Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations

We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the unknown Galerkin BEM error. The required assumptions are weak and allow for piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. In particular, our analysis gives a first contribution to adaptive BEM in the frame of isogeometric analysis (IGABEM), for which we formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments underline the theoretical findings and show that the proposed adaptive strategy leads to optimal convergence.


Introduction
Isogeometric analysis.The central idea of isogeometric analysis 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 computer aided design (CAD) by means of NURBS or T-splines.This concept, originally invented in [HCB05] for finite element methods (IGAFEM) has proved very fruitful in applications [HCB05,SBTR12]; see also the monograph [CHB09].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).Isogeometric BEM (IGABEM) has first been considered in [SBTR12].Unlike standard BEM with piecewise polynomials which is well-studied in the literature, cf. the monographs [SS11,Ste08] and the references therein, the numerical analysis of IGABEM is essentially open.We only refer to [SBTR12,PTC13,TM12] for numerical experiments and to [HAD14] for some quadrature analysis.In particular, a posteriori error estimation has been wellstudied for standard BEM, e.g., [CS95, CS96, Car97, CMPS04, CMS01, Fae00, Fae02] as well as the recent overview article [FFH + 14], but has not been treated for IGABEM so far.The purpose of the present work is to shed some first light on a posteriori error analysis for IGABEM which provides some mathematical foundation of a corresponding adaptive algorithm.

Main result.
Let Ω ⊂ R 2 be a Lipschitz domain and Γ ⊆ ∂Ω be a compact, piecewise smooth part of the boundary with finitely many connected components (see Section 2.2 and Section 2.3).Given a right-hand side f , we consider boundary integral equations in the abstract form V φ(x) = f (x) for all x ∈ Γ, (1.1) where V : H −1/2 (Γ) → H 1/2 (Γ) is an elliptic isomorphism.Here H 1/2 (Γ) is a fractionalorder Sobolev space, and H −1/2 (Γ) is its dual (see Section 2 below).Given f ∈ H 1/2 (Γ), the Lax-Milgram lemma provides existence and uniqueness of the solution φ ∈ H −1/2 (Γ) of the variational formulation of (1.1) ˆΓ V φ(x)ψ(x) dx = ˆΓ f (x)ψ(x) dx for all ψ ∈ H −1/2 (Γ). (1.2) In the Galerkin boundary element method (BEM), the test space H −1/2 (Γ) is replaced by some discrete subspace X h ⊆ L 2 (Γ) ⊆ H −1/2 (Γ).Again, the Lax-Milgram lemma guarantees existence and uniqueness of the solution φ h ∈ X h of the discrete variational formulation ˆΓ V φ h (x)ψ h (x) dx = ˆΓ f (x)ψ h (x) dx for all ψ h ∈ X h , (1.3) and φ h can in fact be computed by solving a linear system of equations.We assume that X h is linked with a partition T h of Γ into a set of connected segments.For each vertex x ∈ N h of T h , let ω h (z) := T ∈ T h : z ∈ T denote the node patch.If X h is sufficiently rich (e.g., X h contains certain splines or NURBS; see Section 4), we prove that with some X h -independent constants C eff , C rel > 0, i.e., the unknown BEM error is controlled by some computable a posteriori error estimator η h .Here, r h := f −V φ h ∈ H 1/2 (Γ) denotes the residual and is the Sobolev-Slobodeckij seminorm.Estimate (1.4) has first been proved by Faermann [Fae00] for closed Γ = ∂Ω and standard spline spaces X h based on the arclength parametrization γ : [0, L] → Γ.In isogeometric analysis, γ is not the arclength parametrization.In our contribution, we generalize and refine the original analysis of Faermann [Fae00]: Our analysis allows, first, closed as well as open parts of the boundary, second, general piecewise smooth parametrizations γ and, third, covers standard piecewise polynomials as well as NURBS spaces X h .
Outline.Section 2 recalls the functional analytic framework, provides the assumptions on Γ and its parametrization γ, and fixes the necessary notation.The proof of (1.4) is given in Section 3 for sufficiently rich spaces X h (Theorem 3.1).In Section 4, we recall the NURBS spaces for IGABEM and prove that these spaces X h satisfy the assumptions (Assumptions (A1)-(A2) in Section 3.1) of the a posteriori error estimate (1.4).Based on knot insertion, we formulate an adaptive algorithm which is capable to control and adapt the multiplicity of the nodes as well as the local mesh-size (Algorithm 4.5).The final Section 5 gives some brief comments on the stable implementation of adaptive IGABEM for Symm's integral equation and provides the numerical evidence for the superiority of the proposed adaptive IGABEM over IGABEM with uniform mesh-refinement.

Preliminaries
The purpose of this section is to collect the main assumptions on the boundary and its discretization as well as to fix the notation.For more details on Sobolev spaces and the functional analytic setting of weakly-singular integral equations, we refer to the literature, e.g., the monographs [HW08, McL00,SS11] and the references therein.
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.2.1.Sobolev spaces.For any measurable subset ω ⊆ Γ, let L 2 (ω) denote the Lebesgue space of all square integrable functions which is associated with the norm u 2 L 2 (ω) := ´ω |u(x)| 2 dx.We define the Hilbert space associated with the Sobolev-Slobodeckij norm For finite intervals I ⊆ R we use analogous definitions.By H −1/2 (ω), we denote the dual space of H 1/2 (ω), where duality is understood with respect to the L 2 (ω)-scalar product, i.e., Gelfand triple and all inclusions are dense and compact.Amongst other equivalent definitions of H 1/2 (ω) are the characterization as trace space of functions in H 1 (Ω) as well as equivalent interpolation techniques.All these definitions provide the same space of functions but different norms, where norm equivalence constants depend only on ω; see, e.g., the monograph [McL00] and references therein.Throughout, we shall use the Sobolev-Slobodeckij norm (2.2), since it is numerically computable.
2.2.Connectedness of Γ.Let the part of the boundary Γ = i Γ i be decomposed into its finitely many connected components Γ i .The Γ i are compact and piecewise smooth as well.Note that this yields existence of some constant c > 0 such that |x − y| ≥ c > 0 for all x ∈ Γ i , y ∈ Γ j , and i = j.Together with and results in norm equivalence 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 from now on assume that Γ is connected.All results of this work remain valid for non-connected Γ.
2.3.Boundary parametrization.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, 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 for t ∈ [a, b).Moreover we assume that γ ′ ℓ (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 L (t)| = 1 = |γ ′r L (t)|, and its periodic extension.Then, elementary differential geometry yields bi-Lipschitz continuity (2.4) (2.5) 2.4.Boundary discretization.The part of the boundary Γ is split into a set T h = {T 1 , . . ., T n } of compact and connected segments T j .The endpoints of the elements of T h form the set of nodes N h := z j : j = 1, . . ., n for closed Γ and N h = z j : j = 0, . . ., n for open Γ.The arclength of each element T ∈ T h is denoted by h T , where h := max T ∈T h h T .Moreover, we define the shape regularity constant For closed Γ, we extend the nodes, elements and their length periodically.We suppose

A posteriori error estimate
3.1.Main theorem.For T ∈ T h , we inductively define the patch ω m h (T ) ⊆ Γ of order m ∈ N 0 by The main result of Theorem 3.1 requires the following two assumptions on T h and X h for some fixed integer m ∈ N 0 : (A1) For each T ∈ T h , there exists some fixed function ψ T ∈ X h with connected support supp(ψ T ) such that

2)
(A2) There exists some constant q ∈ (0, 1] such that With these assumptions, we can formulate the following theorem which states validity of (1.4).For standard BEM and piecewise polynomials based on the arclength parametrization γ L of some closed boundary Γ = ∂Ω, the analogous result is first proved in [Fae00, Theorem 3.1] Theorem 3.1.The residual r h = f − V φ h satisfies the efficiency estimate If the mesh T h and the discrete space X h satisfy assumptions (A1)-(A2), also the reliability estimate holds.The constant C eff > 0 depends only on V , while C rel > 0 holds additionally on Γ, m, κ(T h ), and q.
Remark 3.2.The proof reveals that the efficiency estimate (3.4) is valid for any approximation φ h of φ, while the upper reliability estimate (3.5) requires some Galerkin orthogonality.

Proof of efficiency estimate (3.4).
The elementary proof of the following proposition is already found in [Fae00, page 208].It is found as well in [Gan14, Theorem 2.12].

3.3.
Proof of reliability estimate (3.5).We start with the following lemma.For the elementary (but long) proof, we refer to [Fae00, Lemma 2.3].A detailed proof is also found in [Gan14, Proposition 2.13].
Lemma 3.4.There exists a constant C 1 > 0 such that for all u ∈ H 1/2 (Γ) The constant only depends on Γ and κ(T h ).

Our next goal is to bound
To this end we need the following Poincaré-type inequality from [Fae00, Lemma 2.5].
Lemma 3.5.Let I ⊂ R be a finite interval with length |I| > 0.Then, there holds Then, there exists a constant C 2 > 0 which depends only on Γ, m, κ(T h ), and q such that for all T ∈ T h Proof of Lemma 3.6 for closed Γ = ∂Ω.The assertion is formulated on the boundary itself.Without loss of generality, we may therefore assume that γ = γ L .Since supp(ψ T ) is connected, there is an interval I of length |I| ≤ L with γ(I) = supp(ψ T ).We use Lemma 3.5 and get With the orthogonality (3.7) and Assumption (A2), we see . Using the last two inequalities, we therefore get Together with For m = 0, (2.6), (A1) and (2.5), (3.9) conclude the proof with for m > 0, we use induction on ℓ to prove the following assertion for all ℓ ∈ N: (3.10) For ℓ = 1, (3.10) even holds with equality.The induction hypothesis for ℓ − 1 ≥ 1 is Ǔ (r, s) dr ds. (3.12) For r < t < s ∈ R , we have With the abbreviate notation h k := h Ťk , it hence follows Ǔ (r, s) dr ds dt 1 ds dr dt There holds žj+ℓ−2 − žj−1 This implies Inserting this into (3.12) and using as well as the induction hypothesis (3.11), we obtain This concludes the induction step and thus proves (3.10).There is a j We use (3.9) and (3.10) for ℓ = min{2m , n − 1} to see Finally, we use (2.5) and which concludes the proof.
Proof of Lemma 3.6 for open Γ ∂Ω.The proof works essentially as before, where (3.10) now becomes Details are found in [Gan14, Lemma 2.15].
Then, there exists a constant C 3 > 0 which depends only on Γ, m, κ(T h ), and q such that Proof of Proposition 3.7 for closed Γ = ∂Ω.Without loss of generality, we may assume that γ = γ L .Due to Lemma 3.4, it remains to estimate the term . For m = 0, we see For m > 0, Assumption (A1) and Lemma 3.6 give Let j ∈ {1, . . ., n} with T = T j .We extend the mesh data periodically.With the abbreviate notation h ℓ := h Ťℓ , we see Combining (3.14) and (3.15), we obtain with This concludes the proof.
Proof of Proposition 3.7 for open Γ ∂Ω.The proof works essentially as for Γ = ∂Ω.

Adaptive IGABEM
4.1.B-splines and NURBS.Throughout this subsection, we consider knots Ǩ := (t i ) i∈Z on R with t i−1 ≤ t i for i ∈ Z and lim i→±∞ t i = ±∞.For the multiplicity of any knot t i , we write #t i .We denote the corresponding set of nodes Ň := t i : i ∈ Z = žj : j ∈ Z with žj−1 < žj for j ∈ Z.For i ∈ Z, the i-th B-Spline of degree p is defined inductively by where, for t ∈ R, We also use the notations B Ǩ i,p := B i,p and β Ǩ i,p := β i,p to stress the dependence on the knots Ǩ.The proof of the following theorem is found in [dB86, Theorem 6].
Theorem 4.1.Let I = [a, b) be a finite interval and p ∈ N 0 .Then is a basis for the space of all right-continuous Ň −piecewise polynomials of degree lower or equal p on I and which are, at each knot t i , p − #t i times continuously differentiable if p − #t i ≥ 0.
In addition to the knots Ǩ = (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 non-uniform rational B-Spline of degree p or shortly NURBS as We also use the notation R Ǩ,W i,p := R i,p .Note that the denominator is locally finite and never zero as shown in the following lemma.
Lemma 4.2.For p ∈ N 0 and i, ℓ ∈ Z, the following assertions hold: ) i∈Z be a sequence of knots such that #t i,ℓ = #t i for all i ∈ Z and, W ℓ = (w i,ℓ ) i∈Z a sequence of positive weights.If ( Ǩℓ ) ℓ∈N converges pointwise to Ǩ and (W ℓ ) ℓ∈N converges pointwise to W, then R Ǩℓ ,W ℓ i,p ℓ∈N converges almost everywhere to R Ǩ,W i,p for all i ∈ N.
Proof.The proof for (i)-(v) can be found in [dB86, Section 2, page 9-10] for B-splines.The generalization to NURBS is trivial.(vi) is an immediate consequence of (v).(vii) follows from Theorem 4.1.To prove (viii), we note that for all ℓ ∈ Z and t ∈ R it holds Hence, the assertion is an immediate consequence of the definition of B-splines.For B-splines, (ix) is proved by induction, noting that for all p ′ ∈ N and i ∈ Z, we have This easily implies the convergence of R Ǩℓ i,p .For any p ∈ N 0 , we define the vector spaces as well as Note that the sums are locally finite.An analogous version of the following result is already found in [Fae00] for the special case of B-splines of degrees p = 0, 1, 2 and knot multiplicity #t i = 1 for all i ∈ Z and weight function ϕ = 1.The following generalization to arbitrary NURBS, however, requires a completely new idea.
Proof.We prove the lemma in five steps.
Step 3: Now, we prove the lemma for ϕ = 1.In the definition of M ν we replace the interval I by R to define a superset of We extend the function Φ k,ℓ,ν to We define a closed and bounded and hence compact subset of M ν M 0,1 ν,R := (ž, w) ∈ M ν,R : ž0 = 0, ž1 = 1 .
As in (4.1), one verifies that Φ k,ℓ,ν only attains values in [0, 1) and is continuous.Moreover, due to the substitution rule and Lemma 4.2, (viii), we have for any element of (ž, w) and hence The second set is compact, since it is the image of a closed and bounded set under a continuous mapping.Therefore it attains a maximum smaller than one.This concludes the proof for Step 4: Finally, we are in the position to prove the assertion of the lemma for arbitrary functions ϕ with the desired properties.Let (ž m , w m ) m∈N be a sequence in M ν such that the Φ k,ℓ,ν -values converge to sup(Φ k,ℓ,ν (M ν )).Because of the boundedness of M ν , we can assume convergence of the sequence, where the limit In the first case, we are done because of (4.12) and the continuity of Φ k,ℓ,ν .For the second case, we define Note that a n < b n , and that the sequences (a n ) n∈N and (b n ) n∈N converge to the limit We consider two cases.
Case 1: If ϕ is continuous at the limit Z, it is absolutely continuous on the interval [a n , b n ] for sufficiently big n ∈ N. Hence we have for sufficiently big n ∈ N The second summand converges to zero.We consider the first one.For any C ∈ (0, 1), there holds for sufficiently big n ∈ N Since C was arbitrary, this implies sup Φ k,ℓ,ν (M ν ) ≤ 1 − q κ max , w min , w max , p, 1 < 1.
Case 2: If ϕ is not continuous at the limit Z we proceed as follows.For sufficiently big n ∈ N, ϕ is absolutely continuous on [a n , Z] and on [Z, b n ].By considering suitable subsequences, we can assume that a n < b n ≤ Z, Z ≤ a n < b n or a n ≤ Z ≤ b n , each for all n ∈ N. In the first two cases, we can proceed as in Case 1.In the third case, we argue similarly as in Case 1 to see, with the left-handed limit ϕ ℓ (Z) and the right-handed limit ϕ r (Z) for n ∈ N big enough .
Again, the second summand converges to zero, wherefore it remains to consider the first one.For any C ∈ (0, 1), there holds for sufficiently big n ∈ N due to (4.1) which concludes the proof.
We return to our problem (1.1).If Γ = ∂Ω is closed, each node ž ∈ Ňh may be assigned with a multiplicity #ž ≤ p + 1.This induces a sequence of non decreasing knots Ǩh = (t i ) N i=1 on (a, b].Let W h = (w i ) N i=1 be a sequence of weights on these knots.We extend the knot sequence (b−a)-periodically to (t i ) i∈Z and the weight sequence to (w i ) i∈Z by w n+i := w i for i ∈ Z.For the extended sequences we also write Ǩh and W h .We set If Γ = ∂Ω is open, we assign to each node ž ∈ Ňh a corresponding multiplicity #ž ≤ p + 1 such that #ž 0 = #ž n = p + 1.This induces a sequence of non decreasing knots Ǩh = (t i ) N i=0 on [a, b].Let W h = (w i ) N −p i=1 be a sequence of weights.To keep the notation simple, we extend the sequences arbitrarily to Ǩh = (t i ) i∈Z with t i ≤ t i+1 for i ∈ Z, a > t i → −∞ for i < 0 and b < t i → ∞ for i > N, and W h = (w i ) i∈Z with w i > 0. This allows to define (4.14) Due to Lemma 4.2, (ii) and (iv), this definition does not depend on how the sequences are extended.
With the following theorem we conclude that Theorem 3.1 holds for the span of transformed NURBS functions.
Theorem 4.4.Let p ∈ N 0 and m := ⌈p/2⌉.Then, the space X h := N p ( Ǩh , W h ) is a subspace of L 2 (Γ) which satisfies the assumptions (A1)-(A2) from Section 3.1 with the constant of Lemma 4.3 Proof of Theorem 4.4 for closed Γ = ∂Ω.Lemma 4.2, (i) and (ii), implies Let T be an element of the mesh T h , j ∈ {1, . . ., n} with T = T j , and i ∈ {1, . . ., N} with žj−1 = t i−1 and žj = t i .We define ψT (t) := R i−m,p (t) for t ∈ [a, b) and extend it continuously at b.We set To verify Assumption (A2), we apply Lemma 4.3.Note that R i−m,p is completely determined by the knots in I and their weights.This is due to I ⊇ [t i−m−p−1 , t i+2p−m ] and Lemma 4.2, (iv).The regularity constant of these knots from (4.10) is obviously smaller or equal than κ( Ǩh ).Since γ is piecewise two times continuously differentiable and its left and right derivative vanishes nowhere, |γ ′ | is piecewise continuously differentiable and is bounded from above by some positive constant.With Lemma 4.3 and (4.16), we hence get Consequently, Assumption (A2) is also fulfilled.This concludes the proof.If Ǩ′ contains only one additional knot t ′ (possibly already contained in Ǩ), the coefficients can be calculated explicitly.We assume t i = t ′ i for all i with t i < t ′ .Then, [dB86, Algorithm 11] shows With t N < t N +1 , it is easy to check that B Ǩh i,p | I = 0 for i = 0, . . ., N. Hence, Theorem 4.1 implies w i = w + i for i = 0, . . ., N. It just remains to add the knots t ′ − (b − a), t ′ and t ′ + (b − a).To this end, we can repetitively apply (4.19) to obtain the weights (w ′ i ) N +1 i=1 .Note that this only involves the weights (w + i ) N i=0 are needed.Moreover, the new weights (w ′ i ) N +1 i=1 are just convex combinations of the old ones (w i ) N i=1 .With (4.9), (4.17), and (4.20), we get nestedness For closed Γ ∂Ω, a knot t ′ ∈ (a, b] can analogously be inserted to the knots Ǩh = (t i ) N i=0 .4.3.Adaptive algorithm.In this subsection, we introduce an adaptive algorithm, which uses the local contributions of η h to steer the h-refinement of the mesh T h as well as the increase of the multiplicity of the nodes N h .To respect the iterative character of this procedure, all discrete quantities (as, e.g., T h , φ h , etc.) are indexed by the level ℓ ∈ N 0 of the adaptive process instead of the mesh-size h.Let 0 < θ < 1 be an adaptivity parameter and p ∈ N 0 a polynomial degree.We start with some nodes Ň0 .Each node has a multiplicity lower or equal p + 1, where for open Γ ∂Ω we assume #a = #b = p + 1.This induces knots Ǩ0 .Let W 0 be some initial positive weights.We assume p + 1 ≤ N 0 and for closed Γ = ∂Ω, |T | ≤ |Γ|/4 for all T ∈ T 0 .As the initial trial space, we consider The adaptive algorithm with Dörfler marking reads as follows: Algorithm 4.5.Input: Adaptivity parameter 0 < θ < 1, polynomial order p ∈ N 0 , initial mesh T 0 with knots Ǩ0 , initial weights W 0 .
Adaptive loop: Iterate the following steps, until η ℓ is sufficiently small: (4.23) (iv) If both nodes of an element T ∈ T ℓ belong to M ℓ , T will be marked.
(v) For all other nodes in M ℓ , 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 ℓ , will be marked.(vi) Refine all marked elements T ∈ T ℓ by bisection of the corresponding Ť ∈ Ťℓ .Use further bisections to guarantee that the new mesh T ℓ+1 satisfies κ( Ťℓ+1 ) ≤ 2κ( Ť0 ).(4.24) Update counter ℓ → ℓ + 1. Output: Approximate solutions φ ℓ and error estimators η ℓ for all ℓ ∈ N 0 .
An optimal 1D bisection algorithm which ensures (4.24), is discussed and analyzed in [AFF + 13].Note that boundedness of κ( Ťℓ ) implies as well boundedness of κ(T ℓ ).Moreover, there holds since the new weights are convex combinations of the old weights.Hence, Theorem 3.1 and Theorem 4.4 apply and show efficiency and reliability of the estimator (4.26)

Numerical experiments
In this section, we empirically investigate the performance of Algorithm 4.5 in three typical situations: In Section 5.2 and Section 5.3, we consider a closed boundary Γ = ∂Ω, where the solution is smooth resp.exhibits a generic (i.e., geometry induced) singularity.In Section 5.4, we consider a slit problem.In either example, the exact solution is known and allows us to compute the Galerkin error to underline reliability and efficiency of the proposed estimator.
In each example, the parametrization γ of the part Γ of the boundary is a NURBS curve and thus has the special form (5.1) for all t ∈ [a, b].Here, p ∈ N is the polynomial degree, Ǩ0 and W 0 are knots and weights as in Section 4.3 and (C i ) i∈Z are control points in R 2 which are periodic for closed Γ = ∂Ω.We choose the same polynomial degree p for our approximation spaces X ℓ .Since for the refinement strategy only knot insertion is used, we can apply (4.17) and (4.20) to see for the first and second component of γ (5.2) Hence, this approach reflects the main idea of isogeometric analysis, where the same space is used for the geometry and for the approximation.We compare uniform refinement, where M ℓ = N ℓ and hence all elements are refined, and adaptive refinement with θ = 0.75.forms a basis of N ( Ǩh , W h ).We abbreviate the elements of this basis with Ri and its index set with I.Then, there holds the unique basis representation φ h = i∈I c h,i Ri .The coefficient vector c h is the unique solution of with the symmetric positive definite matrix (5.6) and the right-hand side vector (5.7) The energy norm then reads (5.8) To calculate V h , f h and the H 1/2 -seminorms of the residual r h = f − V φ h , singular integrals and double integrals have to be evaluated.Since, this is hardly possible analytically, we approximate the appearing integrals.To this end, we first write them as sum of integrals over the elements of the mesh Ť .In the spirit of [SS11, Section 5.3], the local integrals which contain singularities, are transformed by Duffy transformations such that either the singularity vanishes or a pure logarithmic singularity of the form log(t) on [0, 1] remains.Finally, the integrals are evaluated over the domain [0, 1] or [0, 1] 2 using tensor-Gauss quadrature with weight function 1 resp.log(t).Since the integrands are smooth up to logarithmic terms, this yields exponential convergence of adapted Gauss quadrature and hence provides accurate approximations.For closed Γ = ∂Ω and arbitrary parametrization γ as in Section 2.3, all details are elaborated in [Gan14, Section 5].denotes the double-layer integral operator.The unique solution of (1.1) is the normal derivative φ = ∂u/∂ν of the weak solution u ∈ H 1 (Ω) of (5.9). of the last refinement step.As expected, the algorithm mainly refines the mesh at the tips t = 0 and t = 1.

Figure 5
Figure 5.7.Experiment with singular solution on slit from Section 5.4.Histogram of number of knots over the parameter domain.
a rational function with nonzero denominator, which can be extended continuously at t ℓ .(ii)Ri,p vanishes outside the interval[t i−1 , t i+p ).It is positive on the open interval (t i−1 , t i+p ).(iii)It holds t i−1 = t i+p if and only if R i,p = 0.(iv) B i,p is completely determined by the p + 2 knots t i−1 , . . ., t i+p .R i,p is completely determined by the 3p+2 knots t i−p−1 , . . ., t i+2p and the 2p+1 weights w i−p , . . ., w i+p .
Therefore, we will also use the notation R(•|t i−p−1 , . . ., t i+2p , w i−p , . . ., w i+p ) := R i,p .(4.4) (v) The NURBS functions of degree p form a partition of unity, i.e. i∈Z R i,p = 1 on R. (4.5) (vi) If all weights are equal, then R i,p = B i,p .Hence, B-splines are just special NURBS functions.
Proof of Theorem 4.4 for open Γ ∂Ω.The proof works analogously as before.Details are found in [Gan14, Theorem 4.14].
.19)For closed Γ = ∂Ω, we consider again knots Ǩh = (t i ) N i=1 and weights W h = (w i ) N i=1 as in the previous subsection.We additionally assume p + 1 ≤ N. Now we insert an additional knot t ′ ∈ (a, b] to the knots Ǩh such that the multiplicities of the new knots Ǩ′ h are still smaller or equal than p + 1.The new knots are extended (b − a)-periodically.We want to find the unique weights (w ′ i ) i∈Z which fulfill