Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations

In a recent work (Feischl et al. in Eng Anal Bound Elem 62:141–153, 2016), we analyzed a weighted-residual error estimator for isogeometric boundary element methods in 2D and proposed an adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the multiplicity of the knots. In the present work, we give a mathematical proof that this algorithm leads to convergence even with optimal algebraic rates. Technical contributions include a novel mesh-size function which also monitors the knot multiplicity as well as inverse estimates for NURBS in fractional-order Sobolev norms.


Isogeometric analysis
The central idea of isogeometric analysis (IGA) is to use the same ansatz functions for the discretization of the partial differential equation at hand, as are used for the representation of the problem geometry. Usually, the problem geometry Ω is represented in CAD by means of non-uniform rational B-splines (NURBS), T-splines, or hierarchical splines. This concept, originally invented in [32] for finite element methods (IGAFEM) has proved very fruitful in applications; see also the monograph [9].
Since CAD directly provides a parametrization of the boundary ∂Ω, this makes the boundary element method (BEM) the most attractive numerical scheme, if applicable (i.e., provided that the fundamental solution of the differential operator is explicitly known). However, compared to the IGAFEM literature, only little is found for isogeometric BEM (IGABEM). The latter has first been considered for 2D BEM in [36] and for 3D BEM in [40]. Unlike standard BEM with piecewise polynomials which is well-studied in the literature, cf. the monographs [39,41] and the references therein, the numerical analysis of IGABEM is widely open. We refer to [35,37,38,42] for numerical experiments, to [44] for fast IGABEM with H-matrices, and to [31] for some quadrature analysis. To the best of our knowledge, a posteriori error estimation for IGABEM, however, has only been considered for simple 2D model problems in the recent own works [24,25]. The present work extends the techniques from standard BEM to non-polynomial ansatz functions. The remarkable flexibility of the IGA ansatz functions to manipulate their smoothness properties motivates the development of a new adaptive algorithm which does not only automatically adapt the mesh-width, but also the continuity of the IGA ansatz function to exploit the additional freedoms and the full potential of IGA. This is the first algorithm which simultaneously steers the resolution and the smoothness of the ansatz functions, and, it may thus be a first step to a full hpk-adaptive algorithm.
For standard BEM with discontinuous piecewise polynomials, a posteriori error estimation and adaptive mesh-refinement are well understood. We refer to [1,11,12] for weighted-residual error estimators and to [19,22] for recent overviews on available a posteriori error estimation strategies. Moreover, optimal convergence of mesh-refining adaptive algorithms has recently been proved for polyhedral boundaries [20,21,26] as well as smooth boundaries [27]. The work [2] allows to transfer these results to piecewise smooth boundaries; see also the discussion in the review article [8].
While this work focusses on adaptive IGABEM, adaptive IGAFEM is considered, e.g., in [16,43]. A rigorous error and convergence analysis in the frame of adaptive IGAFEM is first found in [5] which proves linear convergence for some adaptive IGAFEM with hierarchical splines for the Poisson equation, and optimal rates are announced for some future work.

Model problem
We develop and analyze an adaptive algorithm for the following model problem: Let Ω ⊂ R 2 be a Lipschitz domain with diam(Ω) < 1 and Γ ⊆ ∂Ω be a compact, piecewise smooth part of its boundary with finitely many connected components. We consider the weakly-singular boundary integral equation for all x ∈ Γ, (1.1) where the right-hand side f is given and the density φ is sought. We note that (1.1) for Γ = ∂Ω is equivalent to the Laplace-Dirichlet problem To approximate φ, we employ a Galerkin boundary element method (BEM) with ansatz spaces consisting of p-th order NURBS. The convergence order for uniform partitions of Γ is usually suboptimal, since the unknown density φ may exhibit singularities, which are stronger than the singularities in the geometry. In [24], we analyzed a weighted-residual error estimator and proposed an adaptive algorithm which uses this a posteriori error information to steer the h-refinement of the underlying partition as well as the local smoothness of the NURBS across the nodes of the adaptively refined partitions. It reflects the fact that it is a priori unknown, where the singular and smooth parts of the density φ are located and where approximation by nonsmooth resp. smooth functions is required. In [24], we observed experimentally that the proposed algorithm detects singularities and possible jumps of φ and leads to optimal convergence behavior. In particular, we observed that the proposed adaptive strategy is also superior to adaptive BEM with discontinuous piecewise polynomials in the sense that our adaptive NURBS discretization requires less degrees of freedom to reach a prescribed accuracy.

Contributions
We prove that the adaptive algorithm from [24] is rate optimal in the sense of [8]: Let μ be the weighted-residual error estimator in the -th step of the adaptive algorithm. First, the adaptive algorithm leads to linear convergence of the error estimator, i.e., μ +n ≤ Cq n μ for all , n ∈ N 0 and some independent constants C > 0 and 0 < q < 1. Moreover, for sufficiently small marking parameters, i.e. aggressive adaptive refinement, the estimator decays even with the optimal algebraic convergence rate. Here, the important innovation is that the adaptive algorithm does not only steer the local refinement of the underlying partition (as is the case in the available literature, e.g., [8,20,21,26,27]), but also the multiplicity of the knots. In particular, the present work is the first available optimality result for adaptive algorithms in the frame of isogeometric analysis. Additionally, we can prove at least plain convergence if the adaptive algorithm is driven by the Faermann estimator η analyzed in [25] instead of the weighted-residual estimator μ , which generalizes a corresponding result for standard adaptive BEM [23]. Technical contributions of general interest include a novel mesh-size function h ∈ L ∞ (Γ ) which is locally equivalent to the element length (i.e., h| T length(T ) for all elements T ), but also accounts for the knot multiplicity. Moreover, for 0 < σ < 1, we prove a local inverse estimate h σ Ψ L 2 (Γ ) ≤ C Ψ H −σ (Γ ) for NURBS on locally refined meshes. Similar estimates for piecewise polynomials are shown in [15,29,30], while [3] considers NURBS but integer-order Sobolev norms only.
Throughout, all results apply for piecewise smooth parametrizations γ of Γ and discrete NURBS spaces. In particular, the analysis thus covers the NURBS ansatz used for IGABEM, where the same ansatz functions are used for the discretization of the integral equation and for the resolution of the problem geometry, as well as spline spaces and even piecewise polynomials on the piecewise smooth boundary Γ which can be understood as special NURBS.

Outline
The remainder of this work is organized as follows: Sect. 2 fixes the notation and provides the necessary preliminaries. This includes, e.g., the involved Sobolev spaces (Sect. 2.2), the functional analytic setting of the weakly-singular integral equation (Sect. 2.3), the assumptions on the parametrization of the boundary Γ (Sect. 2.4), the discretization of the boundary (Sect. 2.5), the mesh-refinement strategy (Sect. 2.6), Bsplines and NURBS (Sect. 2.7), and the IGABEM ansatz spaces (Sect. 2.8). Section 3 states our adaptive algorithm (Algorithm 3.1) from [24] and formulates the main theorems on linear convergence with optimal rates for the weighted-residual estimator μ (Theorem 3.2) and on plain convergence for the Faermann estimator η (Theorem 3.4). The linear convergence for the μ -driven algorithm is proved in Sect. 4. The proof requires an inverse estimate for NURBS in a fractional-order Sobolev norm (Proposition 4.1) as well as a novel mesh-size function for B-spline and NURBS discretizations (Proposition 4.2) which might be of independent interest. The proof of optimal convergence behaviour is given in Sect. 5. In Sect. 6, we show convergence for the η -driven algorithm.
For the empirical verification of the optimal convergence behavior of Algorithm 3.1 for μ -as well as η -driven adaptivity and a comparison of IGABEM and standard BEM with discontinuous piecewise polynomials, we refer to the numerous numerical experiments in our preceding work [24].

General notation
Throughout, | · | denotes the absolute value of scalars, the Euclidean norm of vectors in R 2 , the measure of a set in R (e.g., the length of an interval), or the arclength of a curve in R 2 . The respective meaning will be clear from the context. We write A B to abbreviate A ≤ cB with some generic constant c > 0 which is clear from the context. Moreover, A B abbreviates A B A. Throughout, mesh-related quantities have the same index, e.g., N is the set of nodes of the partition T , and h is the corresponding local mesh-width etc. The analogous notation is used for partitions T + resp. T etc.

Sobolev spaces
For any measurable subset Γ 0 ⊆ Γ , let L 2 (Γ 0 ) denote the Lebesgue space of all square integrable functions which is associated with the norm u 2 L 2 (Γ 0 ) := Γ 0 |u(x)| 2 dx. We define for any 0 < σ ≤ 1 the Hilbert space associated with the Sobolev-Slobodeckij norm where ∂ γ denotes the arclength derivative. For finite intervals I ⊆ R, we use analogous We note that H σ (Γ ) ⊂ L 2 (Γ ) ⊂ H −σ (Γ ) form a Gelfand triple and all inclusions are dense and compact. Amongst other equivalent definitions of H σ (Γ 0 ) are for example interpolation techniques. All these definitions provide the same space of functions but different norms, where norm equivalence constants depend only on Γ 0 ; see, e.g., the monographs [33,34] and the references therein. Throughout our proofs, we shall use the Sobolev-Slobodeckij norm (2.2), since it is numerically computable.
In the Galerkin boundary element method, the test space H −1/2 (Γ) is replaced by some discrete subspace X ⊂ L 2 (Γ) ⊂ H −1/2 (Γ). Again, the Lax-Milgram lemma guarantees existence and uniqueness of the solution Φ ∈ X of the discrete variational formulation (2.6) Below, we shall assume that X is linked to a partition T of Γ into a set of connected segments.

Boundary parametrization
Let Γ = i Γ i be decomposed into its finitely many connected components Γ i . Since the Γ i are compact and piecewise smooth as well, it holds see, e.g., [25,Section 2.2]. The usual piecewise polynomial and NURBS basis functions have connected support and are hence supported by some single Γ i each. Without loss of generality and for the ease of presentation, we may therefore assume throughout that Γ is connected. All results of this work remain valid for non-connected Γ . We assume that either Γ = ∂Ω is parametrized by a closed continuous and piecewise two times continuously differentiable path γ : [a, b] → Γ such that the restriction γ | [a,b) is even bijective, or that Γ ∂Ω is parametrized by a bijective continuous and piecewise two times continuously differentiable path γ : [a, b] → Γ . In the first case, we speak of closed Γ = ∂Ω, whereas the second case is referred to as open Γ ∂Ω. For closed Γ = ∂Ω, we denote the (b − a)-periodic extension to R also by γ . For the left and right derivative of γ , we assume that γ (t) = 0 for t ∈ (a, b] and γ r (t) = 0 for t ∈ [a, b). Moreover we assume that γ (t) + cγ r (t) = 0 for all c > 0 and t ∈ [a, b] resp. t ∈ (a, b). Finally, let γ L : [0, L] → Γ denote the arclength parametrization, i.e., |γ L (t)| = 1 = |γ r L (t)|, and its periodic extension. Elementary differential geometry yields bi-Lipschitz continuity where C Γ > 0 depends only on Γ . A proof is given in [ (2.8)

Boundary discretization
In the following, we describe the different quantities which define the discretization. Nodes z j = γ (ž j ) ∈ N . Let N := z j : j = 1, . . . , n and z 0 := z n for closed Γ = ∂Ω resp. N := z j : j = 0, . . . , n for open Γ ∂Ω be a set of nodes. We suppose that Multiplicity #z j and knots K ,Ǩ . Let p ∈ N 0 be some fixed polynomial order. Each node z j has a multiplicity #z j ∈ {1, 2 . . . , p + 1} with #z 0 = #z n = p + 1. This induces knots The patch ω (z) of some node z ∈ N resp. the patch ω (T ) are illustrated in blue resp. green and ω m (U) := ω m U .

Mesh-refinement
Suppose that we are given a deterministic mesh-refinement strategy ref ( For the proof of our main result, we need the following assumptions on ref(·). (M1) There exists a constantκ max ≥ 1 such that the local mesh-ratios (2.11) are uniformly boundedκ

Assumption 2.1 For an arbitrary initial mesh
These assumptions are especially satisfied for pure h-refinement based on local bisection [1] as well as for the concrete strategy used in [24,25]. The latter strategy looks as follows: Let (i) If both nodes of an element T ∈ T belong to M , the element T will be marked.
(ii) For all other nodes in M , the multiplicity will be increased if it is less or equal to p + 1, otherwise the elements which contain one of these nodes z ∈ M , will be marked. (iii) Recursively, mark further elements T ∈ T for refinement if there exists a marked element T ∈ T with T ∩ T = ∅ andȟ ,T >κ 0ȟ ,T . (iv) Refine all marked elements T ∈ T by bisection and hence obtain [T + ].
Proof For any partition T of Γ and any subset of marked elements S ⊆ T , let ref(T , S ) be the partition obtained from the recursive bisection in step (iii)-(iv) above. This local h-refinement procedure has been analyzed in [1]. According to [1,Theorem 2.3], the recursion is well-defined and guaranteesκ ≤ 2κ 0 for all T ∈ ref(T 0 ).
To see (M2), [1, Theorem 2.3] guarantees the existence of some coarsest common refinement The corresponding nodes just satisfy N ⊕ N + = N ∪ N + . There exists a finite sequence of meshes T = T 1 , where # resp. # + denote the multiplicity in K resp. K + and, e.g., # + z := 0 if z ∈ N \N + . There obviously holds

Moreover, [T ⊕ T + ] is clearly a refinement of [T + ] as well.
Finally we consider (M3). As before we have as the number of multiplicity increases during the j-th refinement. There holds The term |T |−|T 0 | can be estimated by C −1 j=0 |S j | with some constant C > 0 which depends only on the initial partition of the parameter domain, see [1,Theorem 2.3], and hence by 2C

B-splines and NURBS
Throughout this subsection, we consider knotsǨ := (t i ) i∈Z on R with multiplicity #t i which satisfy t i−1 ≤ t i for i ∈ Z and lim i→±∞ t i = ±∞. LetŇ := t i : i ∈ Z = ž j : j ∈ Z denote the corresponding set of nodes withž j−1 <ž j for j ∈ Z. For i ∈ Z, the i-th B-spline of degree p is defined inductively by (Fig. 2) where, for t ∈ R, We also use the notations BǨ i, p := B i, p and βǨ i, p := β i, p to stress the dependence on the knotsǨ. The following lemma collects some basic properties of B-splines.
In addition to the knotsǨ = (t i ) i∈Z , we consider positive weights W := (w i ) i∈Z with w i > 0. For i ∈ Z and p ∈ N 0 , we define the i-th NURBS by We also use the notation RǨ ,W i, p := R i, p . Note that the denominator is locally finite and positive.
For any p ∈ N 0 , we define the B-spline space as well as the NURBS space (2.20)

Ansatz spaces
Let [T 0 ] be a given initial mesh with corresponding knots K 0 such that For the extended sequences, we also writeǨ 0 and W 0 and set Due to Lemma 2.3, this definition does not depend on how the sequences are extended. Let [T ] ∈ [T] be a mesh with knots K . Via knot insertion from K 0 to K , one obtains unique corresponding weights W . These are chosen such that the denominators of the NURBS functions do not change. In particular, this implies nestedness where the spaces X resp. X + are defined analogously to (2.21)-(2.22). Moreover, the weights are just convex combinations of W 0 , wherefore For further details, we refer to, e.g., [25, Section 4.2].

For each mesh [T ] ∈ [T]
, define the node-based error estimator where the refinement indicators read Here, we must additionally suppose f ∈ H 1 (Γ ) to ensure that μ is well-defined. It has been proved in [24] that μ is reliable, i.e., where C rel > 0 depends only on p, w min , w max , γ , andκ max . We note that the weighted-residual error estimator in the form μ back to the works [6,13], where reliability (3.2) is proved for standard 2D BEM with piecewise constants on polyhedral geometries, while the corresponding result for 3D BEM is found in [12]. We consider the following adaptive algorithm which employs the Dörfler marking strategy (3.3) from [17] to single out nodes for refinement. Our main result is that the proposed algorithm is linearly convergent, even with the optimal algebraic rate. For a precise statement of this assertion, let [T N ] := [T ] ∈ [T] : |K | − |K 0 | ≤ N be the finite set of all refinements having at most N knots more than [T 0 ]. Following [8], we introduce an estimator-based approximation class In explicit terms, this just means that an algebraic convergence rate of O(N −s ) for the estimator is possible, if the optimal meshes are chosen. The following theorem is the main result of our work: , so that the weighted-residual error estimator μ from (3.1) is well-defined and that Algorithm 3.1 is driven by μ . We suppose that the Assumption 2.1 on the mesh-refinement holds true. Then, for each 0 < θ ≤ 1, there exist constants 0 < q lin < 1 and C lin > 0 such that Algorithm 3.1 is linearly convergent in the sense of μ +n ≤ C lin q n lin μ for all , n ∈ N 0 . (3.5) In particular, this implies convergence Moreover, there is a constant 0 < θ opt < 1 such that for all 0 < θ < θ opt , there exists a constant C opt > 0 such that, for all s > 0, it holds The constants q lin , C lin depend only on p, w min , w max , γ, θ, andκ max from (M1). The constant θ opt depends only on p, w min , w max , γ, and (M1)-(M3), and C opt depends additionally on θ . The proof of Theorem 3.2 is given in Sects. 4 and 5. The ideas essentially follow those of [8], where an axiomatic approach of adaptivity for abstract problems is found. We note, however, that [8] only considers h-refinement, while the present formulation of Algorithm 3.1 steers both, the h-refinement and the knot multiplicity increase.
If Algorithm 3.1 is steered by the Faermann estimator with the refinement indicators instead of μ , we can prove at least plain convergence of the estimator to zero. In contrast to the weighted-residual estimator which requires additional regularity f ∈ H 1 (Γ ), the Faermann estimator η allows a right-hand side f ∈ H 1/2 (Γ ). Moreover, η estimator is efficient and reliable where C eff > 0 depends only on Γ , while C rel > 0 depends additionally on p,κ max , w min , w max and γ ; see [25,Theorem 3.1 and 4.4]. This equivalence of error and estimator puts some interest on the following convergence theorem which is, however, weaker than the statement of Theorem 3.2. According to (3.9), this is equivalent to (4. 2) The constant C inv only depends onκ max , p, w min , w max , γ, and σ .
Proof The proof is done in four steps. First, we show that h σ ψ L 2 (Γ ) ψ H −σ (Γ ) holds for all ψ ∈ L 2 (Γ ) which satisfy a certain assumption. In the second step, we prove an auxiliary result for polynomials which is needed in the third one, where we show that all ψ ∈ X satisfy the mentioned assumption. In the last step, we apply a recent result of [2], which then concludes the proof.
Step 1 Let X ⊂ L 2 (Γ ) satisfy the following assumption: There exists a constant q ∈ (0, 1) such that for all T ∈ T and all ψ ∈ X there exists some connected subset Δ(T, ψ) ⊆ T of length |Δ(T, ψ)| ≥ q|T | such that ψ does not change its sign on Δ(T, ψ) and min x∈Δ(T,ψ) Then, there exists a constant C > 0 which depends only on q andκ , such that Note that (4.4) and (4.5) hold for P Δ(T,ψ) with I simply replaced by Δ(T, ψ) and with (·) replaced by the arclength derivative ∂ Γ . By definition of the dual norm, it holds (4.7) First, we estimate the numerator in (4.7): It remains to estimate the denominator in (4.  [18] h −σ χ 2 This yields where the hidden constant depends only onκ max , σ , and γ . With we finish the first step. Step 2 For some fixed polynomial degree p ∈ N 0 , there exists a constant q 1 ∈ (0, 1) such that for all polynomials F of degree p on Note that M is a compact subset of L ∞ ([0, 1]) and that differentiation (·) is a continuous mapping on M due to finite dimension. This especially implies boundedness sup F∈M F ∞ ≤ C 4 < ∞. We may assume C 4 > 2. For given F ∈ M, we define the interval I as follows: Without loss of generality, we assume that the maximum of |F| is attained at some t 1 ∈ [0, 1/2] and that F(t 1 ) = 1. We set t 3 := t 1 +C −1 4 ∈ (t 1 , 1] and t 2 := t 1 + C −1 4 /2 ∈ (t 1 , 3/4] and I := [t 1 , t 2 ]. Then, |I | = 1/(2C 4 ) and for all t ∈ I it holds Altogether, we thus have and conclude this step.
Step 3 We show that X satisfies the assumption of Step 1 and hence conclude (4.10) The functionψ has the form F/w with a polynomial F of degree p and the weight function w, which is also a polynomial of degree p and which satisfies w min ≤ w ≤ w max . Hence, (4.10) is especially satisfied if After scaling to the interval [0, 1], we can apply Step 2 and conclude this step. Altogether, this proves (4.1).
The proof of linear convergence (3.5) will be done with the help of some auxiliary (and purely theoretical) error estimator ρ . The latter relies on the following definition of an equivalent mesh-size function which respects the multiplicity of the knots.

Proposition 4.2 Assumption 2.1 (M1) implies the existence of a modified mesh-size function h : [T] → L ∞ (Γ ) with the following properties:
There exists a constant C wt > 0 and 0 < q ctr < 1 which depend only onκ max , p and γ such that for all [
For any [T ] ∈ [T], we define the auxiliary estimator which employs the novel mesh-size function h from Proposition 4.2. Obviously the estimators μ and ρ are locally equivalent (4.16) where the hidden constants depend only onκ max , p, and γ . The proof of the following lemma is inspired by [26,Proposition 3.2] resp. [8,Lemma 8.8], where only h-refinement is considered.
Proof The proof is done in several steps.
Step 1 With the inverse estimate (4.2), there holds the following stability property for with a constant C > 0 which depends only on C wt , C inv , and γ .

Proof of Theorem 3.2, optimal convergence (3.7)
As in the previous section, we define an auxiliary error estimator. For each [T ] ∈ [T], let Note that the estimators μ and ρ are again locally equivalent T ∈T z∈T ρ 2 (T ) for all z ∈ N and T ∈ T with z ∈ T, (5.2) where the hidden constant depends only onκ max . Analogous versions of the next two lemmas are already proved in [26,Proposition 4.2 and 4.3] for h-refinement and piecewise constants; see also [8,Propostion 5.7] for discontinuous piecewise polynomials and h-refinement. The proof for Lemma 5.1 is essentially based on Proposition 4.1. The proof of Lemma 5.2 requires the construction of a Scott-Zhang type operator (5.9) which is not necessary in [8,26], since both works consider discontinuous piecewise polynomials.

Lemma 5.1 (Stability of ρ) Let [T ] ∈ [T] and [T + ] ∈ ref(T ). For
where C stab > 0 depends only on the parametrization γ and the constant C inv of Proposition 4.1.
Proof For all subsets Γ 0 ⊆ Γ , it holds The choice Γ 0 = S shows stability and we conclude the proof.

Lemma 5.2 (Discrete reliability of ρ)
There exist constants C rel , C ref > 0, which depend only onκ max , p, w min , w max , and γ, such that for all refinements as well as where C sz depends only onκ max , p, w max , and γ .
Proof All NURBS basis functions which are non-zero on T , have support in ω p (T ).
With this, (i) follows easily from the definition of P ,I . For stability (ii), we use 0 ≤ R i, p ≤ 1 and (5.8) to see Overall, the hidden constants depend only onκ max , p, w max , and γ .
Since we use a different mesh-refinement strategy, we cannot directly cite the following lemma from [8]. However, we may essentially follow the proof of [8, Proposition 4.12] verbatim. Details are left to the reader.

Lemma 5.4 (Optimality of Dörfler marking) Define
For all 0 < θ < θ opt there is some 0 < q opt < 1 such that for all refinements the following implication holds true The constant q opt depends only on θ and the constants C stab of Lemma 5.1 and C rel of Lemma 5.2.
The next lemma reads similarly as [8,Lemma 3.4]. Since we use a different meshrefinement strategy and our estimator ρ does not satisfy the reduction axiom (A2), we cannot directly cite the result. However, the idea of the proof is the same. Indeed, one only needs a weaker version of the mentioned axiom.

Lemma 5.5 (Quasi-monotonicity of ρ) For all refinements [T + ] ∈ ref([T ]) of [T ] ∈ [T], there holds
where C mon > 0 depends only on the parametrisation γ and the constants C inv of Proposition 4.1 and C rel of Lemma 5.2.
Proof We split the estimator as follows For the first sum, we use (5.4), (T + \T ) = (T \T + ), andȟ + ≤ȟ to estimate For the second sum, we use Lemma 5.1 to see We end up with Lemma 5.2 concludes the proof.
The optimality in Theorem 3.2 essentially follows from the following lemma. It was inspired by an analogous version from [8,Lemma 4.14]. and With the constants C rel , C mon , and q opt from Lemmas 5.2, 5.4 and 5.5, it holds C 1 = 2C rel and C 2 = (C mon q −1 opt ) 1/2 .
Step 1 There exists [T δ ] ∈ [T] with ρ δ ≤ δ and |K δ | − |K 0 | ≤ φ 1/s By the definition of φ A s , we see By the definition of [T N ], we see Step 2 We consider the overlay [T + ] := [T ] ⊕ [T δ ] of (M2). Quasi-monotonicity shows Step 3 Finally, the assumptions on the refinement strategy are used. The overlay estimate and Step 1 give Combining the last two estimates, we end up with This proves (5.18) with C 1 = 2C rel and C 2 = α −1/2 . By (5.20) we can apply Lemma 5.4 and see (5.19).
So far, we have only considered the auxiliary estimator ρ . In particular, we did not use Algorithm 3.1, but only the refinement strategy ref(·) itself. For the proof of optimal convergence (3.7), we proceed similarly as in [8,Theorem 8.4 (ii)].
Since the chosen set M j of Algorithm 3.1 has essentially minimal cardinality, we see with (5.18) that With the mesh-closure estimate of (M3), we get Linear convergence (3.5) shows μ ≤ C lin q − j lin μ j for all j = 0, . . . , . Hence, This concludes the proof. , Additionally let be an auxiliary error estimator with local contributions ρ ( f, z) ≥ 0. Then, there holds the following convergence result. (A2) ρ ( f ) is contractive on R ( f ): There is a constant C 2 such that for all ∈ N 0 , m ∈ N and all δ > 0, it holds In addition, we suppose for all f ∈ H * validity of: The only challenging part is the proof of (A2) for fixed f ∈ H 1 (Γ ). We proceed similarly as in the proof of [23, Theorem 3.1]. In the following, we skip the dependence of f . The heart of the matter are the estimates h +1 ≤ q ctr h on ω (M ) and h +1 ≤ h on Γ , which follow from Proposition 4. Hence, the estimator ρ satisfies Here, the factor 1/2 on the left-hand side stems from the fact that each node patch consists (generically) of two elements. This fact also shows h 1/2 ∂ Γ ( f − V Φ ) 2 L 2 (Γ ) = ρ 2 /2. The Young inequality (c + d) 2 ≤ (1 + δ)c 2 + (1 + δ −1 )d 2 for c, d ≥ 0, together with the triangle inequality shows (1 − q ctr ) ρ (M ) 2 /2 ≤ ρ 2 /2 − 1 1 + δ ρ 2 +m /2 Finally, we use Proposition 4.1 and see with a constant C inv > 0 which depends only on C inv and h h . This yields and concludes the proof of (A2) with C 2 = max( 1 1−q ctr , 2C 2 inv ).