Preprint, 1703.10016, 2017

ASC Report No. 20/2018 Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods

T. Fuhrer,¨ G. Gantner, D. Praetorius, and S. Schimanko

Institute for Analysis and Scientific Computing Vienna University of Technology — TU Wien www.asc.tuwien.ac.at ISBN 978-3-902627-00-1 Most recent ASC Reports 19/2018 A. Arnold, C. Klein, and B. Ujvari WKB-method for the 1D Schrödinger equation in the semi-classical limit: enhanced phase treatment 18/2018 A. Bespalov, T. Betcke, A. Haberl, and D. Praetorius Adaptive BEM with optimal convergence rates for the Helmholtz equation 17/2018 C. Erath and D. Praetorius Optimal adaptivity for the SUPG finite element method 16/2018 M. Fallahpour, S. McKee, and E.B. Weinmüller Numerical simulation of flow in smectic liquid crystals 15/2018 A. Bespalov, D. Praetorius, L. Rocchi, and M. Ruggeri Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs 14/2018 J. Burkotova, I. Rachunkova, S. Stanek, E.B. Weinmüller, S. Wurm On nonsingular BVPs with nonsmooth data. Part 1: Analytical results 13/2018 J. Gambi, M.L. Garcia del Pino, J. Mosser, and E.B. Weinmüller Post-Newtonian equations for free-space laser communications between space- based systems 12/2018 T. Führer, A. Haberl, D. Praetorius, and S. Schimanko Adaptive BEM with inexact PCG solver yields almost optimal computational costs 11/2018 X. Chen and A. Jüngel Weak-strong uniqueness of renormalized solutions to reaction-cross-diffusion systems 10/2018 C. Erath, G. Gantner, and D. Praetorius Optimal convergence behavior of adaptive FEM driven by simple (h-h/2)-type error estimators

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THOMAS FUHRER,¨ GREGOR GANTNER, DIRK PRAETORIUS, AND STEFAN SCHIMANKO

Abstract. We define and analyze (local) multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions. Hypersingular and weakly-singular integral equations are considered. We prove that the condition number of the preconditioned systems of linear equations is independent of the mesh-size and the refinement level. Therefore, the computational complexity, when using appropriate iterative solvers, is optimal. Our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results.

1. Introduction In the last decade, the isogeometric analysis (IGA) had a strong impact on the field of scientific computing and numerical analysis. We refer, e.g., to the pioneering work [HCB05] and to [CHB09, BdVBSV14] for an introduction to the field. The basic idea is to utilize the same ansatz functions for approximations as are used for the description of the geometry by some computer aided design (CAD) program. Here, we consider the case, where the geometry is represented by rational splines. For certain problems, where the fundamental solution is known, the boundary element method (BEM) is attractive since CAD programs usually only provide a parametrization of the boundary ∂Ω and not of the volume Ω itself. Isogeometric BEM (IGABEM) has first been considered for 2D BEM in [PGK+09] and for 3D BEM in [SSE+13]. We refer to [SBTR12, PTC13, SBLT13, NZW+17] for numerical experiments, to [HR10, TM12, MZBF15, DHP16, DHK+18, DKSW18] for fast IGABEM based on wavelets, fast multipole, -matrices resp. 2-matrices, and to [HAD14, KHZvE17, ACD+17, FK18] for some quadratureH analysis. H Recently, adaptive IGABEM has been analyzed in [FGP15, FGHP16, FGK+18] for rational resp. hierarchical splines in 2D and optimal algebraic convergence rates have been proven in [FGHP17, Sch16] (see also [GPS18]) for rational splines in 2D resp. in [Gan17] for hierarchical splines in 3D. In 2D, the corresponding adaptive algorithms allow for both h-refinement as well as regularity reduction via knot multiplicity increase. Usually, it is as- sumed that the resulting systems of linear equations are solved exactly. In practice, however, iterative solvers are used and their effectivity hinges on the condition number of the Galerkin

Date: August 16, 2018. 2010 Mathematics Subject Classification. 65N30, 65F08, 65N38. Key words and phrases. preconditioner, multilevel additive Schwarz, isogeometric analysis, boundary element methods. Acknowledgement. The authors are supported by the Austrian Science Fund (FWF) through the research projects Optimal isogeometric boundary element method (grant P29096) and Optimal adaptivity for BEM and FEM-BEM coupling (grant P27005), the doctoral school Dissipation and dispersion in nonlinear PDEs (grant W1245), and the special research program Taming complexity in PDE systems (grant SFB F65). 1 matrices. It is well-known that the condition number of Galerkin matrices corresponding to the discretization of certain integral operators depend not only on the number of degrees of freedom but also on the ratio hmax/hmin of the largest and smallest element diameter, which can become arbitrarily large on locally refined meshes; see, e.g., [AMT99] for the case of affine boundary elements and lowest-order ansatz functions. Therefore, the construction of optimal preconditioners is a necessity. We say that a preconditioner is optimal, if the condition number of the resulting preconditioned matrices is independent of the mesh-size function h, the number of degrees of freedom and the refinement level. In this work, we consider simple additive Schwarz methods. The central idea of our local multilevel diagonal preconditioners is to use newly created nodes and old nodes whose multiplicity has changed, to define local diagonal scalings on each refinement level. This allows us to prove optimality of the proposed preconditioner and the computational complexity for applying our preconditioner is linear with respect to the number of degrees of freedom on the finest mesh. In particular, this extends our prior works [FFPS15, FFPS17, FHPS18] on local multilevel diagonal preconditioners for hypersingular integral equations and weakly- singular integral equations for affine geometries in 2D and 3D and lowest-order discreta- tizations. Other results on Schwarz methods for BEM with affine boundaries are found in [Cao02, TS96, TSM97], mainly for uniform mesh-refinements and in [AM03] for some specially local refined meshes. For the higher order case, we refer, e.g., to [Heu96, FMPR15]. Diagonal preconditioners for BEM are covered in [AMT99, GM06]. Another preconditioner technique that leads to uniformly bounded condition numbers is based on the use of integral operators of opposite order. The case of closed boundaries is analyzed in [SW98], whereas open boundaries are treated in the recent works [HJHUT14, HJHUT16, HJHUT17]. The recent work [SvV18] deals with the opposite order operator preconditioning technique in Sobolev spaces of negative order. To the best of our knowledge, the preconditioning of IGABEM, even on uniform meshes, is still an open problem. For isogeometric finite elements (IGAFEM), a BPX-type preconditioner is analyzed in [BHKS13] on uniform meshes, where the authors consider general pseudodifferential operators of positive order. Recently, a BPX-type preconditioner with local smoothing has been analyzed in [CV17] for locally refined T-meshes. Other multilevel preconditioners for IGAFEM have been studied in [GKT13, ST16, HTZ17, Tak17] for uniform resp. in [HJKZ16] for hierarchical meshes, and domain decomposition methods can be found in [BdVCPS13, BdVPS+14, BdVPS+17] for uniform meshes. Model problem. Let Ω be a bounded simply connected Lipschitz domain in R2, with piecewise smooth boundary ∂Ω and let Γ ∂Ω be a connected subset with Lipschitz boundary ∂Γ. Neumann screen problems on Γ yield⊆ the weakly-singular integral equation

∂ ∂ Wu(x) := G(x, y) u(y) dy = f(x) for all x Γ (1.1) −∂νx ˆΓ ∂νy ∈ with the hypersingular integral operator W and some given right-hand side f. Here, νx denotes the outer normal unit vector of Ω at some point x Γ, and ∈ 1 G(x, y) := log x y (1.2) −2π | − | 2 is the fundamental solution of the Laplacian. Similarly, Dirichlet screen problems lead to the weakly-singular integral equation

Vφ(x) := G(x, y) φ(y) dy = g(x) (1.3) ˆΓ with the weakly-singular integral operator V and some given right-hand side g. Outline. The remainder of the work is organized as follows: Section 2 provides the functional analytic setting of the boundary integral operators, the deﬁnition of the mesh, B-splines and NURBS together with their basic properties. Auxiliary results that are used in the proof of our main results are stated in Section 3. In Section 4, we deﬁne our local multilevel diagonal preconditioner for the hypersingular integral operator on closed and open boundaries and prove its optimality (Theorem 4.1). Then, in Section 5, we extend our local multilevel diagonal preconditioner to the weakly-singular case and give a proof of its optimality (Theo- rem 5.2). Finally, in Section 6 we restate the abstract results for additive Schwarz operators in matrix formulation (Corollary 6.1). Moreover, numerical examples for closed and open boundaries are presented and some aspects of implementation are discussed.

2. Preliminaries

2.1. Notation. Throughout and without any ambiguity, denotes the absolute value of |·| scalars, the Euclidean norm of vectors in R2, the measure of a set in R (e.g., the length of an interval), or the arclength of a curve in R2. 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., • is the set of nodes of the partition , and h is the corresponding local mesh-width etc.N The analogous T• • notation is used for partitions ◦ resp. ℓ etc. We sometimes use to transform notation on the boundary to the parameterT domain.T The most important symbo· ls are listed in Table 1. 2.2. Sobolev spaces. The usual Lebesgue and Sobolev spacesb on Γ are denoted by L2(Γ) = H0(Γ) and H1(Γ). We introduce the corresponding seminorm on any measurable subset Γ Γ via 0 ⊆ 1 v 1 := ∂ v 2 for all v H (Γ), (2.1) | |H (Γ0) k Γ kL (Γ0) ∈ with the arclength derivative ∂Γ. We have that 2 2 2 1 v 1 = v 2 + v 1 for all v H (Γ), (2.2) k kH (Γ) k kL (Γ) | |H (Γ) ∈ Moreover, H1(Γ) is the space of H1(Γ) functions, which have a vanishing trace on the relative boundary ∂Γ equipped with the same norm. On Γ, Sobolev spaces of fractional order 0 <σ

Name Description First appearance Name Description First appearance B•,i,p B-spline Section 2.8 b R NURBS Section 2.8 B•,i,q B-spline on Γ Section 2.9 b•,i,p R NURBS on Γ Section 2.9 B•,i,p continuous B-spline on Γ Section 2.9 •,i,p ∗ R•,i,p continuous NURBS on Γ Section 2.9 B•,i,p dual B-spline Section 3.1 b R∗ dual NURBS Section 3.1 gen(·) generation fct. Section 3.3 b•,i,p t•,i knot Section 2.8 bh•,T length in [a, b] Section 2.6 T• mesh Section 2.6 h•,T arclength Section 2.6 w•,i weight Section 2.8 h•(·),h•(·) mesh-size functions Section 2.6 b w(·) denominator Section 2.9 huni(m) uniform mesh-size in [a, b] Section 3.3 b wmin,wmax bounds for weights Section 2.9 Iℓ index set Section 4 e W• weight vector Section 2.9 J• Scott-Zhang operator Section 3.1 X• NURBS space for W Section 2.9 K• knot vector in [a, b] Section 2.6 b Xℓ space of new NURBS Section 4 K• knot vector Section 2.6 e Xℓ,i one-dim. subspace Section 4 N set of new knots Section 3.1 e◦\• Y• spline space for V Section 5 K admissible knot vectors Section 2.7 0 Yℓ space of new splines Section 5 level•(·) level fct. Section 3.3 e z•,j node Section 2.6 N• number of knots Section 2.6 zℓ,i new knot Section 4 N• set of nodes Section 2.6 e κ• local mesh-ratio in [a, b] Section 2.6 o 1ifΓ= ∂Ω, 0 else Section 2.9 b κmax bound for local mesh-ratio in [a, b] Section 2.7 p positive polynomial order Section 2.6 b V Πuni(m) projection on unif. space Section 3.3 P Schwarz operator for V Section 5 m eL ω• (·) patch Section 2.6 PW Schwarz operator for W Section 4 eL #• multiplicity Section 2.6 refine(·) set of reﬁned knot vectors Section 2.7

±σ ±σ ±σ with v H±σ(Γ) . v ±σ for all v H (Γ). We note that H (Γ) = H (Γ) for 0 <σ< k k k kHe (Γ) ∈ 1/2 with equivalent norms. Moreover, it holds for Γ = ∂Ω that H±σ(∂Ω) = H±σ(∂Ω) even with equal norms for all 0 < σ 1. Finally,e the treatment ofe the closed boundary Γ = ∂Ω ±σ ≤ ±σ requires the definition of H0 (∂Ω) = v H (∂Ω) : v , 1 ∂Ω =0e for all 0 σ 1. Details and equivalent definitions of the∈ Sobolev spacesh are,i found, e.g., in [McL00,≤ ≤ SS11]. 2.3. Hypersingular integral equation. For 0 σ 1, the hypersingular integral ≤ ≤ operator W : Hσ(Γ) Hσ−1(Γ) is well-defined, linear, and continuous. Recall that Γ and ∂Ω are supposed to be→ connected. For Γ $ ∂Ωe and σ =1/2, W : H1/2(Γ) H−1/2(Γ) is symmetric and elliptic. Hence, → 1/2 u , v W := Wu , v for all u, v H (Γ), (2.4) hh ii e h iΓ ∈ 1/2 H W defines an equivalent scalar product on (Γ) with correspondinge norm . For Γ = ∂Ω, the operator W is symmetric and elliptic up to the constant|| · || functions, i.e., W : H1/2(∂Ω) H−1/2(∂Ω) is elliptic. Ine particular, 0 → 0 1/2 u , v W := Wu , v + u , 1 v , 1 for all u, v H (Γ), (2.5) hh ii h i∂Ω h i∂Ωh i∂Ω ∈ 1/2 1/2 defines an equivalent scalar product on H (∂Ω) = H (∂Ω) with norm W. e || · || With this notation and provided that f H−1/2(Γ)incaseofΓ = ∂Ω, the strong form (1.1) ∈ 0 is equivalently stated in variational form: Find u He1/2(Γ) such that ∈ 1/2 u , v W = f , v for all v H (Γ). (2.6) hh ii h iΓ e ∈ Therefore, the Lax-Milgram lemma applies and hence (1.1) admits a unique solution u e ∈ H1/2(Γ). More details and proofs are found, e.g., in [McL00, SS11, Ste08]. 4 e 2.4. Weakly-singular integral equation. For 0 σ 1, the weakly-singular integral ≤ ≤ operator V : Hσ−1(Γ) Hσ(Γ) is well-defined, linear, and continuous. For Γ = ∂Ω, we suppose diam(Ω) < 1. → For σ =1/2,eV : H−1/2(Γ) H1/2(Γ) is symmetric and elliptic. In particular, → −1/2 φ , ψ V := Vφ , ψ for all φ, ψ H (Γ), (2.7) e hh ii h iΓ ∈ −1/2 defines an equivalent scalar product on H (Γ) with corresponding norm V. With this e notation, the strong form (1.3) with data g H1/2(Γ) is equivalently stated||by · || ∈ e −1/2 φ , ψ V = g , ψ for all ψ H (Γ). (2.8) hh ii h iΓ ∈ Therefore, the Lax-Milgram lemma applies and hence (1.3) admits a unique solution φ e ∈ H−1/2(Γ). More details and proofs are found, e.g., in [McL00, SS11, Ste08]. 2.5. Boundary parametrization. We assume that either Γ = ∂Ω is parametrized by a closede continuous and piecewise continuously differentiable path γ :[a, b] Γ with a < b → such that the restriction γ [a,b) is even bijective, or that Γ $ ∂Ω is parametrized by a bijective continuous and piecewise 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′ℓ (t) = 0 for t (a, b] and γ′r (t) =0 6 ∈ 6 for t [a, b). Moreover, we assume that γ′ℓ (t)+ cγ′r (t) = 0 for all c > 0 and t [a, b] ∈ 6 ∈ resp. t (a, b). Finally, let γarc : [0, Γ ] Γ denote the arclength parametrization, i.e., ′ℓ ∈ ′r | | → γarc(t) =1= γarc(t) , and its periodic extension. Elementary differential geometry yields bi-Lipschitz| | continuity| | 3 −1 γarc(s) γarc(t) s t 4 Γ , for closed Γ, C | − | CΓ for s, t R, with | − | ≤ | | (2.9) Γ ≤ s t ≤ ∈ s = t [0, Γ ], for open Γ, | − | ( 6 ∈ | | where CΓ > 0 depends only on Γ. A proof is given in [Gan14, Lemma 2.1] for closed Γ = ∂Ω. For open Γ $ ∂Ω, the proof is even simpler. 2.6. Boundary discretization. In the following, we describe the different quantities, which define the discretization. Nodes z•,j = γ(z•,j) ∈ N• and number of nodes n•. Let := z : j = N• •,j 1,...,n and z := z for closed Γ = ∂Ω resp. := z : j = 0,...,n for • •,0 •,n• N• •,j • open Γ $ ∂Ω be a setb of nodes. We suppose that z•,j = γ(z•,j) for some z•,j [a, b] with 1 ∈ a = z•,0 < z•,1 < z•,2 < < z•,n• = b such that γ [z•,j−1,z•,j ] C ([z•,j−1, z•,j]). ··· | b b ∈ Multiplicity #•z•,j, knot vector K• and number of knots N•. Let p N be b b ∈ someb fixedb positiveb polynomialb order. Each interior node z•,j hasb a multiplicityb #•z•,j 1, 2 ...,p and # z = # z = p + 1. This induces knots ∈ { } • •,0 • n• =(z ,...,z ,...,z ,...,z ), (2.10) K• •,k •,k •,n• •,n• #•z•,k−times #•z•,n• −times with k =1forΓ= ∂Ω resp. k =0forΓ| {z$ ∂}Ω. We| define{z the number} of knots in γ((a, b]) as n•

N• := #•z•,j. (2.11) j=1 X 5 Elements T•,j, partition T•. Let = T ,...,T be a partition of Γ into T• { •,1 •,n• } compact and connected segments T•,j = γ(T•,j) with T•,j = [z•,j−1, z•,j]. −1 Local mesh-sizes h•,T , h•,T and h•, h•. For T , we define h := γ (T ) as ∈ T• •,T | | its length in the parameter domain, and hb•,T := T bas itsb arclength.b We define the local | | mesh-width functions hb , h L∞(Γ) byb h = h and h = h . b • • ∈ •|T •,T •|T •,T Local mesh-ratio κ•. We define the local mesh-ratio by

b b ′b ′ κ• := max h•,T /h•,T ′ : T, T • with T T = . (2.12) b ∈ T ∩ 6 ∅ m m Patches ω• (z) and ω• (Γ0). For each set Γ0 Γ, we inductively define for m N0 b b b ⊆ ∈ Γ0 if m =0, m ω (Γ0) := ω (Γ ) := T : T Γ = if m =1, •  • 0 ∈ T• ∩ 0 6 ∅ ω (ωm−1(Γ )) if m> 1.  • • S0 m m For points z Γ, we abbreviate ω•(z) := ω•( z ) and ω (z) := ω ( z ). ∈  { } • • { } 2.7. Admissible knot vectors. Throughout, we consider knot vectors • as in Section 2.6 with uniformly bounded local mesh-ratio , i.e., we suppose the existenceK of κ 1 with max ≥ κ• κmax. (2.13) ≤ b Let and be knot vectors (2.13). We say that is finer than and write K• K◦ K◦ K• K◦ ∈ refine( •) if • is a subsequence of b◦ suchb that ◦ is obtained from • via iterative dyadic bisectionsK K in the parameter domainK and multiplicityK increases. Formally,K this means that • ◦ with #•z #◦z for all z • ◦, and that for all T ◦ there exists ′ N ⊆ N ≤ ′ −1 ∈′ N ∩N−j −1 ∈ T T • and j N0 with T T and γ (T ) =2 γ (T ) . Throughout, we suppose that all∈ considered T ∈ knot vectors ⊆ with (2.13)| are| finer| than some| fixed initial knot vector . K• K0 We call such a knot vector admissible. The set of all these knot vectors is abbreviated by K. 2.8. B-splines and NURBS. Throughout this subsection, we consider knots := (t ) K• •,i i∈Z on R with multiplicity #•t•,i, which satisfy that t•,i−1 t•,i for i Z and limi→±∞ t•,i = . ≤ ∈ ±∞ Let • := t•,i : i Z = z•,j : j Z denote the corresponding set ofb nodes with N ∈ ∈ z•,j−1 < z•,j for j Z. For i Z, the i-th B-spline of degree q is defined inductively by ∈ ∈ b b B•,i,0 := χ[t ,t ), b b •,i−1 •,i (2.14) B•,i,q := β•,i−1,qB•,i,q−1 + (1 β•,i,q)B•,i+1,q−1 for q N, b − ∈ where, for t R, ∈ b b b t−t•,i if t•,i = t•,i+q, t•,i+q −t•,i β•,i,q(t) := 6 (2.15) (0 if t•,i = t•,i+q. The following lemma collects basic properties of B-splines. Proves are found, e.g., in [dB86].

Lemma 2.1. For an interval I = [a, b) and q N0, the following assertions (i)–(vii) hold: ∈ (i) The set B•,i,q I : i Z, B•,i,q I = 0 is a basis for the space of all right-continuous | ∈ | 6 •-piecewise polynomials of degree lower or equal q on I, which are, at each knot t•,i, Nq # t btimes continuouslyb differentiable if q # t 0. − • •,i − • •,i ≥ c 6 (ii) For i Z, B•,i,q vanishes outside the interval [t•,i−1, t•,i+q). It is positive on the open interval∈ (t , t ) and a polynomial of degree q on each interval (z , z ) •,i−1 •,i+q j−1 j ⊆ (t•,i−1, t•,i+q) forb j Z. ∈ (iii) For i Z, B•,i,q is completely determined by the q +2 knots t•,i−1,...,t•,i+q,b whereforeb we also∈ write b B( t ,...,t ) := B (2.16) ·| •,i−1 •,i+q •,i,q (iv) The B-splines of degree q form a (locally finite) partition of unity, i.e., b b B•,i,q =1 on R. (2.17) Xi∈Z b (v) For i Z with t•,i−1 < t•,i = = t•,i+q < t•,i+q+1, it holds that ∈ ··· B (t )=1 and B (t )=1. (2.18) •,i,q •,i− •,i+1,q •,i (vi) Suppose the convention q/0 := 0. For q 1 and i Z, it holds for the right derivative b ≥ b ∈ q q B′r = B B . (2.19) •,i,q t t •,i,q−1 − t t •,i+1,q−1 •,i+q−1 − •,i−1 •,i+q − •,i ′ (vii) Let t (tℓ−1, tbℓ] for some ℓ Z and letb ◦ be the refinementb of •, obtained by adding ′ ∈ ∈ K K t . Then, for all coefficients (a•,i)i∈Z, there exists (a◦,i)i∈Z such that b b a•,iB•,i,q = a◦,iB◦,i,q (2.20) Xi∈Z Xi∈Z With the multiplicity # t′ of t′ in theb knots , theb new coefficients can be chosen as ◦ K◦ ′ a•,i if i ℓ q + #◦t 1, ′ ′ b ≤ − ′ − a◦,i = (1 β (t ))a + β (t )a if ℓ q + # t i ℓ, (2.21)  − •,i−1,q •,i−1 •,i−1,q •,i − ◦ ≤ ≤ a•,i−1 if ℓ +1 i. ≤ If one assumes #•ti q +1 for all i Z, these coefficients are unique. Note that these three cases are equivalent≤ to t ∈ t′, t < t′ < t , resp. t′ t . •,i+q−1 ≤ •,i−1 •,i+q−1 ≤ •,i−1 Remark 2.2. Let j Z and (δij)i∈ be the corresponding Kronecker sequence. Choosing ∈ Z (a•,i)i∈Z = (δij)i∈Z in Lemma 2.1 (vii), one sees that B•,j,q is a linear combination of B◦,j,q ′ and B◦,j+1,q, where B•,j,q = B◦,j,q if j ℓ q + #◦t 2 and B•,j,q = B◦,j+1,q if ℓ +1 j. ≤ − −b ≤b In addition to the knots =(t ) , we consider fixed positive weights := (w ) b b Kb• •,i i∈Z b b W• •,i i∈Z with w•,i > 0. For i Z and q N0, we define the i-th NURBS by ∈ ∈ b w•,iB•,i,q R•,i,q := . (2.22) k∈ w•,kB•,k,q Z b Note that the denominator is locallyb finiteP and positive. b For any q N0, we define the B-spline space ∈

q ( •) := aiB•,i,q : ai R (2.23) S K ∈ ( i∈Z ) X 7 b b b as well as the NURBS space q q ( •) ( •, •) := aiR•,i,q : ai R = S K . (2.24) S K W ∈ ( i∈ ) k∈ w•,kB•,k,q XZ Zb b We define for 0b<σ 0, 0 <σ< 1, and K,wminb,wmax > 0. Suppose that the weights • are boundedb by w and w , i.e., w inf w bsup w w , and that theW local min max min ≤ i∈Z •,i ≤ i∈Z •,i ≤ max zj −zj−1 zj −zj−1 mesh-ratio on is bounded by K, i.e., sup max b b , b b : j K. Then, R zj−1−zj−2 zj+1−zj Z b b b b ∈ ≤ there exists a constant Cscale > 0, which dependsn only on q,wmin,wmax, and Ko, such that for all i Z with suppR•,i,q > 0, it holds that ∈ | | 1−2σ 2 suppR•,i,q Cscale R•,i,q Hσ(suppR ). (2.25) b | | ≤ | | b•,i,q Proof. The proof is split into twob steps. b Step 1: First, we suppose that wi = 1 for all i Z and hence R•,i,q = B•,i,q. The definition of the B-splines implies their invariance with respect∈ to affine transformations of the knots: b b B(t t0,...,tq+1)= B(ct + s ct0 + s,...,ctq+1 + s) for all t0 tq+1,s,t R and c> 0. | | ≤···≤ ∈ With the abbreviation S := suppB = [t , t ], it hence holds that b b •,i,q •,i−1 •,i+q 2 2 B•,i,q(r) B•,i,q(s) B σ = | −b | dsdr | •,i,q|H (S) ˆ ˆ r s 1+2σ S S | − | b t•,i−t•b,i−1 t•,i+q−1−t•,i−1 b 1 1 B(r 0, ,..., , 1) B(s ... ) 2 1−2σ | | |S| |S| − | | (2.26) = S 1+2σ dsdr | | ˆ0 ˆ0 r s b 1 1 | − | b 1−2σ 2 S inf B(r 0, t1,...,tq, 1) B(s 0, t1,...,tq, 1) ds dr, ≥| | 0≤t1≤···≤tq ≤1 ˆ0 ˆ0 | | − | | where for the last inequality we haveb used that r s 1.b We use a compactness argument to conclude the proof. Let (t ,...,t ) be| − a convergent| ≤ minimizing sequence for the k,1 k,q k∈N infimum in (2.26). Let (t ,...,t ) be the corresponding limit. With the definition of the ∞,1 ∞,q B-splines one easily verifies that

B(r 0, tk,1,...,tk,q, 1) B(r 0, t∞,1,...,t∞,q, 1) for almost every r R. | → | ∈ The dominated convergence theorem implies that the infimum is attained at (t∞,1,...,t∞,q). b b Lemma 2.1 (ii) especially implies that B( 0, t∞,1,...,t∞,q, 1) is not constant. Therefore the infimum is positive, and we conclude the·| proof. Step 2: Recall that R = w•,iBb•,i,qb . As in Step 1, we transform suppR onto the •,i,q Pi+q w B •,i,q j=i−q •,j b•,j,q interval [0, 1]. Hence, it suffices to prove, with the compact interval I := [0 Kq, 1+ Kq], that the infimum b −b 1 1 2 w1B(r 0, t1,...,tq, 1) w1B(s 0, t1,...,tq, 1) inf 1+q | 1+q | dsdr t−q,...,t−1,t1,...,tq,tq+2,...,t1+2q ∈I ˆ ˆ − 0 0 j=1−q wjB(r tj−1,...,tj+q) j=1−q wjB(s tj−1,...,tj+q) w1−q ,...,w1+q ∈[wmin,wmax] | | b b is bigger than 0. This can be P proved analogously as before. P b 8 b 2.9. Ansatz spaces. Throughout this section, we abbreviate γ −1 with γ−1 if Γ $ ∂Ω is |[a,b) N0−p an open boundary. Additionally to the initial knots 0 K, suppose that 0 =(w0,i) K ∈ W i=1−p are given initial weights with w = w , where N = for closed Γ = ∂Ω resp. 0,1−p 0,N0−p 0 |K0| N0 = 0 (p +1) for open Γ $ ∂Ω. In the weakly-singular case we assume w0,i = 1 for |K | − i = 1 p,...,N0 p. We extend the corresponding knot vector in the parameter domain, − N0 − N0 0 = (t0,i) if Γ = ∂Ω is closed resp. 0 = (t0,i) if Γ $ ∂Ω is open, arbitrarily to K i=1 K i=−p (t ) with t = = t = a, t t , lim t = . For the extended sequence 0,i i∈Z −p ··· 0 0,i ≤ 0,i+1 i→±∞ 0,i ±∞ web also write . We define the weight functionb K0 N0−p b w := w B . (2.27) 0,k 0,k,p|[a,b] kX=1−p b Let • K be an admissible knotb vector. Outside of the interval (a, b], we extend the K ∈ corresponding knot sequence in the parameter domain exactly as before and write again K• for the extension as well. This guarantees that forms a subsequence of . Via K• K0 K• knot insertion from 0 to •,b Lemma 2.1 (i) proves the existence and uniqueness of weights N•−p K K • =(w•,i)i=1−p such that b b W b b N0−p N•−p w = w B = w B . (2.28) 0,k 0,k,p|[a,b] •,k •,k,p|[a,b] kX=1−p kX=1−p By choosing these weights,b we ensure thatb the denominator ofb the considered rational splines does not change. Lemma 2.1 (v) states that B (a)=1= B (b ), which implies •,1−p •,N•−p − that w•,1−p = w•,N•−p. Further, Lemma 2.1 (iv) and (vii) show that w := min( ) min( ) w b max( ) max(b ) := w . (2.29) min W0 ≤ W• ≤ ≤ W• ≤ W0 max In the weakly-singular case there even holds that w = 1 for i = 1 p,...,N p, and •,i − • − w = 1. Finally, we extend • arbitrarily tob (w•,i)i∈Z with w•,i > 0, identify the extension with and set for the hypersingularW case W• b p( , ) := V γ−1 : V p( , ) (2.30) S K• W• • ◦ • ∈ S K• W• and for the weakly-singular case b b b b p−1( ) := V γ−1 : V p−1( ) (2.31) S K• • ◦ • ∈ S K• Lemma 2.1 (iii) shows that the definition does not depend on how the sequences are extended. We define the transformed basis functionsb b b b R := R γ−1 and B := B γ−1. (2.32) •,i,p •,i,p ◦ •,i,p−1 •,i,p−1 ◦ Later, we will also need the notation B•,i,p, which we define analogously. We introduce the ansatz spaceb for the hypersingular caseb

p 1/2 V• ( •, •): V•(γ(a)) = V•(γ(b )) H (Γ) if Γ = ∂Ω, • := ∈ Sp K W − ⊂ 1/2 (2.33) X V• ( •, •): V•(γ(a))=0= V•(γ(b )) H (Γ) if Γ $ ∂Ω, ( ∈ S K W − ⊂ and for the weakly-singular case e p−1 −1/2 • := ( •) H (Γ). (2.34) Y S K9 ⊂ e Note that, in contrast to the hypersingular case, we only allow for non-rational splines in the weakly-singular case. We exploit this restriction in Lemma 5.1 below, which states that ∂ = . For rational splines, this assertion is in general false. We abbreviate ΓX• Y• R + R for i =1 p, R := ℓ,1−p,p ℓ,Nℓ−p,p − (2.35) ℓ,i,p R for i =1 p. ( ℓ,i,p 6 − We deﬁne Bℓ,i,p analogously. Further, we set 0 if Γ = ∂Ω, o := (2.36) (1 if Γ $ ∂Ω. Lemma 2.1 (i) and (v) show that = span R : i =1 p + o,...,N p 1 , (2.37) X• •,i,p − • − − as well as = span B : i =1 (p 1),...,N 1 (p 1) . (2.38) Y• •,i,p−1 − − • − − − In both cases, the corresponding sets form a basis of resp. . Note that the spaces X• Y• X• and satisfy that H1(Γ) and L2(Γ). Lemma 2.1 (i) implies nestedness Y• X• ⊂ Y• ⊂ refine • ◦ and • ◦ for all •, ◦ K with ◦ ( •). (2.39) X ⊆ X eY ⊆Y K K ∈ K ∈ K 3. Auxiliary results for hypersingular case

3.1. Scott-Zhang-type projection. In this section, we recall a Scott-Zhang-type operator from [GPS18]. Let • K. In [BdVBSV14, Section 2.1.5], it is shown that, for i K ∈ ∈ 1 p,...,N p , there exist dual basis functions B∗ L2(a, b) such that { − • − } •,i,p ∈ suppB∗ = suppB = [t , t ], (3.1) •,i,p •,i,p •,i−1b •,i+p b ∗ 1, if i = j, Bb•,i,p(t)B•,j,p(tb)dt = δij = (3.2) â (0, else, and b b ∗ p −1/2 B 2 9 (2p + 3) suppB . (3.3) k •,i,pkL (a,b) ≤ | •,i,p| Each dual basis function depends only on the knots t•,i−1,...,t•,i+p. Therefore, we also write b b B∗ = B∗( t ,...,t ). (3.4) •,i,p ·| •,i−1 •,i+p With the denominator w from (2.28), define b b ∗ ∗ R•,i,p := B•,i,pw/w•,i. (3.5) b This immediately proves that bb b b ∗ R•,i,p(t)R•,j,p(t)dt = δij (3.6) â and b b ∗ p −1/2 R 2 . 9 (2p + 3) suppR , (3.7) k •,i,pkL (a,b) | •,i,p| 10 b b where the hidden constant depends only on wmin and wmax. We define the Scott-Zhang-type operator J : L2(Γ) • → X• N•−p−1 b R∗ +R∗ b•,1−p,p b•,N•−p,p v γ dt if i =1 p, J•v := α•,i(v)R•,i,p with α•,i(v) := a 2 ◦ − (3.8) ´ b ∗ i=1−p+o ( a R•,i,p v γ dt if i =1 p. X ´ ◦ 6 − N•−p b ∗ A similar operator, namely I• := i=1−p abR•,i,p v γ dt R•,i,p, has been analyzed in ´ ◦ [BdVBSV14, Section 3.1.2]. However,PI• is not applicable here for two reasons: First, for Γ= ∂Ω, it does not guarantee that I•v is continuousb at γ(a)= γ(b). Second, for Γ $ ∂Ω, it does not guarantee that I•v(γ(a))=0= I•v(γ(b)). refine 2 Lemma 3.1. Let •, ◦ K with ◦ ( •). Then, each v L (Γ) satisfies that K K ∈ K ∈ K ∈ (J J )v span R : i 1 p + o,...,N p 1 , suppR = , (3.9) ◦ − • ∈ ◦,i,p ∈{ − ◦ − − } ◦,i,p ∩ N◦\• 6 ∅ where e := z : # z > # z . (3.10) N◦\• N◦ \N• ∪ ∈N◦ ∩N• ◦ • Proof. We only provee the lemma for closed Γ = ∂Ω. For open Γ $ ∂Ω, the proof even simplifies. We split the proof into two steps. Step 1: We consider the case where ◦ is obtained from • by insertion of a single knot t′ [a, b] in the parameter domain. LetK b = t′ (t , t ]K with some ℓ 1,...,N p . ∈ 6 ∈ •,ℓ−1 •,ℓ ∈{ • − } Note that N◦ = N• +1. It holds that b b (J J )v = α (v)R α (v)R ◦ − • ◦,1−p ◦,1−p,p − •,1−p •,1−p,p N•−p b N•−p−1 b (3.11) + R∗ v γ dt R R∗ v γ dt R . ˆ ◦,i,p ◦ ◦,i,p − ˆ •,i,p ◦ •,i,p i=2−p a i=2−p a X X We split the second sum into b b

′ N•−p−1 b ℓ−p+#◦t −2 b R∗ v γ dt R = R∗ v γ dt R ˆ •,i,p ◦ •,i,p ˆ •,i,p ◦ •,i,p i=2−p a i=2−p a X X bℓ b b N•−p−1 b + R∗ v γ dt R + R∗ v γ dt R . ˆ •,i,p •,i,p ˆ •,i,p •,i,p ′ a ◦ a ◦ i=max(2−p,ℓX−p+#◦t −1) i=Xℓ+1 b b Remark 2.2 and the choice (a•,j)j∈Z := (w•,j)j∈Z in Lemma 2.1 (vii) show the following: first, for 1 p i ℓ p + # t′ 2, it holds that B = B and w = w , whence − ≤ ≤ − ◦ − •,i,p ◦,i,p •,i ◦,i R•,i,p = R◦,i,p; second, for ℓ +1 i N• p, it holds that B•,i,p = B◦,i+1,p and w•,i = w◦,i+1, whence R = R . Moreover,≤ ≤ for 1 − p i ℓ bp + # tb′ 2, it holds that •,i,p ◦,i+1,p − ≤ ≤ − ◦ − b b R∗ = B∗( t ,...t )w/w = B∗( t ,...t )w/w = R∗ , •,i,p ·| •,i−1 •,i+p •,i ·| ◦,i−1 ◦,i+p ◦,i ◦,i,p and for ℓ +1 i N• p that b ≤ ≤b − b b b b R∗ = B∗( t ,...t )w/w = B∗( t ,...t )w/w = R∗ . •,i,p ·| •,i−1 •,i+p •,i ·| ◦,i ◦,i+1+p ◦,i+1 ◦,i+1,p 11 b b b b b b Hence, (3.11) simplifies to (J J )v = α (v)R α (v)R (3.12) ◦ − • ◦,1−p ◦,1−p,p − •,1−p •,1−p,p ℓ+1 b ℓ b + R∗ v γ dt R R∗ v γ dt R . ˆ ◦,i,p ◦,i,p ˆ •,i,p •,i,p ′ a ◦ − ′ a ◦ i=max(2−p,ℓX−p+#◦t −1) i=max(2−p,ℓX−p+#◦t −1) Remark 2.2 and Lemma 2.1 (ii)b imply that b R : i = max(2 p,ℓ p + # t′ 1, ) ...,ℓ +1 ◦,i,p − − ◦ − R : i = max(2 p,ℓ p + # t′ 1),...,ℓ (3.13) ∪ •,i,p − − ◦ − span R : i = max(2 p,ℓ p + # t′ 1),...,ℓ +1 span R : γ(t′) suppR ⊆ ◦,i,p − − ◦ − ⊆ ◦,i,p ∈ ◦,i,p ∗ ∗ We have already seen that R•,N•−p,p = R◦,N•−p+1,p and R = R if N• p •,N•−p,p ◦,N•−p+1,p − ≥ ℓ +1, and that R = R and R∗ = R∗ if 3 ℓ + # t′. This shows that the •,1−p,p ◦,1−p,p •,1−p,p ◦,1−p,p ≤ ◦ first summands in (3.12) cancel each other if N p ℓ+1 and 3 ℓ+# t′. Otherwise there • − ≥ ≤ ◦ holds that ℓ = 1b or ℓ = Nb p and theb functionsb R , R and R are in the • − ◦,1−p,p ◦,2−p,p ◦,N•−p,p last set of (3.13). Since R•,1−p,p is a linear combination of these functions, we conclude that (J J )v span R : γ(t′) suppR . ◦ − • ∈ ◦,i,p ∈ ◦,i,p Step 2: Let ◦ K be arbitrary with • ◦ and let ◦ = (M) , (M−1),..., (1), (0) = be a sequenceK ∈ of knot vectors suchK that≤K each isK obtainedK byK insertion ofK oneK single K• K(k) knot γ(t(k)) in (k−1). Note that these meshes do not necessarily belong to K, as the κ-mesh K property (2.13) can be violated. However, the corresponding Scott-Zhang operator J(k) for can be defined just as above and Step 1 holds analogously. There holds that K(k) b M (J J )v = (J J )v. (3.14) ◦ − • (k) − (k−1) Xk=1 This and Step 1 imply that

M (J J )v span R : (γ(t ) suppR . (3.15) ◦ − • ∈ (k),i,p (k) ∈ (k),i,p k=1 X Remark 2.2 shows that any basis function B(k),i,p with k < M is the linear combina-

tion of B(k+1),i,p and B(k+1),i+1,p. Moreover, Lemma 2.1 (ii) shows that suppB(k+1),i,p b ∪ suppB(k+1),i+1,p suppB(k),i,p. We conclude that b ⊆ b b R span R : suppR suppR . b (k),i,pb∈ (k+1),j,p (k+1),j,p ⊆ (k),i,p Together with (3.15), this shows that

M (J J )v span R : γ(t ) suppR ◦ − • ∈ (M),i,p (k) ∈ (M),i,p k=1 X = span R : suppR = , ◦,i,p ◦,i,p ∩ N◦\• 6 ∅ and concludes the proof. 12 e The following proposition is derived in [GPS18]. For closed Γ = ∂Ω, the proof is also found in [Sch16, Lemma 3.1].

Proposition 3.2. For • K, the corresponding Scott-Zhang operator J• satisﬁes the following properties: K ∈ (i) Local projection property: For all v L2(Γ) and all T it holds that ∈ ∈ T• (J v) = v if v p p := V p : V . (3.16) • |T |T |ω•(T ) ∈ X•|ω•(T ) •|ω•(T ) • ∈ X• (ii) Local L2-stability: For all v L2(Γ) and all T , it holds that ∈ ∈ T• J v 2 C v 2 p . (3.17) k • kL (T ) ≤ szk kL (ω• (T )) (iii) Local H1-stability: For all v H1(Γ) and all T , it holds that ∈ ∈ T• J•v H1(T ) Csz v H1(ωp(T )). (3.18) e | |e ≤ | | • (iv) Local approximation property: For all v H1(Γ) and all T , it holds that ∈ ∈ T• −1 h• (1 J•)v L2(T ) Csz v H1(ωp(T )). (3.19) k − k ≤e | | • The constant Csz > 0 depends only on κmax,p,wmin,wmax, and γ. 3.2. Inverse inequalities. In this section, we state some inverse estimates for NURBS from [GPS18], which are well-known forb piecewise polynomials [GHS05, AFF+15].

Proposition 3.3. Let • K and 0 σ 1. Then, there hold the inverse inequalities K ∈ ≤ ≤ −σ V• σ Cinv h• V• L2(Γ) for all V• •, (3.20) k kHe (Γ) ≤ k k ∈ X and 1−σ h• ∂ΓV• L2(Γ) Cinv V• σ for all V• •. (3.21) k k ≤ k kHe (Γ) ∈ X The constant Cinv > 0 depends only on κmax,p,wmin,wmax, γ, and σ.

3.3. Uniform meshes. We consider a sequence uni(m) K of uniform knot vectors: b K ∈ Let uni(0) := 0 and let uni(m+1) be obtained from uni(m) by uniform reﬁnement, i.e., all elementsK of K are bisectedK in the parameter domainK into son elements with half length, Tuni(m) where each new knot has multiplicity one. Deﬁne h := max γ−1(T ) as well as uni(0) T ∈T0 | | h := 2−mh for each m 1. (3.22) uni(m) uni(0) ≥ Note that h is equivalent to the usual local mesh-size function on , i.e., h uni(m) Tuni(m) uni(m) ≃ T for all T uni(m) and all m 0, where the hidden constants depend only on 0 and γ. | | ∈ T ≥ 2 T Moreover, let uni(m) denote the associated discrete space with corresponding L -orthogonal projection Π X : L2(Γ) . Note that the discrete spaces are nested, i.e., uni(m) → Xuni(m) Xuni(m) uni(m) uni(m+1) for all m 0. X The next⊆ X result follows by≥ the approximation property of Proposition 3.2 (iv) and the inverse inequality (3.20) of Proposition 3.3 in combination with [Bor94]. Lemma 3.4. Let 0 <σ< 1. Then, ∞ −2σ 2 2 σ huni(m) (1 Πuni(m))v L2(Γ) Cnorm v Hσ(Γ) for all v H (Γ), (3.23) k − k ≤ k k e ∈ m=0 X where the constant Cnorm > 0 depends only on 0, κmax,p,wmin,wmax, γe, and σ. 13T b 2 Proof. The L -best approximation property of Π yields that v Π v 2 uni(m) k − uni(m) kL (Γ) ≤ v J v 2 . With Proposition 3.2, we have that k − uni(m) kL (Γ) (3.19) −m 1 v Πuni(m)v L2(Γ) . huni(m) v H1(Γ) = huni(0)2 v H1(Γ) for all v H (Γ). (3.24) k − k k k e k k e ∈ The approximation property (3.24) (also called Jackson inequality) together with the inverse inequality (3.20) (also called Bernstein inequality) from Proposition 3.3e allow to apply [Bor94, Theorem 1 and Corollary 1] with X = H1(Γ) and α = 1. The latter proves that ∞ 2 −2σ 2 −2σ 2 σ v σ huni(0) v L2(Γ) + huni(m) (1 Πeuni(m))v L2(Γ) for all v H (Γ). k kHe (Γ) ≃ k k k − k ∈ m=0 X This concludes the proof. e

3.4. Level function. Let • K. For given T •, let T0 0 denote its unique ancestor K ∈ ∈ T ∈ T −1 −1 such that T T0 and define with the corresponding elements T = γ (T ), T0 = γ (T0) in the parameter⊆ domain the generation of T by b b log( T / T0 ) gen(T ) := | | | | N0, log(1/2) ∈ b b i.e., gen(T ) denotes the number of bisections of T0 0 in the parameter domain needed to obtain the element T . To each node z , we∈ T associate ∈ T• ∈N• level (z) := max gen(T ): T and z T for all z . (3.25) • ∈ T• ∈ ∈N• The function level•( ) provides a link between the mesh • and the sequence of uniformly refined meshes · . A simple proof of the following resultT is found in [Füh14, Lemma 6.11]. Tuni(m) Lemma 3.5. Let • K and z • and m := level•(z). Then, it holds that z uni(m) and K ∈ ∈N ∈N C−1 h T C h for all T with z T. (3.26) level uni(m) ≤| | ≤ level uni(m) ∈ T• ∈ The constant C > 0 depends only on , κ , and γ. level T0 max 4. Local multilevel diagonal preconditioner for the hypersingular case b

Throughout this section, let ( ℓ)ℓ∈N0 be a sequence of refined knot vectors, i.e., ℓ, ℓ+1 refine K K K ∈ K with ℓ+1 ( ℓ), and let L N0. We set K ∈ K ∈ := and ω ( ) := ω ( ). (4.1) N0\−1 N0 −1 · 0 · For ℓ N0, abbreviate the corresponding index set from (3.9) ∈ e := i 1 p + o,...,N p 1 : suppR = , (4.2) Iℓ ∈{ − ℓ − − } ℓ,i,p ∩ Nℓ\ℓ−1 6 ∅ and define the spaces e e := span R : i = with := span R . (4.3) Xℓ ℓ,i,p ∈ Iℓ Xℓ,i Xℓ,i { ℓ,i,p} i∈I Xeℓ e e Note that 0 = 1 p + o,...,N0 p 1 and 0 = 0. For all ℓ N0 and i ℓ, fix a node I { − − − } X X ∈ ∈ I e z with z e suppR . e (4.4) ℓ,i ∈ Nℓ\ℓ−1 ℓ,i ∈ ℓ,i,p 14 e e e For V and ℓ =0, 1,...,L, we define (see Lemma 3.1) L ∈ XL V ℓ := (J J )V , where J := 0. (4.5) L ℓ − ℓ−1 L ∈ Xℓ −1 For all i , we set with the abbreviation α (V ℓ) from (3.8) ∈ Iℓ e ℓ,ie L ℓ,i ℓ (4.6) e VL := αℓ,i(VLe) Rℓ,i,p. By the duality property (3.6) and the decomposition (4.3), we have the decompositions e L L ℓ ℓ,i ℓ ℓ,i VL = VL and VL = VL = VL (4.7) ℓ=0 ℓ=0 Xi∈Ieℓ X X Xi∈Ieℓ e e and hence L = . (4.8) XL Xℓ,i ℓ=0 X Xi∈Ieℓ With the one-dimensional , W-orthogonal projections onto defined by hh· ·ii Pℓ,i Xℓ,i 1/2 u,V W = u, V W for all u H (Γ),V , (4.9) hhPℓ,i ℓ,iii hh ℓ,iii ∈ ℓ,i ∈ Xℓ,i the space decomposition (4.8) gives rise to the additive Schwarz operator e L W = . (4.10) PL Pℓ,i ℓ=0 i∈I X Xeℓ e A similar operator for continuous piecewise affine ansatz functions on affine geometries has been investigated in [FFPS17]. Indeed, the proof of the following main result (for the hypersingular case) is essentially inspired by the corresponding proof of [FFPS17, Theorem 1].

Theorem 4.1. The additive Schwarz operator W : H1/2(Γ) satisiﬁes that PL → XL W 2 W W 2 λ V W V , V W λ V W for all V , (4.11) min|| L|| ≤ hhPL L Lii ≤ emax|| eL|| L ∈ XL where the constants λW ,λW > 0 depend only on , κ ,p,w ,w , and γ. min max e T0 max min max We split the proof into two parts. In Section 4.1, we show the lower bound. The upper bound is proved in Section 4.2. b 4.1. Proof of Theorem 4.1 (lower bound). In the remainder of this section, we will show that the decomposition (4.7) of VL is stable, i.e.,

L ℓ,i 2 2 V W . V W. (4.12) || L || || L|| ℓ=0 X Xi∈Ieℓ It is well known from additive Schwarz theory [Lio88, Wid89, Zha92, TW05] that this proves the lower bound in Theorem 4.1; see, e.g., [Zha92, Lemma 3.1]. We start with two auxiliary lemmas. In the following, we set uni(m) := uni(0) if m< 0. X X15 Lemma 4.2. Let ℓ N and q N. There exists a constant C1(q) N0 such that for all z with m = level∈ (z), it holds∈ that ∈ ∈Nℓ ℓ V ωq (z) : V uni(m−C (q)) V ωq (z) : V ℓ−1 (4.13) | ℓ−1 ∈ X 1 ⊆ | ℓ−1 ∈ X The constant C1(q) depends only on κmax,γ and q. We abbreviate C1 := C1(2p + 1). Proof. We show that b ωq (z) ωq (z). (4.14) Nuni(m−C1(q)) ∩ ℓ−1 ⊆Nℓ−1 ∩ ℓ−1 Let τ ℓ such that z τ. Let T ℓ−1 be the father element of τ, i.e., τ T . We note that gen(∈ TT ) = gen(τ) or∈ gen(T ) = gen(∈ Tτ) + 1 and hence ⊆ gen(τ) gen(T ) 1. | − | ≤ Moreover, there exists a constant C N, which depends only on κmax,γ and q such that ∈ ′ ′ ′ q gen(T ) gen(T ) C for all T ℓ−1 with T ωℓ−1(z), | − | ≤ ∈ T b⊆ i.e., the diﬀerence in the element generations within some q-th order patch is uniformly bounded. This implies that gen(τ) gen(T ′)+ C + 1 for all T ′ with T ′ ωq (z). ≤ ∈ Tℓ−1 ⊆ ℓ−1 By deﬁnition of levelℓ(z), we thus infer that C1(q) := C +1 > 0 yields that m = level (z) min gen(T ′): T ′ and T ′ ωq (z) + C (q). (4.15) ℓ ≤ ∈ Tℓ−1 ⊆ ℓ−1 1

For m C1(q) 0, we have that uni(m−C1(q)) = 0, and the assertion is clear. Therefore, we − ≤ X ′ X ′ q suppose that m C1(q) 1. Let T ℓ−1 with T ωℓ−1(z). According to (4.15), it holds − ′ ≥ ∈ T ⊆ ′ that m C1(q) gen(T ). Therefore, there exists a father element Q uni(m−C1(q)) with T − ≤ ′ ∈ T q ⊆ Q. Suppose that (4.14) does not hold true. Then there is some z uni(m−C1(q)) ωℓ−1(z), q ′ ∈N ∩ ′ which is not contained in ℓ−1 ωℓ−1(z). Therefore, z is in the interior of some T ℓ−1 with T ′ ωq (z) and henceN also∩ in the interior of the father Q of T ′∈. This T ⊆ ℓ−1 ∈ Tuni(m−C1(q)) contradicts z uni(m−C1(q)) and concludes the proof of (4.14). ∈N ′ By the deﬁnition of uni(m−C1(q)), we even have for the multiplicities that #uni(m−C1(q))z ′ ′ X q ≤ #ℓ−1z for z uni(m−C1(q)) ωℓ−1(z). With Lemma 2.1 (i) and the fact that the denominator w in (2.28) is∈N ﬁxed, this proves∩ the assertion.

Lemma 4.3. For each m N0 and z L, it holds that m(z) C2, where ∈ ∈N |Z | ≤ (z) := (ℓ, i): ℓ 0,...,L , i , level (z )= m, z = z . (4.16) Zm ∈{ } ∈ Iℓ ℓ ℓ,i ℓ,i The constant C2 > 0 depends only on p. e e e Proof. As we only use bisection or knot multiplicity increase (with maximal multiplicity

p), it holds that ℓ 1,...,L : z ℓ\ℓ−1 p. This shows that only a bounded number of diﬀerent| ℓ∈appears { in} the set∈ N of (4.16).| ≤ For ﬁxed ℓ 0,...,L , (4.4) and ∈ { } ωp+1(z) ω2(p+1)(z) yield that e ℓ−1 ⊆ ℓ (4.4) i : level (z )= m, z = z i : supp(R ) ωp+1(z) ∈ Iℓ ℓ ℓ,i ℓ,i ⊆ ℓ,i,p ⊆ ℓ−1 2(p+1) i : supp(Rℓ,i,p) ωℓ ( z) . e e e ⊆ ⊆ The cardinality of the last set is bounded by a constant C2 > 0 that depends only on p. 16 Proof of lower bound in (4.11). The proof is split into two steps. Step 1: We show (4.19). The norm equivalence W H1/2(Γ), the inverse inequality −1/2 || · || ≃k·k ℓ,i (3.20) for NURBS and h R 2 . 1 prove for the functions V of (4.6) that k ℓ ℓ,i,pkL (Γ) L ℓ,i 2 2 V W . α (J J )V . || L || ℓ,i ℓ − ℓ−1 L The Cauchy-Schwarz inequality, and the property (3.7) of the dual basis functions imply that

ℓ,i 2 −1 2 V W . supp(Rℓ,i,p) (Jℓ Jℓ−1)VL 2 . (4.17) || L || | | k − kL (supp(Rℓ,i,p))

We abbreviate m = levelℓ(zℓ,i). Proposition 3.2 (i) and Lemma 4.2 together with nestedness imply for almost every x ωp+1(z ) that Xℓ−1 ⊆ Xℓ ∈ ℓ−1 ℓ,i e (JℓΠuni(m−C1)VL)(x)=(Πuni(m−C1 )VL)(x)=(Jℓ−1Πuni(m−C1)VL)(x). e 2 This together with (4.4) and local L -stability of Jℓ and Jℓ−1 (Proposition 3.2 (ii)) shows that 2 2 (Jℓ Jℓ−1)VL 2 = (Jℓ Jℓ−1)(1 Πuni(m−C ))VL 2 k − kL (supp(Rℓ,i,p)) k − − 1 kL (supp(Rℓ,i,p)) 2 (Jℓ Jℓ−1)(1 Πuni(m−C1))VL 2 p+1 L (ω (zℓ,i)) (4.18) ≤k − − k ℓ−1 e 2 . (1 Πuni(m−C1))VL 2 2p+1 . L (ω (zℓ,i)) k − k ℓ−1 e Further, Lemma 3.5 shows that h supp(R ) . Hence, (4.17) and (4.18) prove that uni(m) ≃| ℓ,i,p | ℓ,i 2 −1 2 V W . h (1 Πuni(m−C1))VL 2 2p+1 . (4.19) L uni(m) L (ω (zℓ,i)) || || k − k ℓ−1 e

Step 2: We show (4.12), which concludes the lower bound in (4.11). Step 1 gives that

L ∞ L ℓ,i 2 ℓ,i 2 V W = V W || L || || L || ℓ=0 m=0 ℓ=0 X Xi∈Ieℓ X X Xi∈Ieℓ levelℓ(zℓ,i)=m e ∞ L −1 2 . 2p+1 huni(m) (1 Πuni(m−C1))VL L2(ω (z )). k − k ℓ−1 ℓ,i m=0 ℓ=0 e X X Xi∈Ieℓ levelℓ(zℓ,i)=m e There exists a constant C3 N, which depends only on p, κmax,γ, and 0, such that for z with level (z)= m, it∈ holds that T ∈Nℓ ℓ ω2p+1(z) ωC3 (z). b (4.20) ℓ−1 ⊆ uni(m) Hence,

L ∞ L i,ℓ 2 −1 2 V W . h (1 Π )V C L uni(m) uni(m−C1) L L2(ω 3 (z )) || || k − k uni(m) ℓ,i ℓ=0 m=0 ℓ=0 e X Xi∈Ieℓ X X Xi∈Ieℓ levelℓ(zℓ,i)=m e ∞ (4.16) −1 2 = h (1 Π )V C . uni(m) uni(m−C1) L L2(ω 3 (z)) k − k uni(m) m=0 z∈N X XL (ℓ,i)X∈Zm(z) 17 If z L and (ℓ, i) m(z), it follows that z ℓ with levelℓ(z) = m by deﬁnition. Lemma∈ N 3.5 implies that∈z Z . This and Lemma∈ N 4.3 give that ∈Nuni(m) ∞ −1 2 h (1 Π )V C uni(m) uni(m−C1) L L2(ω 3 (z)) k − k uni(m) m=0 X zX∈NL (ℓ,i)X∈Zm(z) ∞ −1 2 = h (1 Π )V C uni(m) uni(m−C1) L L2(ω 3 (z)) k − k uni(m) m=0 z∈N ∩N X LXuni(m) (ℓ,i)X∈Zm(z) (4.16) ∞ −1 2 . h (1 Π )V C uni(m) uni(m−C1) L L2(ω 3 (z)) k − k uni(m) m=0 z∈N ∩N X LXuni(m) ∞ −1 2 h (1 Π )V C uni(m) uni(m−C1) L L2(ω 3 (z)) ≤ k − k uni(m) m=0 z∈N X Xuni(m) ∞ −1 2 . h (1 Π )V 2 uni(m)k − uni(m−C1) LkL (Γ) m=0 X The deﬁnition Πuni(m) = Πuni(0) for m< 0 yields that ∞ ∞ −1 2 −1 2 h (1 Π )V 2 . h (1 Π )V 2 uni(m)k − uni(m−C1) LkL (Γ) uni(m)k − uni(m) LkL (Γ) m=0 m=0 X X Combining the latter three estimates, Lemma 3.4 leads us to

L ∞ (3.23) ℓ,i 2 −1 2 2 2 V W . h (1 Π )V 2 . V 1/2 V W. || L || uni(m)k − uni(m) LkL (Γ) k LkH (Γ) ≃ || L|| ℓ=0 i∈I m=0 X Xeℓ X This proves (4.12) and yields the lower bound in (4.11).

4.2. Proof of Theorem 4.1 (upper bound). For m N0, let uni(m,p) K be the knot ∈ K ∈ vector with uni(m,p) = uni(m) and #z = p for all z uni(m,p) γ(a),γ(b) . By Lemma 3.5, it holds thatT T , where ∈N \{ } NL ⊆Nuni(M,p) M := max levelL(z). (4.21) z∈NL

The deﬁnition of uni(m,p) yields that L uni(M,p). Moreover, we can rewrite the additive Schwarz operatorX as X ⊆ X M L W = with := . (4.22) PL Qm Qm Pℓ,i m=0 ℓ=0 i∈I X X Xeℓ levelℓ(zℓ,i)=m e e There holds the following type of strengthened Cauchy-Schwarz inequality.

Lemma 4.4. For all 0 m M, m( ) , ( ) W defines a symmetric positive semi-definite 1/2 ≤ ≤ hhQ · · ii bilinear form on H (Γ). For k N0, it holds that ∈ −(m−k) 2 mVuni(k,p) , Vuni(k,p) W C42 Vuni(k,p) W for all Vuni(k,p) uni(k,p). (4.23) hhQ e ii ≤ || || ∈ X The constant C4 > 0 depends only on 0, κmax,p,wmin,wmax and γ. T 18 b Proof. Symmetry and positive semi-definiteness follow by the symmetry and positive semi- definiteness of the one-dimensional projectors . To see (4.23), we only consider closed Pℓ,i Γ= ∂Ω and split the proof into two steps. For open Γ $ ∂Ω the proof works analogously. Step 1: Let ℓ 0,...,L and i with level (z ) = m. We want to estimate ∈ { } ∈ Iℓ ℓ ℓ,i V , V W. From the definition (4.9) of , we infer that hhPℓ,i uni(k,p) uni(k,p)ii Pℓ,i e e 2 Vuni(k,p) , Rℓ,i,p W V , V W = hh ii . (4.24) ℓ,i uni(k,p) uni(k,p) 2 hhP ii R W || ℓ,i,p||

Lipschitz continuity of γ gives that Rℓ,i,p 1/2 . Rℓ,i,p H1/2(supp(R )) . Rℓ,i,p H1/2(Γ)). | |H (supp(Rbℓ,i,p)) | | ℓ,i,p | | Hence, Lemma 2.3 with σ =1/2 shows that 1 . Rℓ,i,p H1/2(Γ). This implies that b | | 1/2 1/2 1/2 R 2 . supp(R ) . supp(R ) R 1/2 . supp(R ) R W. k ℓ,i,pkL (Γ) | ℓ,i,p | | ℓ,i,p | | ℓ,i,p|H (Γ) | ℓ,i,p | || ℓ,i,p|| With the Cauchy-Schwarz inequality and suppR h (Lemma 3.5), this gives that | ℓ,i,p| ≃ uni(m) W 2 2 2 Vuni(k,p) , Rℓ,i,p Γ Vuni(k,p) , 1 Γ Rℓ,i,p , 1 Γ V , V W . h i + h i h i ℓ,i uni(k,p) uni(k,p) 2 2 hhP ii R W R W || ℓ,i,p|| || ℓ,i,p|| W 2 2 . supp(Rℓ,i,p) Vuni(k,p) 2 + supp(Rℓ,i,p) Vuni(k,p) 2 L (supp(Rℓ,i,p)) L (Γ) | | k k | |k k (4.25) h 1/2 uni(m) W 2 2 . h Vuni(k,p) 2 + supp(Rℓ,i,p) Vuni(k,p) 2 uni(k) L (supp(Rℓ,i,p)) L (Γ) huni(k) k k | |k k 1/2 −(m−k) W 2 2 =2 huni(k) Vuni(k,p) 2 + supp(Rℓ,i,p) Vuni(k,p) L2(Γ) . k kL (supp(Rℓ,i,p)) | |k k Step 2: We stress that the choice (4.4) of zℓ,i and (4.20) show that

(4.4) (4.20) supp(R ) ωp+1(z )e ω2p+1(z ) ωC3 (z ). ℓ,i,p ⊆ ℓ−1 ℓ,i ⊆ ℓ−1 ℓ,i ⊆ uni(m) ℓ,i Thus, the deﬁnition of and Step 1 yield that Qm e e e L V , V W = V , V W hhQm uni(k,p) uni(k,p)ii hhPℓ,i uni(k,p) uni(k,p)ii ℓ=0 X Xi∈Ieℓ levelℓ(zℓ,i)=m e (4.25) L −(m−k) 1/2 2 C3 2 . 2 h WV C + ω (z ) V 2 uni(k) uni(k,p) L2(ω 3 (z )) uni(m) ℓ,i uni(k,p) L (Γ) k k uni(m) ℓ,i | |k k ℓ=0 e X Xi∈Ieℓ levelℓ(zℓ,i)=m e e (4.16) −(m−k) 1/2 2 C3 2 = 2 h WV C + ω (z) V 2 . uni(k) uni(k,p) L2(ω 3 (z)) uni(m) uni(k,p) L (Γ) k k uni(m) | |k k zX∈NL (ℓ,i)X∈Zm(z) If z and (ℓ, i) (z), it follows z with level (z)= m. Lemma 3.5 implies that ∈NL ∈ Zm ∈Nℓ ℓ z uni(m). Hence, we can replace in the upper sum L by L uni(m). With Lemma 4.3, ∈N 19 N N ∩N we further see that

1/2 2 C3 2 h WV C + ω (z) V 2 uni(k) uni(k,p) L2(ω 3 (z)) uni(m) uni(k,p) L (Γ) k k uni(m) | |k k z∈NLX∩Nuni(m) (ℓ,i)X∈Zm(z) 1/2 2 C3 2 . h WV C + ω (z) V 2 uni(k) uni(k,p) L2(ω 3 (z)) uni(m) uni(k,p) L (Γ) k k uni(m) | |k k z∈N Xuni(m) 1/2 2 2 . h WV 2 + V 2 . k uni(k) uni(k,p)kL (Γ) k uni(k,p)kL (Γ) Note that boundedness W : H1(Γ) L2(Γ) as well as the inverse inequality (3.21) prove that → 1/2 2 2 2 h WV 2 . h V 1 . V 1/2 . k uni(k) uni(k,p)kL (Γ) uni(k)k uni(k,p)kH (Γ) k uni(k,p)kH (Γ) Putting the latter three inequalities together shows that −(m−k) 2 2 −(m−k) 2 V , V W . 2 V 1/2 + V 2 2 V W. hhQm uni(k,p) uni(k,p)ii k uni(k,p)kH (Γ) k uni(k,p)kL (Γ) ≃ || uni(k,p)|| This ﬁnishes the proof. The rest of the proof of the upper bound in Theorem 4.1 follows essentially as in [TS96, Lemma 2.8] and is only given for completeness; see also [FFPS17, Section 4.6].

1/2 Proof of upper bound in (4.11). For k N0 let uni(k,p) : H (Γ) uni(k,p) denote the ∈ G → X Galerkin projection onto with respect to the scalar product , W, i.e., Xuni(k,p) hh· ·ii 1/2e v, V W = v, V W for all v H (Γ),V . (4.26) hhGuni(k,p) uni(k,p)ii hh uni(k,p)ii ∈ uni(k,p) ∈ Xuni(k,p) Note that uni(k,p) is the orthogonal projection onto uni(k,p) with respect to the energy norm G X e W. Moreover, we set uni(−1,p) := 0. The proof is split into three steps. ||Step · || 1: Let V G . Lemma 3.5 and the boundedness of the local mesh-ratio L ∈ XL ⊆ Xuni(M,p) by κmax yield the existence of a constant C N0, which depends only on 0, κmax,γ, and p, such that ωp+2(z) for all nodes∈ z with level (z)= mT . Lemma 2.1 (i) Nℓ ∩ ℓ−1 ⊆Nuni(m+C,p) ∈Nℓ ℓ henceb proves that ℓ,i uni(m+C,p) for all m 0,...,M ,ℓ 0,...,L , andb i ℓ with level (z )= m. Therefore,X ⊆ X the range of is∈{ a subspace} of ∈{ . This} shows∈ I that ℓ i,ℓ Qm Xuni(m+C,p) e mVL , VL W = mVL , uni(m+C,p)VL W. (4.27) e hhQ ii hhQ G ii

Step 2: In Lemma 4.4, we saw that m( ) , ( ) W defines a symmetric positive semi-definite bilinear form and hence satisfies a Cauchy-SchwarzhhQ · · ii inequality. This and (4.27) yield that

m+C V , V W = V , V W = V , ( )V W hhQm L Lii hhQm L Guni(m+C,p) Lii hhQm L Guni(k,p) − Guni(k−1,p) Lii Xk=0 m+C 1/2 1/2 V , V W ( )V , ( )V W . ≤ hhQm L Lii hhQm Guni(k,p) − Guni(k−1,p) L Guni(k,p) − Guni(k−1,p) Lii Xk=0 For the second scalar product, we apply Lemma 4.4 and obtain that

( )V , ( )V W hhQm Guni(k,p) − Guni(k−1,p) L Guni(k,p) − Guni(k−1,p) Lii −(m−k) 2 −(m−k) 2 . 2 ( )V W =2 ( ) V , V W. || Guni(k,p) − Guni(k−1,p) L|| hh Guni(k,p) − Guni(k−1,p) L Lii 20 2 Note that ( uni(k,p) uni(k−1,p)) = uni(k,p) uni(k−1,p), since uni(k,p) uni(k−1,p) is again an orthogonalG projection.− G G − G G − G W Step 3: With the representation (4.22) of L , the two inequalities from Step 2, and the Young inequality, we infer for all δ > 0 that P

M e W (4.22) V , V W = V , V W hhPL L Lii hhQm L Lii m=0 X e M m+C −1 M m+C δ −(m−k)/2 δ −(m−k)/2 . 2 V , V W + 2 ( )V , V W. 2 hhQm L Lii 2 hh Guni(k,p) − Guni(k−1,p) L Lii m=0 m=0 X Xk=0 X Xk=0 We abbreviate ∞ 2−k/2 =: K < . Changing the summation indices in the second k=−C ∞ sum, we see with VL L uni(M,p) uni(M+C,p) that P ∈ X ⊆ X ⊆ X M W δ V , V W . K V , V W hhPL L Lii 2 hhQm L Lii m=0 X e −1 M+C M δ −(m−k)/2 + 2 ( )V , V W 2 hh Guni(k,p) − Guni(k−1,p) L Lii Xk=0 m=max(Xk−C,0) δ M δ−1 M+C K V , V W + K ( )V , V W ≤ 2 hhQm L Lii 2 hh Guni(k,p) − Guni(k−1,p) L Lii m=0 X Xk=0 δ M δ−1 = K V , V W + K V , V W 2 hhQm L Lii 2 hhGuni(M+C,p) L Lii m=0 X −1 (4.22) δ W δ = K V , V W + K V , V W. 2hhPL L Lii 2 hh L Lii Choosing δ > 0 suﬃciently smalle and absorbing the ﬁrst-term on the right-hand side on the left, we prove the upper bound in (4.11).

5. Local multilevel diagonal preconditioner for the weakly-singular case Finally, we generalize the results of the previous sections to the weakly-singular integral equation. The main tool in the following is Maue’s formula (see, e.g. [AEF+14]) Wu , v = V∂ u , ∂ v for all u, v H1(Γ). (5.1) h iΓ h Γ Γ iΓ ∈ For similar proofs in the case of piecewise constant ansatz functions, we refer to [TS96] e (uniform meshes) resp. [FFPS15] (adaptive meshes). Throughout this section, let ( ℓ)ℓ∈N0 refine K be a sequence of refined knot vectors, i.e., ℓ, ℓ+1 K with ℓ+1 ( ℓ), and let K K ∈ K ∈ K L N0. For each ΨL L, we consider the unique decomposition ∈ ∈Y Ψ =Ψ00 +Ψ0 , where Ψ00 := Ψ , 1 / Γ and Ψ0 := Ψ Ψ00. (5.2) L L L L h L iΓ | | L L − L Note that Ψ00 00 := span 1 and Ψ0 0 := Ψ : Ψ , 1 =0 . (5.3) L ∈Y { } L ∈YL L ∈YL h L iΓ 21 With hidden constants, which depend only on Γ, it holds that 00 0 Ψ V . Ψ V and Ψ V . Ψ V. (5.4) k L k k Lk k Lk k Lk Recall the spaces , and from Section 4. For ℓ 0,...,L , set XL Xℓ Xℓ,i ∈{ } (4.3) 0 := ∂ = 0 with 0 := ∂ . (5.5) eYℓ ΓXℓ Yℓ,i Yℓ,i ΓXℓ,i i∈I Xeℓ Recall that we only considere non-rationale splines in the weakly-singular case; see Section 2.9. For rational splines, the following lemma is in general false. Lemma 5.1. It holds that 0 = ∂ . (5.6) YL ΓXL For closed Γ= ∂Ω, we even have that 0 = ∂ 0, where 0 := V : V , 1 =0 . (5.7) YL ΓXL XL L ∈ XL h L iΓ Proof. Since Γ is connected, the kernel of ∂Γ() is one-dimensional. Linear algebra yields · 0 0 that NL 1 = dim L = dim L, NL 2 = dim ∂Γ L = dim L = dim L for Γ = ∂Ω resp. − X Y − X 0 X Y NL 1 = dim L, NL 2 = dim L = dim ∂Γ L = dim L for Γ $ ∂Ω. It thus remains to − Y 0 − X X Y prove that ∂Γ L L. The inclusion ∂Γ L L follows directly from Lemma 2.1 (vi). We X ⊆Y 1/2 X ⊆Y stress that for any v H (Γ), it holds that ∂Γv , 1 Γ = 0. Thus, any function in ∂Γ L has vanishing integral mean,∈ which concludes theh proof. i X e Define the orthogonal projections on 00 resp. via Y Yℓ,i 00 00 00 −1/2 00 00 χ , Ψ V = χ , Ψ V for all χ H (Γ), Ψ , hhP ii hh ii ∈ ∈Y (5.8) 0 0 0 −1/2 0 0 χ , Ψ V = χ , Ψ V for all χ H (Γ), Ψ . hhPℓ,i ℓ,iii hh ℓ,iii ∈ e ℓ,i ∈Yℓ,i With Lemma 5.1, we see the decomposition e L L (5.6) (4.8) (5.5) = 00 + 0 = 00 + ∂ = 00 + 0 = 00 + 0 . (5.9) YL Y YL Y ΓXL Y Yℓ Y Yℓ,i ℓ=0 ℓ=0 X X Xi∈Ieℓ with the corresponding additive Schwarz operator e L V := 00 + 0 . (5.10) PL P Pℓ,i ℓ=0 X Xi∈Ieℓ e Theorem 5.2. The additive Schwarz operator V : H−1/2(Γ) satisfies that PL →YL V 2 V V 2 λ Ψ V Ψ , Ψ V λ Ψ V for all Ψ , (5.11) min|| L|| ≤ hhPL L Lii ≤ emax|| eL|| L ∈YL where the constants λV ,λV > 0 depend only on , κ ,p,w ,w , and γ. min max e T0 max min max Proof. We only prove the assertion for closed Γ = ∂Ω. For open Γ $ ∂Ω, the proof works 2 b analogously with W = W( ) , ( ) Γ. Step 1: First, wek·k prove theh lower· · boundi of (5.11). We have to find a stable decomposition 0 0 0 0 for any ΨL L. Due to Lemma 5.1, there exists VL L with ∂ΓVL =ΨL. In Section 4.1, ∈Y 22 ∈ X 0 L L 2 0 2 we provided a decomposition V = V such that V W . V W. L ℓ=0 i∈Iℓ ℓ,i ℓ=0 i∈Iℓ ℓ,i L This provides us with a decomposition e e k k k k P P P P L Ψ0 = ∂ V 0 = Ψ0 , where Ψ0 := ∂ V 0 . L Γ L ℓ,i ℓ,i Γ ℓ,i ∈Yℓ,i ℓ=0 X Xi∈Ieℓ Maue’s formula (5.1) and V 0 , 1 = 0 hence show that h L iΓ L L L 2 2 0 2 0 2 Ψ V = WV , V V W . V W = Ψ V. k ℓ,ik h ℓ,i ℓ,iiΓ ≤ k ℓ,ik k L k k Lk ℓ=0 ℓ=0 ℓ=0 X Xi∈Ieℓ X Xi∈Ieℓ X Xi∈Ieℓ With this and (5.4), we finally conclude that

L 00 2 0 2 2 Ψ V + Ψ V . Ψ V. k L k k ℓ,ik k Lk ℓ=0 i∈I X Xeℓ As in Section 4.1, this proves the lower bound. 0 00 Step 2: For the upper bound of (5.11), let ΨL = ΨL + ΨL L with an arbitrary L 0 0 0 0 ∈ Y 0 decomposition ℓ=0 Ψℓ,i = ΨL, where Ψℓ,i ℓ,i. In particular, it holds that Ψℓ,i = i∈Ieℓ ∈ Y αℓ,i∂ΓBℓ,i,p with some αℓ,i R. We deﬁne P P ∈ L

VL := Vℓ,i with Vℓ,i := αℓ,iBℓ,i,p. ℓ=0 i∈I X Xeℓ It is well known from additive Schwarz theory that the existence of a uniform upper bound 2 L 2 in Theorem 4.1 is equivalent to V W . V W for all decompositions V = L ℓ=0 i∈Iℓ ℓ,i L L k k e k k ℓ=0 Vℓ,i; see, e.g., [Zha92, Lemma 3.1]. Maue’s formula (5.1) yields that i∈Ieℓ P P P P L 0 2 2 2 Ψ V = WV , V V W . V W. (5.12) k Lk h L LiΓ ≤k Lk k ℓ,ik ℓ=0 i∈I X Xeℓ Lemma 2.3 shows that (2.25) 2 2 2 2 2 2 2 2 Vℓ,i , 1 Γ . αℓ,i suppVℓ,i . αℓ,i . αℓ,i Bℓ,i,p H1/2(suppB ) . αℓ,i Bℓ,i,p H1/2(a,b). (5.13) h i | | | | bℓ,i,p | | 2 2 2 Lipschitz continuity of γ proves Bℓ,i,p 1/2 b . Bℓ,i,p 1/2 . Bℓ,i,pb 1/2 . With the | |H (a,b) | |H (Γ) | |H (Γ) 2 W 1/2 equivalence H1/2(Γ) ( ) , ( ) Γ on H (Γ), (5.13) becomes |·| ≃h · · ib 2 2 2 V , 1 . α B 1/2 WV , V . h ℓ,i i ℓ,i| ℓ,i,p|H (Γ) ≃h ℓ,i ℓ,iiΓ 2 By definition of the norm W, we infer V W . WV , V . Hence, (5.12) yields that || · || || ℓ,i|| h ℓ,i ℓ,iiΓ (5.12) L L 0 2 (5.1) 2 Ψ V . WV , V = Ψ V. k Lk h ℓ,i ℓ,iiΓ k ℓ,ik ℓ=0 ℓ=0 X Xi∈Ieℓ X Xi∈Ieℓ 23 We conclude that L 2 00 2 0 2 00 2 2 Ψ V . Ψ V + Ψ V . Ψ V + Ψ V. k Lk k L k k Lk k L k k ℓ,ik ℓ=0 X Xi∈Ieℓ Since Ψ00+ L Ψ =Ψ was an arbitrary decomposition, standard additive Schwarz L ℓ=0 i∈Iℓ ℓ,i L theory proves the uppere bound. P P 6. Numerical experiments In this section, we present a matrix version of Theorem 4.1 and Theorem 5.2. We apply these theorems to define preconditioners for some numerical examples. Throughout this section, let ( ℓ)ℓ∈N0 be a sequence of refined knot vectors, i.e., ℓ, ℓ+1 K with ℓ+1 refine K K K ∈ K ∈ ( ℓ), and let L N0. For the hypersingular equation (1.1), we allow for arbitrary K ∈ positive initial weights 0. Whereas, whenever we consider the weakly-singular integral equation (1.3), we supposeW that all weights in are equal to one, wherefore the denominator W0 satisfies that w = 1. The Galerkin approximations Uℓ ℓ for the hypersingular case resp. Φ for the weakly-singular case satisfy that ∈ X ℓ ∈Yℓ U , V W = f,V resp. Φ , Ψ V = g , Ψ for all V , Ψ . (6.1) hh ℓ ℓii h ℓiΓ hh ℓ ℓii h ℓiΓ ℓ ∈ Xℓ ℓ ∈Yℓ The discrete solutions Uℓ, Φℓ are obtained by solving a linear system of equations

Wℓxℓ = fℓ resp. Vℓyℓ = gℓ, (6.2) where

Nℓ−1−o Nℓ−1

Uℓ = (xℓ)kRℓ,k−p+o,p resp. Φℓ = (yℓ)kBℓ,k−(p−1),p−1, (6.3) Xk=1 Xk=1 and b

Nℓ−1−o Nℓ−1−o Wℓ = Rℓ,k−p+o,p , Rℓ,j−p+o,p W , fℓ = f , Rℓ,j−p+o,p Γ , (6.4) hh ii j,k=1 h i j=1 resp. 

Nℓ−1 Nℓ−1 Vℓ = Bℓ,k−(p−1),p−1 , Bℓ,j−(p−1),p−1 V , gℓ = g , Bℓ,j−(p−1),p−1 Γ . (6.5) hh ii j,k=1 h i j=1 W −1 V −1 For any L b N0, we aimb to derive preconditioners (S ) resp.b (S ) for the Galerkin ∈ L L matrices WL resp. VL. For their definition, we first have to introduce the following trans- W W V formation matrices. For 0 ℓ L, let idℓ→L : ℓ e L, id :e ℓ ℓ and idℓ→L : ≤ ≤ X → X ℓe→ℓ X → X ℓ L be the canonical embeddings, i.e., the formal identities, with matrix represen- W W V Y → Y (NL−1−o)×(Nℓ−1−o) (Nℓ−1−o)×(#Iℓ) (NL−1)×(Nℓ−1) tations idℓ→L R , id R e and idℓ→eL R . ℓe→ℓ V∈ (N −1)×(#I ) ∈ ∈ Further, let id R ℓ eℓ be the matrix that represents the B-spline derivatives in ℓe→ℓ ∈ 0 as B-splines in , i.e., Yℓ Yℓ Nℓ−1 e V ∂ΓBℓ,i(k),p = (id )jkBℓ,j−(p−1),p−1 for k =1,..., # ℓ, (6.6) ℓe→ℓ I j=1 X b 24 e with the monotonuously increasing bijection i( ): 1,..., # . All these matrices can · { Iℓ}→ Iℓ be computed with the help of Lemma 2.1. Finally, let 1 R(NL−1)×(NL−1) be the constant ∈ one matrix, i.e., e e (1) =1 for j, k =1,...,N 1. (6.7) jk L − For any quadratic matrix A, we define the corresponding diagonal matrix diag(A)=(A jk · δjk)j,k. We consider L W −1 W W W T W −1 W T W T (SL ) := idℓ→Lid diag((id ) Wℓid ) (id ) (idℓ→L) , (6.8) ℓe→ℓ ℓe→ℓ ℓe→ℓ ℓe→ℓ Xℓ=0 resp. e L V −1 −1 V V V T V −1 V T V T (SL ) := 1 , 1 V 1 + idℓ→Lid diag((id ) Vℓid ) (id ) (idℓ→L) . (6.9) hh ii ℓe→ℓ ℓe→ℓ ℓe→ℓ ℓe→ℓ Xℓ=0 Notee that, by the partition of unity property from Lemma 2.1 (iv), there holds that

NL−1 1 , 1 V = (V ) . (6.10) hh ii L jk j,kX=1 Instead of solving WLxL = fL resp. VLyL = gL, we consider the preconditioned systems W −1 W −1 V −1 V −1 (SL ) WLxL =(SL ) fL resp. (SL ) VLyL =(SL ) gL. (6.11) W −1 V −1 Elementary manipulations verify that the preconditioned matrices (SL ) WL resp. (SL ) VL e e W e Ve are just the matrix representations of L XL : L L resp. L YL : L L. Theo- rem 4.1 resp. Theorem 5.2 then immediatelyP | proveX the→ Xnext corollary,Pe | whichY states→ Y e uniform boundedness of the condition number ofe the preconditioned systems.e For a symmetric and positive definite matrix A, we denote , A := A , 2, and by A the corresponding norm resp. induced matrix norm. Here, ,h· ·idenotesh the· ·i Euclidean innerk·k h· ·i2 product. The condition number condA of a quadratic matrix B of same dimension as A reads −1 condA(B) := B A B A. (6.12) k k k k W −1 V −1 Corollary 6.1. The matrices (SL ) , (SL ) are symmetric and positive definite with respect to , , and PW := (SW)−1W resp. PV := (SV)−1V are symmetric and positive definite h· ·i2 L L L L L L with respect to , SW resp. e, SV .e Moreover, the minimal and maximal eigenvalues of h· ·ieL h· ·ieL eW e V e e the matrices PL resp. PL satisfy that λW λ (PW) λ (PW) λW , (6.13) e e min ≤ min L ≤ max L ≤ max resp. e e λV λ (PV) λ (PV) λV , (6.14) min ≤ min L ≤ max L ≤ max W W V V with the constants λmin,λmax from Theorem 4.1 and λmin,λmax from Theorem 5.2. In partic- e e W V ular, the condition number of the additive Schwarz matrices PL resp. PL is bounded by W W W V V V condSW (PL ) λmax/λmin resp. condSV (PL ) λmax/λmin. (6.15) eL ≤ eL e ≤ e Recall that these eigenvalue bounds depend only on 0, κmax,p,wmin,wmax and γ. e 25 T e b Proof. We only consider the hypersingular case. The weakly-singular case can be treated W analoguously. Due to (4.8) the operator L XL is positive definite with respect to , W. This proves for any V with correspondingP | coefficient vector z that hh· ·ii L ∈ XL L W −1 W (S ) W z , W ez = V , V W > 0. h L L L L Li2 hhPL L Lii W Symmetry and positive definiteness of PL with respect to , SW follow immediately by e e eL h· ·i W symmetry and positive definiteness of WL. Theorem 4.1 and the fact that PL is just W e the matrix representation of L XL show (6.13). Finally, note that the condition number W P | W e condSW (PL ) is just the ratio of the maximal and the minimal eigenvalue of PL . eL e The corollary can be applied for iterative solution methods such as GMRES [SS86] or e e CG [Saa03] to solve (6.11). Here, the relative residual of the j-th residual depends only on W V the condition number condSW (PL ) resp. condSV (PL ). Hence, Corollary 6.1 proves that the eL eL W −1 V −1 iterative scheme together with the preconditioners (SL ) resp. (SL ) is optimal in the following sense: The number ofe iterations to reducee the relative residual under the tolerance ǫ> 0 is bounded by a constant, which depends only one 0, κmax,p,we min,wmax and γ. T W −1 V −1 Remark 6.2. The application of the preconditioners (SL ) resp. (SL ) on a vector zL can be done efficiently in (N ) operations. Furthermore,b the storage requirements of the O L preconditioners, i.e., the memory consumption of all thee tranformatione matrices id and the diagonal matrices diag( ) in the sum is (NL). This implies the optimal linear complexity of our preconditioners. A· detailed descriptionO of an algorithm, which implements the matrix- vector multiplication, can be found in our recent work [FFPS17, Algorithm 1] for some local multilevel preconditioner for the hypersingular integral operator on adaptively refined meshes, resp. in [Yse86] for some hierarchical basis preconditioner. In the following subsections, we numerically show the optimality of the proposed preconditioners. In all examples, the exact solution is known and singular, wherefore adaptive methods are preferable. To steer the mesh refinement, we apply the following adaptive Algorithm 6.3 proposed in [FGHP16, Algorithm 3.1] for the weakly-singuar case resp. in [Sch16, GPS18] for the hypersingular case. We will fix the used refinement indicators ηℓ(z) in each experiment separately. Algorithm 6.3. Input: Adaptivity parameter 0 < θ 1, polynomial order p N, initial ≤ ∈ knots 0, initial weights 0. AdaptiveK loop: For eachWℓ =0, 1, 2,... iterate the following steps (i)–(vi):

(i) Compute discrete approximation Uℓ ℓ in the hypersingular case resp. Φℓ ℓ in the weakly-singular case. ∈ X ∈ Y (ii) Compute reﬁnement indicators ηℓ(z) for all nodes z ℓ. (iii) Determine a minimal set of nodes such that∈N Mℓ ⊆Nℓ θ η2 η (z)2. (6.16) ℓ ≤ ℓ z∈M Xℓ (iv) If both nodes of an element T belong to , T will be marked. ∈ Tℓ Mℓ (v) For all other nodes z ℓ, the multiplicity will be increased if z satisﬁes that z a, b and # z

0.1 γ(1/6) 1

0.05 γ(1/3) 0.5

0 γ(1/2) γ(1) 0 γ(0) γ(1) γ(1/5) γ(2/5) γ(3/5) γ(4/5)

γ (2/3) −0.5 −0.05

−1 γ(5/6) −0.1

−1.5 −0.1 −0.05 0 0.05 0.1 −1.5 −1 −0.5 0 0.5 1 1.5

Figure 6.1. Geometries and initial nodes for the experiments from Section 6.

(vi) Refine all marked elements T ℓ by bisection (insertion of a node with multiplicity one) of the corresponding element∈ T γ−1(T ) in the parameter domain. Use further bisections to guarantee that the new knots satisfy that Kℓ+1 κ 2κ . (6.17) ℓ+1 ≤ 0 Output: Approximate solutions Uℓ resp. Φℓ and error estimators ηℓ for all ℓ N0. b b ∈ The resulting linear systems are solved by PCG. We compare the preconditioners to simple diagonal preconditioning. In all experiments the initial vector in the PCG-algorithm is set to 0 and the tolerance parameter ǫ> 0 for the relative residual is ǫ = 10−8. 6.1. Adaptive BEM for hypersingular integral equation for Neumann problem on pacman. We consider the boundary Γ = ∂Ω of the pacman geometry 1 π π Ω := r(cos(β), sin(β)):0 r < , β , (6.18) ≤ 10 ∈ −2τ 2τ with τ =4/7, sketched in Figure 6.1. It can be parametrized by a NURBS curve γ : [0, 1] Γ of degree two; see [FGHP16, Section 3.2]. With the 2D polar coordinates (r, β), the function→ P (x, y) := rτ cos(τβ) satisfies that ∆P = 0 and has a generic singularity at the origin. With the adjoint double- − ′ layer operator K , we define with the normal derivative ∂ν P f := (1/2 K′)∂ P. − ν Up to an additive constant, there holds u = P Γ, where u is the solution of the corresponding hypersingular integral equation. | For Algorithm 6.3, we choose NURBS of degree two as ansatz space (i.e., p = 2) and Xℓ the same initial knots 0 and weights 0 as for the geometry representation. Due to numerical stabilityK reasons,W we replace the right-hand side f in each step by f := (1/2 K′)φ . Here, φ is the L2(Γ)-orthogonal projection of φ := (∂ P ) onto the ℓ − ℓ ℓ ν space of transformed piecewise polynomials of degree p on ℓ, i.e., φℓ γ is polynomial 27 T ◦ on all γ−1(T ) with T . This leads to a perturbed Galerkin approximation U pert. To ∈ Tℓ ℓ steer the algorithm, we use the h h/2 based error indicators η (z)2 := h1/2∂ (U pert − ℓ k ℓ Γ fine(ℓ) − pert 2 1/2 2 pert U ) 2 + h (φ φℓ) 2 , where U is the perturbed Galerkin approximation ℓ kL (ωℓ(z)) k ℓ − kL (ωℓ(z)) fine(ℓ) in the space corresponding to the uniformly refined knots . Theseb are generated Xfine(ℓ) Kfine(ℓ) from ℓ via the refinementb steps Algorithm 6.3 (iv)–(vi) with ℓ = ℓ. InK Figure 6.2, we compare the condition numbers of diagonalM preconN ditioning with our proposed additive Schwarz approach. Whereas diagonal preconditioning is suboptimal, we observe optimality for our approach, which numerically verifies our theoretical result in Corollary 6.1. This is also reflected by the number of PCG iterations.

diagonal 140 diagonal add. Schwarz add. Schwarz

3 10 120

100

2 10 60 condition number number of iterations 40

1 10

2 3 2 3 10 10 10 10 number of knots N number of knots N

Figure 6.2. Condition numbers λmax/λmin of the diagonal and the additive Schwarz preconditioned Galerkin matrices as well as the number of PCG iterations for the hypersingular equation on the pacman from Section 6.1.

6.2. Adaptive BEM for weakly-singular integral equation for Dirichlet problem on pacman. Let Ω and P be as in the previous section. With the double-layer operator K and the right-hand side g := (1/2+ K)P , (6.19) |Γ the solution of the weakly-singular integral equation (1.3) is just the normal derivative of P , i.e., φ = ∂ P . For Algorithm 6.3, we choose splines of degree two as ansatz space (i.e., p = ν Yℓ 3 and all weights are equal to one) and the initial knots 0 as for the geometry. To steer the K 1/2 V 2 algorithm, we use the weighted-residual error indicators ηℓ(z) := hℓ ∂Γ(g Φℓ) L (ωℓ(z). Figure 6.3 shows a comparison of the diagonal and the additive Schwak rz preconditioner.− k 6.3. Adaptive BEM for hypersingular integral equation onb slit. We consider the hypersingular integral equation on the slit Γ = [ 1, 1] 0 , sketched in Figure 6.1, which is represented as spline curve γ : [0, 1] Γ of degree− one;×{ } see [FGHP16, Section 3.4]. For → f := 1, the exact solution is u(x, 0) = 2√1 x2. For Algorithm 6.3, we choose splines of degree one as ansatz space (i.e., p = 1− and all weights are equal to one) and the Xℓ initial knots 0 as for the geometry. To steer the algorithm, we use the h h/2 based error K 1/2 − 2 indicators ηℓ(z) := hℓ ∂Γ(Uﬁne(ℓ) Uℓ) L (ωℓ(z)), where Uﬁne(ℓ) is the Galerkin approximation k − k 28 b

160 diagonal diagonal add. Schwarz add. Schwarz

140 4 10

120

3 100 10

2 60 10 condition number number of iterations 40

1 20 10

2 2 10 10 number of knots N number of knots N

Figure 6.3. Condition numbers λmax/λmin of the diagonal and the additive Schwarz preconditioned Galerkin matrices as well as the number of PCG iterations for the weakly-singular equation on the pacman from Section 6.2.

in the space corresponding to the uniformly refined knots . These are generated Xfine(ℓ) Kfine(ℓ) from ℓ via the refinement steps Algorithm 6.3 (iv)–(vi) with ℓ = ℓ. Again, we compare diagonalK preconditioning and the local multilevel diagonal preconditM ioner;N see Figure 6.4.

diagonal 160 diagonal add. Schwarz add. Schwarz

140

120

2 10 100

60 condition number

1 number of iterations 10 40

1 2 3 1 2 3 10 10 10 10 10 10 number of knots N number of knots N

Figure 6.4. Condition numbers λmax/λmin of the diagonal and the additive Schwarz preconditioned Galerkin matrices as well as the number of PCG iterations for the hypersingular equation on the slit from Section 6.3.

6.4. Adaptive BEM for weakly-singular integral equation on slit. Let Γ be again the slit [ 1, 1] 0 . For the weakly-singular integral equation with g := x/2, the cor- − ×{ } − responding solution reads φ(x, 0) = x/√1 x2. For Algorithm 6.3, we choose splines of degree one as ansatz space (i.e., p−= 2 and− all weights are equal to one) and the initial Yℓ knots 0 as for the geometry. To steer the algorithm, we use the weighted-residual error K 1/2 V 2 indicators ηℓ(z) := hℓ ∂Γ(g Φℓ) L (ωℓ(z)). A comparison of the diagonal and the additive Schwarz preconditionerk is found− in Figurek 6.5. b 29

diagonal diagonal add. Schwarz add. Schwarz 200 4 10 180

160

3 10 140

120

100

2 10 80 condition number

number of60 iterations

1 40 10

1 2 3 1 2 3 10 10 10 10 10 10 number of knots N number of knots N

Figure 6.5. Condition numbers λmax/λmin of the diagonal and the additive Schwarz preconditioned Galerkin matrices as well as the number of PCG iterations for the weakly-singular equation on the slit from Section 6.4.

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