Adaptive IGAFEM with optimal convergence rates: T-splines

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations. We employ analysis-suitable T-splines of arbitrary odd degree on T-meshes generated by the refinement strategy of [Morgenstern, Peterseim, Comput. Aided Geom. Design 34 (2015)] in 2D resp. [Morgenstern, SIAM J. Numer. Anal. 54 (2016)] in 3D. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (resp. the sum of energy error plus data oscillations) with optimal algebraic rates.

1. Introduction 1.1. Adaptivity in isogeometric analysis. The central idea of isogeometric analysis (IGA) [HCB05, CHB09, BBdVC + 06] is to use the same ansatz functions for the discretization of the partial differential equation (PDE) as for the representation of the problem geometry in computer aided design (CAD). While the CAD standard for spline representation in a multivariate setting relies on tensor-product B-splines, several extensions of the B-spline model have emerged to allow for adaptive refinement, e.g., (analysis-suitable) T-splines [SZBN03,DJS10,SLSH12,BdVBSV13], hierarchical splines [VGJS11, GJS12,KVVdZvB14], or LR-splines [DLP13,JKD14]; see also [JRK15,HKMP17] for a comparison of these approaches. All these concepts have been studied via numerical experiments. However, to the best of our knowledge, the thorough mathematical analysis of adaptive isogeometric finite element methods (IGAFEM) is so far restricted to hierarchical splines [BG16,BG17,GHP17,BG18,BBGV19]. Recently, linear convergence at optimal algebraic rate has been proved in [BG17] for the refinement strategy of [BG16] based on truncated hierarchical B-splines [GJS12], and in our own work [GHP17] for a newly proposed refinement strategy based on standard hierarchical B-splines. In the latter work, we identified certain abstract properties for the underlying meshes, the mesh-refinement, and the finite element spaces that automatically guarantee linear convergence at optimal rate, and subsequently verified these properties in the case of hierarchical splines. We stress that adaptivity is well understood for standard FEM with globally continuous piecewise polynomials; see, e.g., [Dör96,MNS00,BDD04,Ste07,CKNS08,FFP14,CFPP14] for milestones on convergence and optimal convergence rates. In the frame of adaptive isogeometric boundary element methods (IGABEM), we also mention our recent works [FGP15,FGHP16,FGHP17,Gan17,GPS19].

Model problem.
On the bounded Lipschitz domain Ω ⊂ R d , d ∈ {2, 3}, with initial mesh T 0 and for given f ∈ L 2 (Ω) as well as f ∈ L 2 (Ω) d with f | T ∈ H(div, T ) for 1 all T ∈ T 0 , we consider a general second-order linear elliptic PDE in divergence form with homogenous Dirichlet boundary conditions Lu := −div(A∇u) + b · ∇u + cu = f + divf in Ω, u = 0 on ∂Ω. (1.1) We pose the following regularity assumptions on the coefficients: A(x) ∈ R d×d sym is a symmetric and uniformly positive definite matrix with A ∈ L ∞ (Ω) d×d and A| T ∈ W 1,∞ (T ) for all T ∈ T 0 . The vector b(x) ∈ R d and the scalar c(x) ∈ R satisfy that b, c ∈ L ∞ (Ω). We interpret L in its weak form and define the corresponding bilinear form (1. 2) The bilinear form is continuous, i.e., it holds that Additionally, we suppose ellipticity of · , · L on H 1 0 (Ω), i.e., v , v L ≥ C ell v 2 H 1 (Ω) for all v ∈ H 1 0 (Ω). (1.4) Note that (1.4) is for instance satisfied if A(x) is uniformly positive definite and if b ∈ H(div, Ω) with − 1 2 div b(x) + c(x) ≥ 0 almost everywhere in Ω. Overall, the boundary value problem (1.1) fits into the setting of the Lax-Milgram theorem and therefore admits a unique solution u ∈ H 1 0 (Ω) to the weak formulation (1.5) Finally, we note that the additional regularity f | T ∈ H(div, T ) and A| T ∈ W 1,∞ (T ) for all T ∈ T 0 is only required for the well-posedness of the residual a posteriori error estimator; see Section 2.5.

Outline & Contributions.
The remainder of this work is organized as follows: Section 2 recalls the definition of T-meshes and T-splines of arbitrary odd degree in the parameter domain (Section 2.1) from [BdVBSV13] for d = 2 resp. [Mor16] 1 for d = 3. Moreover, it recalls corresponding refinement strategies (Section 2.2) from [MP15] resp. [Mor16], derives a canonical basis for the T-spline space with homogeneous boundary conditions (Section 2.3), and transfers all the definitions to the physical domain Ω via some parametrization γ : [0, 1] d → Ω (Section 2.4). Subsequently, a standard adaptive algorithm (Algorithm 2.5) of the form driven by some residual a posteriori error estimator (2.36) is given. For T-splines in 2D, this algorithm has already been investigated numerically in [HKMP17]. Finally, our main result Theorem 2.9 is presented. First, it states that the error estimator η • associated with the FEM solution U • ∈ X • ⊂ H 1 0 (Ω) is efficient and reliable, i.e., there exist C eff , C rel > 0 such that where osc • (·) denotes certain data oscillation terms (see (2.38)). Second, it states that Algorithm 2.5 leads to linear convergence with optimal rates in the spirit of [Ste07,CKNS08,CFPP14]: Let η denote the error estimator in the -th step of the adaptive algorithm. Then, there exist C > 0 and 0 < q < 1 such that Moreover, for sufficiently small marking parameters in Algorithm 2.5, the estimator (resp. the so-called total error u − U H 1 (Ω) + osc (U ); see (1.7)) decays even with the optimal algebraic convergence rate with respect to the number of mesh elements, i.e., whenever the rate s > 0 is possible for optimally chosen meshes. The proof of Theorem 2.9 is postponed to Section 3. In [GHP17], we have identified abstract properties of the underlying meshes, the mesh-refinement, the finite element spaces, and the oscillations which imply (an abstract version of) Theorem 2.9. In Section 3, we briefly recapitulate these properties and verify them for our considered T-spline setting. The final Section 4 comments on possible extensions of Theorem 2.9.
1.4. General notation. Throughout, | · | denotes the absolute value of scalars, the Euclidean norm of vectors in R d , as well as the d-dimensional measure of a set in R d . Moreover, # denotes the cardinality of a set as well as the multiplicity of a knot within a given knot vector. 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., X • is the ansatz space corresponding to the mesh T • . The analogous notation is used for meshes T • , T , T etc. Moreover, we use · to transfer quantities in the physical domain Ω to the parameter domain Ω, e.g., we write T for the set of all admissible meshes in the parameter domain, while T denotes the set of all admissible meshes in the physical domain.

Adaptivity with T-splines
In this section, we recall the formal definition of T-splines from [BdVBSV13] for d = 2 resp. [Mor16] for d = 3 as well as corresponding mesh-refinement strategies from [MP15] resp. [Mor16]. We formulate an adaptive algorithm (Algorithm 2.5) for conforming FEM discretizations of our model problem (1.1), where adaptivity is driven by the residual a posteriori error estimator (see (2.36) below). Our main result of the present work Theorem 2.9 states reliability and efficiency of the estimator as well as linear convergence at optimal algebraic rate.
2.1. T-meshes and T-splines in the parameter domain Ω. Meshes T • and corresponding spaces X • are defined through their counterparts on the parameter domain Ω := (0, N 1 ) × · · · × (0, N d ), (2.1) where N i ∈ N are fixed integers for i ∈ {1, . . . , d}. Let p 1 , . . . , p d ≥ 3 be fixed odd polynomial degrees. Let T 0 be an initial uniform tensor-mesh of the form and define the k-th uniform refinement of T 0 inductively by  (2.6) In order to define T-splines, we have to extend the mesh T • on Ω to a mesh on Ω ext , where (2.7) We define T ext 0 analogously to (2.2) and T ext • as the mesh on Ω ext that is obtained by extending any bisection, that takes place on the boundary ∂ Ω during the refinement from T 0 to T • , to the set Ω ext \ Ω; see Figure 2.3. For d = 2, this reads and the remaining ext • (·) terms are defined analogously. Note that the logical expression means that there exists an element at the (lower part of the) boundary ∂ Ω with side [a 1 , b 1 ]. For d = 3, this reads and the remaining ext • (·) terms are defined analogously. Note that the logical expressions mean that there exists an element at the (lower part of the) boundary ∂ Ω with side [a 1 , b 1 ] resp. with sides [a 1 , b 1 ] and [a 2 , b 2 ]. The corresponding skeleton in any direction i ∈ {1, . . . , d} reads (2.9) Recall that p i ≥ 3 are odd. We abbreviate As in the literature, its closure Ω act is called active region, whereas Ω ext \ Ω act is called frame region. The set of nodes N act To each node z = (z 1 , . . . , z d ) ∈ N act • and each direction i ∈ {1, . . . , d}, we associate the corresponding global index vector where sort(·) returns (in ascending order) the sorted vector corresponding to a set of numbers. The corresponding local index vector is the vector of all p i + 2 consecutive elements in I gl •,i (z) having z i as their ((p i + 3)/2)-th (i.e., their middle) entry; see Figure 2.3. Note that such elements always exist due to the definition of I gl •,i (z) and the fact that p i is odd. This induces the global knot vector and the local knot vector where max(·, 0) and min(·, N i ) are understood element-wise (i.e., for each element in I gl •,i (z) resp. I gl •,i (z)). We stress that the resulting global knot vectors in each direction are so-called open knot vectors, i.e., the multiplicity of the first knot 0 and the last knot N i is p i + 1. Moreover, the interior knots coincide with the indices in Ω and all have multiplicity one. For more general index to parameter mappings, we refer to Section 4.2. We define the corresponding tensor-product B-spline B •,z : Ω → R as where B t i | K loc •,i (z) denotes the unique one-dimensional B-spline induced by K loc •,i (z); see, e.g., [DB01] for a precise definition and elementary properties. According to, e.g., [DB01, Section 6], each tensor-product B-spline satisfies that B •,z ∈ C 2 Ω . With this, we see for the space of T-splines in the parameter domain that (2.16) Finally, we define our ansatz space in the parameter domain as Note that this specifies the abstract setting of Section 3.3. For a more detailed introduction to T-meshes and splines, we refer to, e.g., [BdVBSV14, Section 7].
2.3. Basis of X • . First, we emphasize that for general T-meshes T • as in Section 2.1, the set B •,z : z ∈ N act • is not necessarily a basis of the corresponding T-spline space Y • since it is not necessarily linearly independent; see [BCS10] for a counter example. According to [BdVBSV14, Proposition 7.4], a sufficient criterion for linear independence of a set of B-splines is dual-compatibility: We say that B •,z : z ∈ N • is dual-compatible if for all z, z ∈ N • with | B •,z ∩ B •,z | > 0, the corresponding local knot vectors are at least in one direction aligned, i.e., there exists i ∈ {1, . . . , d} such that K loc •,i (z) and K loc •,i (z ) are both sub-vectors of one common sorted vector K.
We stress that admissible meshes yield dual-compatible B-splines, where the local knot vectors are even aligned in at least two directions for d = 3, and thus linearly independent B-splines; see [MP15,Theorem 3.6] together with [BdVBSV14, Theorem 7.16] for d = 2 and [Mor16, Theorem 5.3 and Theorem 6.6] for d = 3. To be precise, [Mor16] defines the space of T-splines differently as the span of B •,z : z ∈ N act • ∩ Ω and shows that this set is dual-compatible. The functions in this set are not only zero on the boundary ∂ Ω, but also some of their derivatives vanish there since the maximal multiplicity in the used local knot vectors is at most p i in each direction; see, e.g., [DB01, Section 6]. Nevertheless, the proofs immediately generalize to our standard definition of T-splines. The following lemma provides a basis of X • .
Lemma 2.2. Let T • ∈ T be an arbitrary admissible T-mesh in the parameter domain Ω.
Proof. Since we already know that the set B •,z : z ∈ N act • \ ∂ Ω act is linearly independent, we only have to show that it generates X • .
Step 1: It is well-known that the B-spline B(·|t 1 , . . . , t p+2 ) induced by a sorted knot vector (t 1 , . . . , t p+2 ) ∈ R p+2 is positive on the interval (t 1 , t p+2 ). It does not vanish at t 1 resp. t p+2 if and only if t 1 = · · · = t p+1 resp. t 2 = · · · = t p+2 . In particular, for all z ∈ N act • , this yields that B •,z | ∂ Ω = 0 if and only if z ∈ ∂ Ω act . This shows that (2.25) Step 2: To see the other inclusion, let V • ∈ X • . Then, there exists a representation of the Let E be an arbitrary facet of the boundary ∂ Ω and E act its extension onto ∂ Ω act , i.e., with i ∈ {1, . . . , d}, e := 0 and e act := −(p i − 1)/2, or e := N i and e act := N i + (p i − 1)/2. Restricting onto E and using the argument from Step 1, we derive that For d = 2, the set B •,z | E : z ∈ N act • ∩ E act coincides (up to the domain of definition) with the set of (d − 1)-dimensional B-splines corresponding to the global knot vector K gl •,i ((0, 0)) if e = 0 and K gl •,i ((N 1 , N 2 )) if e = N i ; see, e.g., [DB01, Section 2] for a precise definition of the set of B-splines associated to some global knot vector. It is well-known that these functions are linearly independent, wherefore we derive that c z = 0 for the corresponding coefficients. (2.26) We have already mentioned that [Mor16, Theorem 5.3 and Theorem 6.6] shows that the local knot vectors of the B-spline basis of Y • are even aligned in at least two directions. In particular, the knot vectors of the B-splines corresponding to the mesh T ext • | E ext are aligned in at least one direction. This yields dual-compatibility and thus linear independence of these B-splines, which concludes that c z = 0 for the corresponding coefficients. Since ∂ Ω act is the union of all its facets and E was arbitrary, this concludes that c z = 0 for all z ∈ N act • ∩ ∂ Ω act and thus the other inclusion in (2.25).
Finally, we study the support of the basis functions of Y • (and thus of X • ). To this end, we define for T • ∈ T and ω ⊆ Ω, the patches of order k ∈ N inductively by The constant k supp depends only on d and (p 1 , . . . , p d ).
Proof. We prove the assertion in two steps.
Step 1: We prove the first assertion. Without loss of generality, we can assume that z ∈ Ω, since otherwise there exists z ∈ N act Step 2: We prove the second assertion. First, let z ∈ N act • ∩ Ω. Then, Step 1 gives that z ∈ supp( B •,z ) ⊆ π ksupp • ( T ). Therefore, we see that the number of such z is bounded by  (p 1 , . . . , p d ). Altogether, we see that the number of z ∈ N act ( T )). Due to (2.23)-(2.24), this term is bounded by some uniform constant k supp ∈ N 0 . Finally, we set k supp := max(k supp , k supp ).
2.4. T-meshes and splines in the physical domain Ω. To transform the definitions in the parameter domain Ω to the physical domain Ω, we assume as in [GHP17, Section 3.6] that we are given a bi-Lipschitz continuous piecewise C 2 parametrization Let U • ∈ X • be the corresponding Galerkin approximation to the solution u ∈ H 1 0 (Ω), i.e., (2.32) We note the Galerkin orthogonality as well as the resulting Céa-type quasi-optimality October 3, 2019 11 2.5. Error estimator. Let T • ∈ T and T 1 ∈ T • . For almost every x ∈ ∂T 1 ∩ Ω, there exists a unique element T 2 ∈ T • with x ∈ T 1 ∩ T 2 . We denote the corresponding outer normal vectors by ν 1 resp. ν 2 and define the normal jump as (2.35) With this definition, we employ the residual a posteriori error estimator where, for all T ∈ T • , the local refinement indicators read (2.36b) We refer, e.g., to the monographs [AO00,Ver13] for the analysis of the residual a posteriori error estimator (2.36) in the frame of standard FEM with piecewise polynomials of fixed order.
(ii) Compute refinement indicators η (T ) for all elements T ∈ T .
(iii) Determine a set of marked elements M ⊆ T with θ η 2 ≤ η (M ) 2 , which has up to the multiplicative constant C min minimal cardinality. (iv) Generate refined mesh T +1 := refine(T , M ). Output: Sequence of successively refined meshes T and corresponding Galerkin approximations U with error estimators η for all ∈ N 0 . Remark 2.6. For our analysis, we assume that U is computed exactly. We mention that in practice the arising linear system is solved iteratively, which requires appropriate preconditioners. For analysis-suitable T-splines, such preconditioners have been recently developed in [CV19].
Let T • ∈ T. For T ∈ T • , we define the L 2 -orthogonal projection P •,T : L 2 (T ) → W | T : W ∈ P(Ω) . For an interior edge E ∈ E •,T := T ∩ T : T ∈ T • ∧ dim(T ∩ T ) = d − 1 , we define the L 2 -orthogonal projection P •,E : L 2 (E) → W | E : W ∈ P(Ω) . Note that E •,T = ∂T ∩ Ω. For V • ∈ X • , we define the corresponding oscillations where, for all T ∈ T • , the local oscillations read (2.38b) We refer, e.g., to [NV12] for the analysis of oscillations in the frame of standard FEM with piecewise polynomials of fixed order.
(2.41) By definition, u As < ∞ (resp. u Bs < ∞) implies that the error estimator η • (resp. the total error) on the optimal meshes T • decays at least with rate O (#T • ) −s . The following main theorem states that Algorithm 2.5 reaches each possible rate s > 0. The proof builds upon the analysis of [GHP17] and is given in Section 3. Generalizations are found in Section 4.
Theorem 2.9. It hold the following four assertions (i)-(iv): (i) The residual error estimator (2.36) satisfies reliability, i.e., there exists a constant C rel > 0 such that The residual error estimator satisfies efficiency, i.e., there exists a constant C eff > 0 such that (2.43) (iii) For arbitrary 0 < θ ≤ 1 and C min ≥ 1, there exist constants C lin > 0 and 0 < q lin < 1 such that the estimator sequence of Algorithm 2.5 guarantees linear convergence in the sense of η 2 +j ≤ C lin q j lin η 2 for all j, ∈ N 0 .
(2.44) (iv) There exists a constant 0 < θ opt ≤ 1 such that for all 0 < θ < θ opt , all C min ≥ 1, and all s > 0, there exist constants c opt , C opt > 0 such that i.e., the estimator sequence will decay with each possible rate s > 0. The constants C rel , C eff , C lin , q lin , θ opt , and C opt depend only on d, the coefficients of the differential operator L, diam(Ω), C γ , and (p 1 , . . . , p d ), where C lin , q lin depend additionally on θ and the sequence (U ) ∈N 0 , and C opt depends furthermore on C min , and s > 0. Finally, c opt depends only on C son , #T 0 , and s.
Remark 2.10. In particular, it holds that C −1 eff u As ≤ u Bs ≤ C rel u As for all s > 0. (2.46) If one applies continuous piecewise polynomials of degree p on a triangulation of some polygonal resp. polyhedral domain Ω as ansatz space, [GM08] proves that u B p/d < ∞. The proof requires that u allows for a certain decomposition and that the oscillations are of higher order; see Remark 2.7. In our case, u As u Bs depends besides the polynomial degrees (p 1 , . . . , p d ) also on the (piecewise) smoothness of the parametrization γ. In practice, γ is usually piecewise C ∞ . Given this additional regularity of γ, one might expect that the result of [GM08] can be generalized such that u As , u Bs < ∞ for s = min i=1,...,d p i /d. However, the proof goes beyond the scope of the present work and is left to future research.
Remark 2.11. Note that almost minimal cardinality of M in Algorithm 2.5 (iii) is only required to prove optimal convergence behavior (2.45), while linear convergence (2.44) formally allows C min = ∞, i.e., it suffices that M satisfies the Dörfler marking criterion. We refer to [CFPP14, Section 4.3-4.4] for details.
Remark 2.12. (a) If the bilinear form · , · L is symmetric, C lin , q lin as well as c opt , C opt are independent of (U ) ∈N 0 ; see [GHP17, Remark 4.1].
(b) If the bilinear form · , · L is non-symmetric, there exists an index 0 ∈ N 0 such that the constants C lin , q lin as well as c opt , C opt are independent of (U ) ∈N 0 , if (2.44)-(2.45) are formulated only for ≥ 0 . We refer to the recent work [BHP17, Theorem 19].
Remark 2.13. Let h := max T ∈T |T | 1/d be the maximal mesh-width. Then, h → 0 as → ∞, ensures that X ∞ := ∈N 0 X = H 1 0 (Ω); see [GHP17, Remark 2.7] for the elementary proof. We note that the latter observation allows to follow the ideas of [BHP17] to show that the adaptive algorithm yields optimal convergence even if the bilinear form · , · L is only elliptic up to some compact perturbation, provided that the continuous problem is well-posed. This includes, e.g., adaptive FEM for the Helmhotz equation; see [BHP17].

Proof of Theorem 2.9
In [GHP17, Section 2], we have identified abstract properties of the underlying meshes, the mesh-refinement, the finite element spaces, and the oscillations which imply Theorem 2.9; see [Gan17, Section 4.2-4.3] for more details. We mention that [GHP17,Gan17] actually only treat the case f = 0, but the corresponding proofs immediately extend to more general f as in Section 1.2. In the remainder of this section, we recapitulate these properties and verify them for our considered T-spline setting. For their formulation, we define for T • ∈ T and ω ⊆ Ω, the patches of order k ∈ N inductively by The corresponding set of elements is (3.2) To abbreviate notation, we let π • (ω) := π 1 • (ω) and Π • (ω) : 3.1. Mesh properties. We show that there exist C locuni , C patch , C trace , C dual > 0 such that all meshes T • ∈ T satisfy the following four properties  Regularity (2.30) of γ shows that it is sufficient to prove (M3) for hyperrectangles T in the parameter domain. There, the trace inequality (M3) is well-known; see, e.g., [Gan17, Proposition 4.2.2] for a proof for general Lipschitz domains. The constant C trace depends only on d and C γ .
Finally, (M4) in the parameter domain follows immediately from the Poincaré inequality. By regularity (2.30) of γ, this property transfers to the physical domain. The constant C dual depends only on d and C γ .
3.2. Refinement properties. We show that there exist C son ≥ 2 and 0 < q son < 1 such that all meshes T • ∈ T satisfy for arbitrary marked elements M • ⊆ T • with corresponding refinement T • := refine(T • , M • ), the following elementary properties (R1)-(R3): (R1) Bounded number of sons. It holds that #T • ≤ C son #T • , i.e., one step of refinement leads to a bounded increase of elements. (R2) Father is union of sons. It holds that T = T ∈ T • : T ⊆ T for all T ∈ T • , i.e., each element T is the union of its successors.
(R3) Reduction of sons. It holds that |T | ≤ q son |T | for all T ∈ T • and all T ∈ T • with T T , i.e., successors are uniformly smaller than their father. By induction and the definition of refine(T • ), one easily sees that (R2)-(R3) remain valid for arbitrary T • ∈ refine(T • ). In particular, (R2)-(R3) imply that each refined element T ∈ T • \ T • is split into at least two sons, wherefore Remark 3.2. In usual applications, the properties (R2)-(R3) are trivially satisfied. The same holds for (R1) on rectangular meshes. However, (R1) is not obvious for standard refinement strategies for simplicial meshes; see [GSS14] for 3D newest vertex bisection for tetrahedral meshes.
(R5) Overlay estimate. For all T • , T ∈ T, there exists a common refinement T • ∈ refine(T • ) ∩ refine(T ) such that Verification of (R1)-(R3). (R1) is trivially satisfied with C son = 2, since each refined element is split into exactly two elements. Moreover, the union of sons property (R2) holds by definition. The reduction property (R3) in the parameter domain is trivially satisfied and easily transfers to the physical domain with the help of the regularity (2.30) of γ; see [GHP17, Section 5.3] for details. The constant 0 < q son < 1 depends only on d and C γ .
Verification of (R4). The proof of the closure estimate (R4) is found in [MP15, Section 6] for d = 2, and in [Mor16, Section 7] for d = 3. The constant C clos depends only on the dimension d and the polynomials orders (p 1 , . . . , p d ).
Verification of (R5). The proof of the overlay property (R5) is found in [MP15, Section 5] for d = 2. For d = 3, the proof follows along the same lines.
3.3. Space properties. We show that there exist constants C inv > 0 and k loc , k proj ∈ N 0 such that the following properties (S1)-(S3) hold for all T • ∈ T: (S1) Inverse estimate.
where the hidden constants depend only on d and C γ . Thus, it is sufficient to prove (S1) in the parameter domain. In general, V • is not a T • -piecewise tensor-polynomial. However, there is a uniform constant k ∈ N 0 depending only on d and (p 1 , . . . , p d ) such that V • | T is a tensor-polynomial on any k-times refined son T ⊆ T with T ∈ T uni(level( T )+k) : To see this, we use Lemma 2.3, which yields that the number of B-splines B In particular, we can apply a standard scaling argument on T . Since T is the union of all such sons and | T | | T |, this yields that where the hidden constant depends only on d and (p 1 , . . . , p d ). Together, (3.4)-(3.5) conclude the proof of (S1), where C inv depends only on d, C γ , and (p 1 , . . . , p d ).
Verification of (S2). We note that in general, i.e., for arbitrary T-meshes, nestedness of the induced T-splines spaces is not evident; see, e.g., [ which also yields the inclusion X • ⊆ X • .
Verification of (S3). We show the assertion in the parameter domain. For arbitrary but fixed k proj ∈ N 0 (which will be fixed later in Section 3.3 to be k proj := k supp ), we set k loc := k proj + k supp with k supp from Lemma 2.3. Let T • ∈ T, T • ∈ refine( T • ), and V • ∈ X • . We define the patch functions π • (·) and Π • (·) in the parameter domain analogously to the patch functions in the physical domain.
. Then, one easily shows that see [GHP17, Section 5.8]. We see that ω = π k loc • ( T ), and, in particular, also ω := π k proj • ( T ) = π k proj • ( T ). According to Lemma 2.2, it holds that as well as We will prove that which will conclude (S3). To show "⊆", let B •,z be an element of the left set. By Lemma 2.3, this implies that supp( B •,z ) ⊆ π k loc • ( T ). Together with (3.7), we see that supp( B •,z ) ⊆ ( T • ∩ T • ). This proves that no element within supp( B •,z ) is changed during refinement. Thus, the definition of T-spline basis functions proves that B •,z = B •,z . The same argument shows the converse inclusion "⊇". This proves validity of (3.8), and thus (S3) follows.
Verification of (S4)-(S6). Given T • ∈ T, we introduce a suitable Scott-Zhang-type operator J • : H 1 0 (Ω) → X • which satisfies (S4)-(S6). To this end, it is sufficient to construct a corresponding operator J • : H 1 0 ( Ω) → X • in the parameter domain, and to define and With these dual functions, it is easy to define a suitable Scott-Zhang-type operator by (3.12) A similar operator has already been defined and analyzed, e.g., in [BdVBSV14, Section 7.1]. Indeed, the only difference in the definition is the considered index set N act October 3, 2019 18 In the second case, we set V • := 0. In the first case, we apply the Poincaré inequality, whereas we use the Friedrichs inequality in the second case. In either case, we obtain that V • ∈ X • and (2.23)-(2.24) show that The hidden constants depend only on T 0 , (p 1 , . . . , p d ), and the shape of the patch π ksupp • ( T ) resp. the shape of π Finally, we prove (S6). Let again T ∈ T • and v ∈ H 1 0 ( Ω). For all V • ∈ X • that are constant on T , the projection property (3.13) implies that (3.14) Arguing as before and using (3.15), we conclude the proof.

Oscillation properties.
There exists C inv > 0 such that the following property (O1) holds for all T • ∈ T: (O1) Inverse estimate in dual norm. For all W ∈ P(Ω), it holds that |T | 1/d W L 2 (T ) ≤ C inv W H −1 (T ) . Moreover, there exists C lift > 0 such that for all T • ∈ T and all T, T ∈ T • with non-trivial (d − 1)-dimensional intersection E := T ∩ T , there exists a lifting operator L •,E : W | E : W ∈ P(Ω) → H 1 0 (T ∪ T ) with the following properties (O2)-(O4): (O2) Lifting inequality. For all W ∈ P(Ω), it holds that´E L 2 (E) . (O4) H 1 -control. For all W ∈ P(Ω), it holds that ∇L •,E (W | E ) 2 L 2 (T ∪T ) ≤ C lift |T ∪ T | −1/d W 2 L 2 (E) . The properties can be proved along the lines of [GHP17, Section 5.11-5.12], where they are proved for polynomials on hierarchical meshes; see also [Gan17, Section 4.5.11-4.5.12] for details. The proofs rely on standard scaling arguments and the existence of a suitable bubble function. The involved constants thus depend only on d, C γ , and (q 1 , . . . , q d ).

Possible Generalizations
In this section, we briefly discuss several easy generalizations of Theorem 2.9. We note that all following generalizations are compatible with each other, i.e., Theorem 2.9 holds analogously for rational T-splines in arbitrary dimension d ≥ 2 on geometries Ω that are initially non-uniformly meshed if one uses arbitrarily graded mesh-refinement. If d = 2, one can even employ rational T-splines of arbitrary degree p 1 , p 2 ≥ 2.

4.1.
Rational T-splines. Instead of the ansatz space X • , one can use rational hierarchical splines, i.e., where W 0 ∈ Y 0 with W 0 > 0 is a fixed positive weight function. In this case, the corresponding basis consists of NURBS instead of B-splines. Indeed, the mesh properties (M1)-(M4), the refinement properties (R1)-(R5), and the oscillation properties (O1)-(O4) from Section 3 are independent of the discrete spaces. To verify the validity of Theorem 2.9 in the NURBS setting, it thus only remains to verify the properties (S1)-(S6) for the NURBS finite element spaces. The inverse estimate (S1) follows similarly as in Section 3.3 since we only consider a fixed and thus uniformly bounded weight function W 0 ∈ Y 0 . The properties (S2)-(S3) depend only on the numerator of the NURBS functions and thus transfer. To see (S4)-(S6), one can proceed as in Section 3.3, where the corresponding Scott-Zhang-type operator J W 0 • : L 2 (Ω) → X W 0 • now reads J W 0 • v := J • (vW 0 )/W 0 for all v ∈ L 2 (Ω). Overall, the involved constants then depend additionally on W 0 .

4.2.
Non-uniform initial mesh. By definition, T 0 is a uniform tensor-mesh. Instead one can also allow for non-uniform tensor-meshes where (a i,j ) N i j=0 is a strictly increasing vector with a i,0 = 0 and a i,N i = N i , and adapt the corresponding definitions accordingly. In particular, for the refinement, the definition (2.18) of neighbors of an element has to be adapted and depends on T 0 . To circumvent this problem, one can transform the non-uniform mesh via some ϕ to a uniform one, perform the refinement there, and then transform the refined mesh back via ϕ −1 . Indeed, for each i ∈ {1, . . . , d}, there exists a continuous strictly monotonously increasing function ϕ i : [0, N i ] → [0, N i ] that affinely maps any interval [a i,j , a i,j+1 ] to [j, j + 1]. Then, the resulting tensor-product ϕ := ϕ 1 ⊗ · · · ⊗ ϕ d : Ω → Ω defined as in (2.15) is a bijection. To prove the mesh properties (M1)-(M4) and the refinement properties (R1)-(R5), one first verifies them on transformed meshes ϕ( T ) : T ∈ T • as in Section 3.1-3.2, and then transforms these results via γ • ϕ −1 to physical meshes T • . The space properties (S1)-(S6) and the oscillation properties (O1)-(O4) follow as in Section 3.3-3.4. Then, all involved constants depend additionally on T 0 .

Arbitrary grading.
Instead of dividing the refined elements into two sons, one can also divide them into m sons, where m ≥ 2 is a fixed integer. Indeed, such a grading parameter n has already been proposed and analyzed in [Mor16] to obtain a more localized refinement strategy. The proofs hold verbatim, but the constants depend additionally on m.
4.4. Arbitrary dimension d ≥ 2. [Mor17, Section 5.4 and 5.5] generalizes T-meshes, T-splines, and the refinement strategy developed in [Mor16] for d = 3 to arbitrary d ≥ 2. We note that the resulting refinement for d = 2 does not coincide with the refinement from [MP15] that we consider in this work. Instead, the latter leads to a smaller mesh closure. However, Theorem 2.9 is still valid if the refinement strategy from [Mor17, Section 5.4 and 5.5]