {T T c : #T #T init N}. N With T c εϱ := {T T c : ϱt ε} for ε > 0, there holds the equality 66 ε A c sϱ = sup ε>0 min #T #T init s. T T c εϱ The minimum in 65 exists, since all T c 717 N are finite sets. The minimum in 66 exists, since the cardinality is a mapping # : T nc 718 N. In either case, the minimizers might not be unique. If T c = T c T init is replaced by T nc = T nc T init, one can define A nc s, T nc 719 N, and T nc 720 ε ϱ similarly, and the assertion 66 holds accordingly Proof. We only consider the set T c of conforming triangulations, the proof for the set T nc of non-conforming triangulations follows along the same lines. For N N 0, define ε N := min T T c N ϱt 0. Step 1. To prove in 66, let ε > 0. If 0 < ε < ε 0, there exists a minimal N N 0 such that min T T c N ϱt ε. In particular, it follows that N > 0, T c N Tc εϱ, and ε < min T T c N 1 ϱt. This yields that 67 ε min T T cϱ#t #T init s min ϱt N s sup N + 1 s min ϱt = A c sϱ. ε T T c N 1 N N 0 T T c N If ε 0 ε, then T init T c ε 0 ϱ T c εϱ and hence the left-hand side of 67 is zero, and 67 thus remains true. Taking the supremum over all ε > 0, we prove in 66. Step 2. To prove in 66, let N N 0. If ε N > 0, the definition of ε N yields that #T #T init N + 1 for all T T c λε N ϱ and all 0 < λ < 1. This proves that N + 1 s min ϱt min #T #T init s ε N 1 T T c N T T c λε ϱ N λ sup ε min ε>0 T T cϱ#t #T init s. ε If ε N = 0, then the left-hand side of 68 is zero, and the overall estimate thus remains true. Taking the supremum over all N N 0, we prove in 66 for the limit λ 1. The following lemma specifies ϱt and hence introduces the precise approximation class of the present work. Lemma 5.2. For s > 0, let 69 A c s := A c sϱ, where ϱt := ηt ; U T [p T ], p T + U T [p T ] Ω for T T c. December 20,

23 Then, ϱ satisfies the assumptions of Lemma 5.1. Moreover, there exists a constant C > 0, which depends only on C stab and C rel, such that 70 ϱt C min ηt ; UT [Q T ], Q T + U T [Q T ] Ω. Q T PT Proof. Let Q T PT. According to 21, we have that U T [p T ] U T [Q T ] V p T Q T P. Since p T is the best approximation of p in PT, it holds that p T Q T P p Q T P. Hence, stability 32 and reliability 30 of the error estimator prove that ϱt = ηt ; U T [p T ], p T + U T [p T ] Ω 32 ηt ; U T [Q T ], Q T + U T [p T ] U T [Q T ] V + p T Q T P + U T [Q T ] Ω ηt ; U T [Q T ], Q T + U T [Q T ] Ω + p Q T P. 30 ηt ; U T [Q T ], Q T + U T [Q T ] Ω. This proves 70. With linear convergence Theorem 4.1, this yields that inf ϱt inf ϱt ijk inf ηijk + U ijk Ω = 0. T T c i,j,k Q i,j,k Q This concludes the proof. Together with Theorem 4.1, the following theorem is the main result of this work. It states optimal convergence of Algorithm 3.7. The proof is given in Section 5.2. Theorem 5.3. Let 0 < ϑ < C 1 div and 0 < θ < θ opt := 1 + C 2 stab C2 drel 1. Suppose that 71 κ 1 < θ 1/2 C stab and θ < sup δ>0 1 κ 1 C stab 2 θ opt 1 + δ 1 κ 2 1Cstab 2, 1 + δ i.e., 0 κ 1 < 1 is sufficiently small. Moreover, let 0 < κ 2, κ 3 < 1 be sufficiently small in the sense of Lemma 4.5, Lemma 4.6, and Lemma 5.6 below. Then, for all s > 0, there exist constants c opt, C opt > 0 such that c opt A c s ηijk + U ijk Ω #Tijk #T init + 1 s 72 Copt 1 + A c s. sup i,j,k Q The constant c opt depends only on the initial triangulation T init and the polynomial degree m, while C opt depends additionally on the domain Ω, the parameters κ 1, κ 2, κ 3, ϑ, d, θ, C mark, s, and f. The following remark relates our definition of the approximation class from Lemma 5.2 to that of [KS08]. We refer to Appendix C for the proof. Remark 5.4. i The seminal work [KS08] employs two approximation classes: A nc s p := A nc s ϱ p for ϱ p P := min p Q P P = p p P P, Q P PP A c su := A c sϱ u for ϱ u T := min u V T V. V T VT Clearly, the definitions of ϱ p and ϱ u satisfy the assumptions of Lemma 5.1. Moreover, 73 A nc s p A c sp := A c sϱ p. December 20,