Robust approximation error estimates and multigrid solvers for isogeometric multi-patch discretizations

Transcription

1 Robust approximation error estimates and multigrid solvers for isogeometric multi-patc discretizations S. Takacs RICAM-Report

2 Robust approximation error estimates and multigrid solvers for isogeometric multi-patc discretizations Stefan Takacs a a Joann Radon Institute for Computational and Applied Matematics (RICAM), Austrian Academy of Sciences Abstract In recent publications, te autor and is coworkers ave sown robust approximation error estimates for B-splines of maximum smootness and ave proposed multigrid metods based on tem. Tese metods allow to solve te linear system arizing from te discretization of a partial differential equation in Isogeometric Analysis in a single-patc setting wit convergence rates tat are provably robust bot in te grid size and te spline degree. In real-world problems, te computational domain cannot be nicely represented by just one patc. In computer aided design, suc domains are typically represented as a union of multiple patces. In te present paper, we extend te approximation error estimates and te multigrid solver to tis multi-patc case. Keywords: Isogeometric Analysis, multi-patc domains, approximation errors, multigrid metods 1. Introduction Te key idea of Isogeometric Analysis (IgA), [19], is to unite te world of computer aided design (CAD) and te world of finite element (FEM) simulation. Spline spaces, suc as spaces spanned by tensor product B-splines or NURBS, are typically used for geometry representation in standard CAD systems. In classical IgA, bot te computational domain and te solution of te partial differential equation (PDE) are represented by spline functions. More complicated domains cannot be represented by just one suc (tensorproduct) spline function. Instead, te wole domain is decomposed into subdomains, in IgA typically called patces, were eac of tem is represented by its own geometry function. Tis is called te multi-patc case, in contrast to te single-patc case. address: stefan.takacs@ricam.oeaw.ac.at (Stefan Takacs)

3 Concerning te approximation error, in early IgA literature, only its dependence on te grid size as been studied, cf. [19, 1]. In recent publications [2, 25, 12] also te dependence on te spline degree as been investigated. Tese error estimates are restricted to te single-patc case. We will extend te results from [25] on approximation errors for B-splines of maximum smootness to te multi-patc case. As a next step, te linear system resulting from te isogeometric discretization of te PDE as to be solved. Several solvers ave been proposed for te multi-patc case, typically establised solution strategies known from te finite element literature, including direct solvers [4] or non-overlapping and overlapping domain decomposition metods [5, 6, 7], FETI-like approaces (called IETI in te IgA context) [20]. Te solution of local subproblems in suc domain decomposition metods is done wit general direct solvers, fast direct solvers exploiting te tensor product structure, cf. [22], or again iterative solvers, like multigrid or multilevel metods, cf. [3] for multigrid metods in te framework of a IETI solver. To apply multigrid metods directly to te system arizing from a multi-patc discretization, is an appealing alternative. If standard smooters known from finite elements (Jacobi, Gauss Seidel) are used, te extension of te multigrid metods to multi-patc IgA discretizations is straigt-forward. However, it is well known tat teir convergence rates deteriorate dramatically if p is increased, cf. [13, 18, 17]. A robust and efficient multigrid solver for te single-patc case was presented in [16]; alternatives include [11, 17]. Based on a robust inverse inequality and a robust approximation error estimate in a large subspace of te wole spline space (from [25]), it was sown tat mass matrices can be used as robust smooters in tis large subspace. For te oter subspaces, particular smooters ave been proposed, wic can capture te outlier frequencies on te one and and wic still ave tensor product structure on te oter and. Te overall smooter is ten obtained by combining tem by an additive Scwarz type approac. Tat multigrid smooter relies on te tensor-product structure of te mass matrix and is, terefore, restricted to te single-patc case. We will set up instances of tat smooter for eac patc and will combine tem in an additive Scwarz type way to obtain a multi-patc multigrid smooter. Tis smooter will be used in a standard multigrid framework living on te wole multi-patc domain. We will discuss te convergence rates of te multigrid solver and its overall computational complexity. Multigrid metods are typically known as optimal metods, wic means tat teir overall computational complexity grows linearly wit te number of unknowns. If also te dependence in te spline degree is of interest, te best we can expect is tat te multigrid metod is not more expensive tan te computation of te residual, wic requires te multiplication wit te stiffness matrix. In two dimensions, te stiffness matrix as O(Np 2 ) non-zero entries, were N is 2

4 te number of unknowns, p is te spline degree, and O( ) is te Landau notation. So, we call te multigrid metod optimal if we can sow tat its overall complexity is not more tan O(Np d ). Te remainder of te paper is organized as follows. First, te model problem and te discretization are discussed in Section 2. Ten, in Section 3, a robust approximation error estimate for te multi-patc domain is given. Tese results are used in Section 4 to set up a multigrid metod for te multi-patc domain. In Section 5, we give numerical experiments for te multigrid metod and in Section 6, we draw conclusions. 2. Preliminaries In tis paper, we consider te following Poisson model problem. For a given function f, we are interested in te function u solving u = f in Ω, u = 0 on Ω, were Ω R 2 is an open, bounded and simply connected Lipscitz domain wit boundary Ω. Te standard weak form of te model problem reads as follows. Given f L 2 (Ω), find u H 1 0 (Ω) suc tat ( u, v) L2(Ω) = (f, v) L2(Ω) for all v H 1 0 (Ω). (1) Here and in wat follows, L 2 (Ω), H 1 (Ω), H 2 (Ω) and H 1 0 (Ω) are te standard Lebesgue and Sobolev spaces wit standard scalar products (, ) L2(Ω), (, ) H1 (Ω) := (, ) L2(Ω), norms L2(Ω), H1 (Ω), H2 (Ω), and seminorm H1 (Ω). Tis problem is solved wit a standard fully matcing multi-patc isogeometric discretization. For sake of completeness and to introduce a notation, we give te details. For simplicity, we restrict ourselves to te two-dimensional case. Assume tat te domain Ω R 2 consists of K patces, denoted by Ω k for k = 1,..., K suc tat te domain Ω is covered by te patces and te patces do not overlap, i.e., Ω = K Ω k and Ω k Ω l = for any k l, (2) k=1 were for any domain T R 2, te symbol T denotes its closure. Eac of tose patces is represented by a bijective geometry function G k : Ω := (0, 1) 2 Ω k := G k ( Ω) R 2, wic can be continuously extended to te closure of Ω. Analogously to [16], we assume tat te geometry function is sufficiently smoot suc tat te following assumption olds. 3

5 Assumption 1. Tere is a constant C G > 0 suc tat geometry functions G k satisfy C 1 G v L 2( Ω) v G 1 k L 2(Ω k ) C G v L2( Ω) for all v L 2 ( Ω) C 1 G v H r ( Ω) v G 1 k H r (Ω k ) C G v H r ( Ω) for all v H r ( Ω), r {1, 2}. As te dependence on te geometry function is not in te focus of tis paper, unspecified constants migt depend on C G. For any patc Ω k, we denote by K k := {G k ((0, 1) 2 )} = {Ω k } its interior, by { } Γ {{0} (0, 1), {1} (0, 1), (0, 1) {0}, (0, 1) {1}} E k := G k (Γ) : suc tat G k (Γ) Ω its edges and by V k := {G k ({(α, β)}) : (α, β) {0, 1} 2 suc tat G k ({(α, β)}) Ω} its vertices, were in bot cases edges and vertices located on te (Diriclet) boundary of Ω are excluded. T k := K k E k V k denotes all pieces of Ω k. Te following assumption excludes anging vertices. Assumption 2. Te intersection of Ω k and Ω l for k l is eiter (a) empty, (b) one common vertex or (c) te union of one common edge and two common vertices. We define te set of all interiors K := K k=1 K k, edges E := K k=1 E k, vertices V := K k=1 V k, pieces T := K k=1 T k = K E V and observe tat using Assumption 2, we obtain tat te pieces form a partition of Ω: Ω = T and S T = for any S, T T, S T. T T Finally, we assume tat te number of neigbors of eac patc is uniformly bounded. Assumption 3. Assume tat none of te vertices T V contributes to more tan C N patces, i.e., {k : T Ω k } C N. Now, aving a representation of te domain, we introduce te isogeometric function space. For te univariate case, te space of spline functions of degree p N := {1, 2,...} and size = m 1 wit m N is given by S p, := { v C p 1 (0, 1) : v ((j 1),j ] P p for all j = 1,..., m }, were P p is te space of polynomials of degree p and C p 1 (0, 1) is te space of all p 1 times continuously differentiable functions. 4

6 We denote te standard basis for S p,, as introduced by te Cox-de Boor formula, cf. [10], by Φ p, := ( B (i) p, )n i=1, were n = m + p is te dimension of te spline space. Note tat only te first basis function B (1) p, = max{0, (1 x/)p } contributes to te left boundary. Analogously, only te last basis function contributes to te rigt boundary. We assign corresponding Greville points 0 = x (1) p, < x(2) p, < < x(n) p, = 1 to te basis functions. On te parameter domain Ω, we introduce for eac patc tensor-product B- spline functions V k := S p, S p, (3) wit basis Φ (i) k := ( B are given by B (i+n (j 1)) k (x, y) = k )n2 i=1 B (i) p,, were te basis functions and te Greville points (j) (j 1)) (x) B (y) and x(i+n = ( x (i) p, k p,, x(j) p, ). (4) For sake of simplicity of te notation, we do not indicate te dependence of p,, or m on te patc index k and te spacial direction. On te pysical domain Ω k, we define te ansatz functions using te pull-back principle V k := {u H 1 (Ω k ) : u G k V k } (5) and obtain te basis by Φ k := (B (i) k )n2 i=1 and B(i) k := points by x (i) k = G k( x (i) k ). (i) B k G 1 k and te Greville We require tat te function spaces are fully matcing on te interfaces. Assumption 4. For any T E being a common edge of te patces Ω k and Ω l (i.e., T Ω k Ω l ), we assume tat te basis functions of te two patces and te corresponding Greville points matc, i.e., for all i tere is some j suc tat B (i) k T = B (j) l T and x (i) k = x(j) l (6) olds, were T is te trace operator. Te multi-patc function space V is given by V := {u H0 1 (Ω) : u Ωk V k for k = 1,..., K}. For tis space, we introduce a set of global basis functions by Φ := {φ x (i) k : k {1,..., K}, i {1,..., n 2 } suc tat x (i) k were te basis functions φ x V are suc tat φ x Ωk = Ω}, (7) { (i) B k were i is suc tat x (i) k = x if x Ω k for all k = 1,..., K. 0 if x Ω k 5

7 Note tat te condition x (i) k Ω in (7) excludes te basis functions assigned to te boundary Ω and guarantees tat te omogenous Diriclet boundary conditions are satisfied. By numbering te basis functions in Φ arbitrarily, we obtain Φ = {φ i : i = 1,..., N} and a basis Φ := (φ i ) N i=1 of V. Note tat by construction only te basis functions wose Greville points are located on an edge (or te corresponding vertices) contribute to tat edge and only te basis function wose Greville point is located on an vertex contributes to tat vertex. So, for any piece T T, we collect te corresponding functions: Φ (T ) := {φ x Φ : x T }. We use a standard Galerkin sceme to discretize (1) and obtain te following discretized problem: Find u V suc tat ( u, v ) L2(Ω) = (f, v ) L2(Ω) for all v V. (8) Using te basis Φ, we obtain a standard matrix-vector problem: Find u R N suc tat A u = f. (9) Here and in wat follows, A := [( φ i, φ j ) L2(Ω)] N i,j=1 is te standard stiffness matrix, M := [(φ i, φ j ) L2(Ω)] N i,j=1 is te standard mass matrix, u = [u i ] N i=1 is te coefficient vector representing u wit respect to te basis Φ, i.e., u = N i=1 u iφ i, and f = [(f, φ i ) L2(Ω)] N i=1 is te coefficient vector obtained by testing te rigt-and-side functional wit te basis functions. Before we proceed, we introduce a convenient notation. Definition 5. Any generic constant c > 0 used witin tis paper is understood to be independent of te grid size, te spline degree p and te number of patces K, but it migt depend on te sape of Ω, and on te constants C G and C N. We use te notation a b if tere is a generic constant c suc tat a cb and te notation a b if a b and b a. For symmetric positive definite matrices A and B, we write A B if u Au u Bu for all vectors u. Te notations A B and A B are defined analogously. Following te standard line of arguments, te Lax Milgram lemma and Friedrics inequality indicate existence and uniqueness of a solution u H 1 0 (Ω) for te continuous problem (1) and of a solution u V for te discrete problem (8). Cea s lemma yields u u 2 H 1 (Ω) inf u v 2 H v V 1 (Ω), i.e., tat te discretization error is bounded by a constant times te approximation error, wic motivates to discuss approximation error estimates in te next section. 6

8 3. Robust multi-patc spline approximation In tis paper, we extend te robust L 2 H 1 and H 1 H 2 -approximation error estimates from [25] to multi-patc domains. For tis purpose, we introduce a projector into te spline space wic is interpolatory on te boundary. Tis is first done in te one dimensional case (Section 3.1) and ten extended to te two-dimensional case (Section 3.2). Based on tat projector, a projector for multi-patc domains is introduced (Section 3.3). All of te projectors satisfy te usual p-robust approximation error estimates Te one dimensional case First, we define an augmented H 1 -scalar product. Definition 6. Te scalar product (, ) H 1 D (0,1) is given by (u, v) H 1 D (Ω) := (u, v) H 1 (0,1) + u(0)v(0). (10) As te scalar product does not ave a kernel, it induces a norm u 2 := HD 1 (0,1) (u, u) H 1 D (0,1) and te following definition introduces an unique projector. Definition 7. Te projector Π p, : H 1 (0, 1) S p, is te HD 1 -ortogonal projection, i.e., for any u H 1 (0, 1), te spline u p, := Π p, u satisfies (u u p,, v p, ) H 1 D (0,1) = 0 for all v p, S p,. (11) We observe tat te original function and te spline function coincide on bot boundary points and tat tey are ortogonal in (, ) H1 (0,1). Lemma 8. For all u H 1 (0, 1), te spline u p, := Π p, u satisfies and u(0) = u p, (0), u(1) = u p, (1) (12) (u u p,, v p, ) H1 (0,1) = 0 for all v p, S p,. (13) Proof. Te first statement is obtained by plugging v(x) := 1 into (11). For te second statement, we plug v(x) := x into (11) and obtain 0 = (u u p,, v) H 1 D (0,1) = u(0) u p, (0) + = u(1) u p, (1). 1 0 u (x) u p,(x) dx For te last statement (13), observe tat (11) togeter wit (10) yields (u u p,, v p, ) H 1 (0,1) + (u(0) u p, (0))v p, (0) = 0 for all v p, S p,, wic sows togeter wit (12) te desired result. 7

9 From (13), we immediately obtain te H 1 -stability: Π p, u H 1 (0,1) u H 1 (0,1). (14) Moreover, we obtain te usual approximation error estimates. Teorem 9. For all u H 2 (0, 1), grid sizes and spline degrees p N, we obtain u Π p, u H 1 (0,1) 2 u H 2 (0,1). (15) Proof. We ave u Π p, u H 1 (0,1) = inf up, S p, u u p, H 1 (0,1) because Π p, minimizes te H 1 -seminorm. For te case < p 1, te estimate directly follows from [25, Teorem 7.3]. For > p 1, we use tat te space of global polynomials is a subspace of te spline space. So, [23, Teorem 3.17] yields (for M = 1, Ω = Ω 1 = (0, 1), k 1 = s 1 = 1) u Π p, u H1 (0,1) 2 1 (p(p + 1)) 1/2 u H2 (0,1). Using p 1 <, we obtain also for tis case te desired result. Teorem 10. For all u H 1 (0, 1), grid sizes and spline degrees p N, we obtain u Π p, u L2(0,1) 2 u H 1 (0,1). (16) Proof. Tis estimate is sown by a classical Aubin Nitsce duality trick. Let v H 2 (0, 1) suc tat v(0) = v(1) = 0 and v = u Π p, u. Ten we obtain using integration by parts (te boundary terms vanis due to Lemma 8) tat u Π p, u L2(0,1) = (u Π p,u, u Π p, u) L2(0,1) u Π p, u L2(0,1) = (u Π p,u, v ) L2(0,1) v L2(0,1) = (u Π p,u, v) H1 (0,1) v H 2 (0,1) sup w H 2 (0,1) (u Π p, u, w) H1 (0,1). w H 2 (0,1) Using Teorem 9, we obtain furter u Π p, u L2(0,1) 2 sup w H 2 (0,1) (u Π p, u, w) H 1 (0,1). w Π p, w H1 (0,1) Wit te ortogonality relation (13), te Caucy-Scwarz inequality, and te stability estimate (14), we finally conclude u Π p, u L2(0,1) 2 sup w H 2 (0,1) (u Π p, u, w Π p, w) H1 (0,1) w Π p, w H 1 (0,1) 2 u Π p, u H1 (0,1) 2 u H1 (0,1). Te projector can be represented by a dual basis. Lemma 11. For all grid sizes and spline degrees p N, tere are dual basis functions λ (i) p, S p, for i = 1,..., n suc tat Π p, u = n (u, λ (i) p, ) (i) HD 1 (0,1) B p, for all u H 1 (0, 1). i=1 8

10 Proof. Let u H 1 (i) (0, 1) be arbitrary but fixed. As ( B we can expand Π p, u = n i=1 u i 0 = (u Π p, u, B (j) p, ) H 1 D (0,1) B (i) p, = (u, p, )n i=1 is a basis of S p,,. By plugging tis into (13), we obtain B (j) p, ) H 1 D (0,1) n (i) (j) u i ( B p,, B p, ) HD }{{ 1 (0,1), } a i,j := for j = 1,..., n. As te HD 1 (Ω)-scalar product induces a norm (and not only a seminorm), te stiffness matrix [a i,j ] n i,j=1 is non-singular. So, tere is an inverse matrix [w i,j ] n i,j=1 and we obtain u i = n w i,j (u, j=1 wic finises te proof. ( (j) B p, ) HD 1 (0,1) = u, n j=1 i=1 w i,j B(j) p, } {{ } λ (i) p, := ) H 1 D (0,1), 3.2. Te two-dimensional case For te two-dimensional case on te parameter domain Ω = (0, 1) 2, we define te projector Π k : H 2 ( Ω) V k using te idea of tensor-product projection. First, we define te following two projectors on u H 2 ( Ω): (Π x p,u)(, y) := Π p, u(, y) for all y (0, 1), (Π y p, u)(x, ) := Π p,u(x, ) for all x (0, 1), and observe tat tese operators commute. Lemma 12. We ave Π x p, Πy p, = Πy p, Πx p,. Proof. Let ξ, η and ξη be te corresponding partial derivatives. Lemma 11 guarantees te existence of a dual bases. So, n ( 1 ) Π x p,u(x, y) = ξ u(ξ, y) ξ λ (i) p, (ξ) dξ + u(0, y)λ(i) p, (0) B (i) p, (x), i=1 and straigt forward computations yield Π y p, Πx p,u(x, y) n n ( 1 = i=1 j= ξη u(ξ, η) ξ λ (i) p, (ξ) ηλ (j) p, (η) dξ dη ξ u(ξ, 0) ξ λ (i) p, (ξ) λ(j) p, (0) dξ + 1 ) + u(0, 0) λ (i) p, (0) λ(j) p, (0) B(i) (j) p, (x) B p, (y). 0 η u(0, η) λ (j) p, (0) ηλ (i) p, (η) dη Observe tat tis term is symmetric in x and y. So Π y p, Πx p, = Πx p, Πy p,. 9

11 As Π x p, Πy p, = Πy p, Πx p,, te projector Π k := Π x p,π y p, (17) maps into V k, te intersection of te image spaces of tese two projectors. Teorem 13. For all u H 2 ( Ω), grid sizes and spline degrees p N, we obtain u Π k u H 1 ( Ω) 2 u H 2 ( Ω). (18) Proof. First we sow x (u Π k u) 2 L 2( Ω) 2( xxu 2 L 2( Ω) + xyu 2 L 2( Ω) ), were x, xx and xy are te corresponding partial derivatives. Using Π k = Π x p, Πy p,, te triangle inequality, te H1 -stability of Π p,, (14), we obtain x (u Π k u) L2( Ω) x(u Π x p,u) L2( Ω) + xπ x p,(u Π y p, u) L 2( Ω) Using Teorems 9 and 10, we obtain furter x (u Π x p,u) L2( Ω) + x(u Π y p, u) L 2( Ω). x (u Π k u) L2( Ω) 2 xx u L2( Ω) + 2 xy u L2( Ω) 2( xx u 2 L 2( Ω) + xyu 2 L 2( Ω) )1/2. Using Π k = Π y p, Πx p,, we obtain using te same arguments also wic yields y (u Π k u) L2( Ω) 2( xyu 2 L 2( Ω) + yyu 2 L 2( Ω) )1/2, u Π k u 2 H 1 ( Ω) = x(u Π k u) 2 L 2( Ω) + y(u Π k u) 2 L 2( Ω) 4 2 ( xy u 2 L 2( Ω) + 2 xyu 2 L 2( Ω) + yyu 2 L 2( Ω) ) = 42 u 2 H 2 ( Ω) and finises te proof. Teorem 14. For all u H 2 ( Ω), we obtain tat u and Π k u coincide at te corners of Ω and Π k u, restricted on any edge Γ of Ω, coincides wit te projector Π p,, applied to te restriction of u to tat edge. So, e.g., for Γ = {0} (0, 1), olds. ( Π k u)(0, ) = Π p, (u(0, )) Proof. Tis is a direct consequence of Lemma 8 and (17). 10

12 3.3. Te multi-patc case Assume to ave a fully matcing multi-patc discretization as introduced in Section 2 and let K H 2 (Ω) := {u H 1 (Ω) : u Ωk H 2 (Ω k )}, u 2 H 2 (Ω) := u 2 H 2 (Ω k ) be a usual bent Sobolev space wit corresponding norm. We obtain tat te projectors Π k are compatible. Lemma 15. For eac u H 2 (Ω) H 1 0 (Ω), tere is exactly one u V suc tat u G k = Π k (u G k ) for all k = 1,..., K. (19) Proof. First observe tat (19) specifies te value of u for all patces Ω k and tat te definition coincides wit te pull-back definition (5) of V k. So, we obtain uniqueness and we obtain tat te restriction of u to any patc Ω k yields a function in V k. It remains to sow tat u H0 1 (Ω), i.e., tat it is continuous and tat it satisfies te Diriclet boundary conditions. Teorem 14 implies tat te projector Π k is interpolatory on vertices, so u is continuous at te vertices. For edges, Teorem 14 implies tat te projector Π k coincides wit te univariate interpolation, so u is also continuous across te edges. Tis sows continuity. Finally, observe tat u satisfies by assumption te omogenous Diriclet boundary conditions. Again, on te boundary Π k u coincides wit te univariate interpolation. As u 0 can be represented exactly by means of splines, we obtain tat te univariate interpolation and, terefore, also u vanis on te boundary (satisfies te Diriclet boundary conditions). So, we define te operator Π : H 2 (Ω) H 1 0 (Ω) V suc tat k=1 ( Π u) G k = Π k (u G k ) for all k = 1,..., K. (20) Tis projector Π satisfies a standard error estimate. Teorem 16. For all u H 2 (Ω), grid sizes and spline degrees p N, we obtain u Π u H 1 (Ω) u H 2 (Ω). Proof. Assumption 1 yields w H 1 (Ω k ) w G k H 1 ( Ω) and w G k H 2 ( Ω) w H2 (Ω k ). Using (20) and Teorem 13, we obtain u Π u H1 (Ω k ) (u Π u) G k H1 ( Ω) u G k Π k (u G k ) H1 ( Ω) w G k H2 ( Ω) u H 2 (Ω k ). By taking te sum over all patces, we obtain te desired result. 11

13 Obviously, te projector Π is not te H 1 -ortogonal projector, but te estimate for te H 1 -ortogonal projection immediately follows. Note tat H 1 (Ω) is a norm on V, so te following definition guarantees uniqueness. Definition 17. Te projector Π : H 1 (Ω) V is te H 1 -ortogonal projection, i.e., for any u H 1 (Ω), te spline u := Π u satisfies (u u, v ) H 1 (Ω) = 0 for all v V. Teorem 18. For all u H 2 (Ω), grid sizes and spline degrees p N, we obtain u Π u H1 (Ω) u H2 (Ω). Proof. Te minimization property of te projector and Teorem 16 yields Te Poincare inequality yields furter u Π u H1 (Ω) u H2 (Ω). v Π v H1 (Ω) ( v H2 (Ω) + (v, 1) L2(Ω)) for all v H 2 (Ω), so also for v := u (u, 1) L2(Ω). As (I Π )(u (u, 1) L2(Ω)) = (I Π )u and u (u, 1) L2(Ω) H 2 (Ω) = u H 2 (Ω), tis finises te proof. Using a standard full elliptic regularity result, we obtain also a corresponding L 2 H 1 -estimate. Assumption 19. For every f L 2 (Ω), te solution u H 1 0 (Ω) of te model problem (1) satisfies u H 2 (Ω) and u H 2 (Ω) C R f L2(Ω). Suc an estimate is satisfied for domains wit smoot boundary, cf. [21], and for convex polygonal domains, cf. [8, 9]. In all cases, te constant C R only depends on te sape of te computational domain Ω, so C R 1. Teorem 20. Assume to ave Assumption 19. Ten, for all u H 2 (Ω), grid sizes and spline degrees p N, we obtain u Π u L2(Ω) u H1 (Ω). (21) Proof. Tis estimate is sown by a classical Aubin Nitsce duality trick. Let v H 1 0 (Ω) be suc tat (v, w) H 1 (Ω) = (u Π u, w) L2(Ω) for all w H 1 (Ω). Observe tat Assumption 19 implies v H 2 (Ω) and v H 2 (Ω) = v H 2 (Ω) u Π u L2(Ω). Using tis and Teorem 18, we obtain u Π u L2(Ω) = (u Π u, u Π u) L2(Ω) u Π u L2(Ω) (u Π u, v) H 1 (Ω) v H2 (Ω) (u Π u, w) H sup 1 (Ω) w H 2 (Ω) w H2 (Ω) sup w H 2 (Ω) (u Π u, w) H 1 (Ω). w Π w H1 (Ω) 12

14 Te H 1 -ortogonality of te projector and te Caucy-Scwarz inequality imply wic was to sow. u Π u L2(Ω) sup w H 2 (Ω) (u Π u, w Π w) H1 (Ω) w Π w H 1 (Ω) u Π u H1 (Ω) u H1 (Ω), 4. A multigrid solver In tis section, we develop a robust multigrid metod for solving te linear system (9). We assume to ave a ierarcy of grids obtained by uniform refinement. For two consecutive grid levels (H = 2), we ave V H V, i.e., nested discretizations. For tose, we define IH to be te canonical embedding from V H into V and te restriction matrix I H to be its transpose. Starting from an initial approximation u (0), te next iterate u(1) is obtained by te following two steps: Smooting: For some fixed number ν of smooting steps, compute ( ) u (0,µ) := u (0,µ 1) + τl 1 f A u (0,µ 1) for µ = 1,..., ν, (22) were u (0,0) := u (0). Te coice of te matrix L and of te damping parameter τ > 0 will be discussed below. Coarse-grid correction: Compute te defect and restrict it to te coarser grid: ( ) r (1) H := IH f A u (0,ν). Compute te correction p (1) H by approximately solving te coarse-grid problem Prolongate p (1) H A H p (1) H = r(1) H. (23) and add te result to te previous iterate: u (1) := u(0,ν) + I H p (1) H. If te problem (23) on te coarser grid is solved exactly (two-grid metod), te coarse-grid correction is given by ( ) u (1) := u(0,ν) + IH A 1 H IH f A u (0,ν). (24) In practice, te problem (23) is approximately solved by recursively applying one step (V-cycle) or two steps (W-cycle) of te multigrid metod. On te coarsest grid level, te problem (23) is solved exactly using a direct metod. 13

15 4.1. An additive smooter For te single-patc case, we ave proposed te subspace-corrected mass smooter in [16]. For te multi-patc case, we propose L := T T P T L T P T, (25) were P T and L T are cosen as follows. Te matrices P T represent te canonical embedding from Φ (T ) in Φ. By construction, tis is a full-rank N Φ (T ) binary matrix, were eac column as exactly one non-zero entry. L T are local smooters. For T K, we coose L 1 T to be te subspacecorrected mass smooter. For T E V, we coose i.e., L 1 T be be an exact solver. L T := P T A P T, (26) Tis coice of L T is feasible because for any T E, te matrix L T as a dimension of O(n) and for any T E te matrix L T is just a 1-by-1 matrix. Note tat te construction of te subspace corrected mass smooter requires for eac patc tat m > p, i.e., tat te number of intervals per direction is larger tan p; for patces were tis is not satisfied, one can coose L T := P T A PT. Note tat te matrices P T realize a partition of te degrees of freedom (like a patc-wise Jacobi iteration), so L is a (in general: reordered) block-diagonal matrix tat can be inverted by inverting te blocks. So, we obtain u (0,µ) := u (0,µ 1) + τ T T P T L 1 T P T } {{ } L 1 = ( f A u (0,µ 1) In [16], we ave sown for te te single-patc case tat a multigrid solver wit te subspace-corrected mass smooter converges robustly. Here, we recall tese results, were te presentation of te results is sligtly altered suc tat we can prove te results for te multi-patc case smootly in te sequel. Te following teorem is a sligt variation of te standard multigrid teory as developed by Hackbusc [14]. Teorem 21. Assume tat te conditions of Teorem 20 old and tat L satisfies ca L c(a + 2 M ). (27) Ten te two-grid metod converges for te coice τ (0, c ] and ν > ν 0 := τ 1 c(1 + 4c 2 A ) wit rate q = ν 0/ν, i.e., u (1) ). A 1 f A + 2 M q u (0) A 1 f A + 2 M, were c A is te constant idden in te estimate in Teorem

16 Proof. We use [17, Teorem 3]. First, observe tat Teorem 20 implies (I Π H )u 2 M c 2 AH 2 u 2 A 4c 2 A 2 u 2 A, were Π H is te A -ortogonal projector or, equivalently, te H 1 -ortogonal projector. Because projectors are stable, we also obtain and using (27) also (I Π H )u 2 A u 2 A, (I Π H )u 2 L c (I Π H )u 2 A + 2 M c(1 + 4c 2 A) u 2 A, i.e., te first condition (approximation error estimate) in [17, Teorem 3] wit C A = c(1 + 4c 2 A ). Now, observe tat te first inequality in (27) coincides wit second condition (inverse inequality) in [17, Teorem 3] wit C I = c 1. Finally, [17, Treorem 3] sows te desired statement. In [17, Teorem 4], it was sown tat under te assumptions of [17, Teorem 3] also a W-cycle multigrid metod converges. Now, we sow tat te conditions of Teorem 21 old patc-wise for te subspace-corrected mass smooter. For tis purpose, we define te piece-local stiffness and mass matrices by A T := P T A P T and M T := P T M P T. Remember tat te domain Ω consists of te patces Ω k for k = 1,..., K. So, we define A k and M k to be te stiffness and mass matrix obtained by restricting te integration to te patces, i.e., A k := [( φ i, φ j ) L2(Ω k )] N i,j=1 and M k := [(φ i, φ j ) L2(Ω k )] N i,j=1 and observe K K A = A k and M = M k. (28) k=1 Analogously to A T and M T, we define A k,t := P T A k P T and M k,t := P T M k P T. Finally, we define stiffness and mass matrices on te parameter domain by k=1 Â k := [( (φ i G k ), (φ j G k )) L2( Ω) ]N i,j=1, Mk := [(φ i G k, φ j G k ) L2( Ω) ]N i,j=1, K K Â := Â k, M := M k, Â k,t := P T Â k PT, and Mk,T := P T Mk PT k=1 k=1 and observe tat tey are similar to te corresponding matrices on te pysical domain. 15

17 Lemma 22. We ave A k Âk, A Â, A T ÂT, A k,t Âk,T, and analogous results for M k, M, M T, and M k,t. Proof. We ave using Assumption 1 u 2 A k = u 2 H 1 (Ω k ) u G k 2 H 1 ( Ω) = u 2 Â k, wic sows te first statement. Te second one is obtained by summing over k, te tird one is obtained as A Â implies A T = PT A P T PT ÂPT = ÂT, and te fourt is obtained as A k Âk implies A k,t = PT A kp T PT ÂkPT = Â k,t. Te statements for te mass matrix are completely analogous. Te following Lemma follows directly from wat as been sown in [16, Section 4.2]. Lemma 23. For all grid sizes and spline degrees p N, te relation A T L T A T + 2 M T olds for all T T. (29) Proof. For T T, te estimate as been sown in te proofs of [16, Lemmas 8 and 9]. For T E V, we ave L T = A T, so te desired statement immediately follows. Now we sow tat L, as defined in (25), satisfies te condition of Teorem 21 wit c being robust and wit c depending linearly on te spline degree, i.e., We sow tis by sowing A L p(a + 2 M ). (30) A T T P T A T P T, (31) P T (A T + 2 M T )P T, (32) P T A T PT L T T T T P T (A T + 2 M T )PT p(a + 2 M ). (33) T T Note tat (32) follows directly from (25) and Lemma 23. Te oter two inequalities are sown in te sequel. Lemma 24. For all grid sizes and spline degrees p N, te inequality (31) olds. Proof. Using T T P T PT = I, we obtain u 2 A = P T PT 2 u = (PT A P S PS u, PT u ). A T T S T T T 16

18 Note tat Assumption 3 implies tat for any T T, te number of S T suc tat P T A P S 0 is bounded. So, we obtain using te Caucy-Scwarz inequality tat u 2 A PT u 2 PT A P T = PT u 2 A T, T T T T wic finises te proof. For sowing (33), we need some trace estimates. Te following lemma is a standard result, wic is given to keep te paper self-contained. Lemma 25. u(0) 2 u 2 L 2(0,1) + u L 2(0,1) u H 1 (0,1) olds for all u H 1 (0, 1). Proof. Let u H 1 (0, 1) be arbitrary but fixed and note tat u is continuous. We ave for all t (0, 1) tat olds. So, u(0) 2 = t 0 u(s)u (s)ds + u(t) 2 u(0) 2 = 1 t u(s)u (s)ds + u(t) 2 dt 1 0 u L2(0,s) u L2(0,s)dt + u 2 L 2(0,1) u L2(0,1) u L2(0,1)dt + u 2 L 2(0,1) = u L 2(0,1) u L2(0,1) + u 2 L 2(0,1), wic finises te proof. Observe tat on eac patc Ω k, we obtain te following stability estimates. Lemma 26. For all k {1,..., K} and all T V k, te inequality olds. P T (A k,t + 2 M k,t )P T p(a k + 2 M k ) Proof. Let k and T be arbitrary but fixed. Note tat te parameter domain was defined to be Ω = (0, 1) 2. Assume witout loss of generality tat tat vertex T corresponds to te vertex T = (0, 0) on te parameter domain. Define Γ := {0} (0, 1) to be an edge tat touces tat vertex. Define on te parameter domain te norms û 2 Q( Ω) := û 2 H 1 ( Ω) + 2 û 2 L 2( Ω), û 2 Q( Γ) := p 1 û 2 H 1 ( Γ) + p 1 û 2 L 2( Γ), (34) and observe tat Lemma 22 implies u 2 A k + 2 M k u 2 Â k + 2 M k = û 2 Q( Ω), (35) 17

19 were ere and in wat follows û := u G k. Now we compute PT u A k,t and PT u M k,t. Note tat tere is just one basis function assigned to te vertex. Due to te tensor-product structure, tis basis function is B (1) k (1) (1) (x, y) = B p, (x) B p, (y) = max{0, (1 x/)p } max{0, (1 y/) p }. As (1) B k (0, 0) = 1 and all oter basis functions vanis on (0, 0), we obtain PT u 2 A k,t PT u 2 (1) = 2 B Â k,t p, 2 (1) H 1 (0,1) B p, 2 L û 2(0,1) (0, 0) 2, PT u 2 M k,t PT u 2 Mk,T (1) = B p, 4 L û 2(0,1) (0, 0) 2. (36) Straigt-forward computations yield (1) B p, 2 L = 2(0,1) 2p + 1 p (1) and B p, 2 H 1 (0,1) = p 2 (2p 1) p. (37) So, P T u 2 A k,t + 2 M k,t Observe tat Lemma 25, and ab a 2 + b 2 imply ( 2 ) p p + 2 p 2 û (0, 0) 2 û (0, 0) 2. (38) û (0, 0) 2 û 2 L 2( Γ) + û L2( Γ) û H 1 ( Γ) (1 + p 1 ) û 2 L 2( Γ) + p 1 û 2 H 1 ( Γ) û 2 Q( Γ). (39) Now, we sow û 2 Q( Γ) p û 2 Q( Ω). (40) Using Lemma 25, we immediately obtain û (0, y) 2 û (, y) 2 L 2(0,1) + û (, y) L2(0,1) û (, y) H1 (0,1). By integrating over y, using te Caucy Scwarz inequality and ab a 2 + b 2, we obtain furter 1 û 2 L 2(Γ) û (, y) 2 L + û 2(0,1) (, y) L2(0,1) û (, y) H 1 (0,1)dy Analogously, we obtain 0 û 2 L 2(Ω) + û L2(Ω) x û L2(Ω) (1 + 1 ) û 2 L 2(Ω) + xû 2 L 2(Ω) û 2 H 1 (Ω) + 1 û 2 L 2(Ω). (41) û 2 H 1 (Γ) = yû 2 L 2(Γ) yû 2 L 2(Ω) + yû L2(Ω) y x û L2(Ω). 18

20 Using a standard inverse inequality, cf. [23, Teorem 3.91], and ab a 2 + b 2, we obtain furter û 2 H 1 (Γ) yû 2 L 2(Ω) + p2 1 y û L2(Ω) x û L2(Ω) p 2 1 û 2 H 1 (Ω). (42) By combining (34), (41) and (42), we obtain û 2 Q( Γ) p û 2 H 1 (Ω) + p 2 û 2 L 2(Ω) = p û 2 Q( Ω), wic finises te proof of (40). Using (38), (39), (40) and (35), we obtain P T u 2 A k,t + 2 M k,t û (0, 0) 2 û 2 Q( Γ) p û 2 Q( Ω) p u 2 A k + 2 M k, wic finises te proof. Lemma 27. For all k {1,..., K} and all T E k, te inequality olds. P T (A k,t + 2 M k,t )P T p(a k + 2 M k ) Proof. Let k and T be arbitrary but fixed. Note tat te parameter domain was defined to be Ω = (0, 1) 2. Assume witout loss of generality tat tat edge T corresponds to te edge Γ := {0} (0, 1) on te parameter domain. We define on te parameter domain te norms û 2 and û Q( Ω) 2 as in (34) and use Q( Γ) again û := u G k. Due to te tensor-product structure, te basis functions contributing to te edge ave te form B (i) k (1) (i) (x, y) = B p, (x) B p, (y) = max{0, (1 x/)p } (i) B p, (y) for i = 1,..., n. Note tat among tose, te first and te last one are associated to te corresponding vertices (0, 0) and (0, 1). Only te basis functions in between belong to Φ (T ). Analogously to (36), we ave P T u 2 A k,t P T u 2 Â k,t (1) = û û (0, 0) B p, û (n) (0, 1) B p, 2 (1) B H 1 ( Γ) p, 2 L 2(0,1) (1) + û û (0, 0) B p, û (n) (0, 1) B p, 2 (1) B L 2( Γ) p, 2 H 1 (0,1), (43) P T u 2 M k,t P T u 2 Mk,T (1) = û û (0, 0) B p, û (n) (0, 1) B p, 2 (1) B L 2( Γ) p, 2 L, (44) 2(0,1) 19

21 were superfluous contributions from te vertices ave been subtracted. Again, using te triangle inequality and (37), we obtain PT u 2 A k,t + 2 M k,t = p û (1) û (0, 0) B p, û (n) (0, 1) B p, 2 H 1 ( Γ) + p û (1) û (0, 0) B p, û (n) (0, 1) B p, 2 L 2( Γ) p û 2 H 1 ( Γ) + p û 2 L 2( Γ) + û (0, 0) 2 + û (0, 1) 2. Using te definition of û Q( Γ) and (39), we obtain furter P T u 2 A k,t + 2 M k,t û 2 Q( Γ) and using (40) and (35) finally wic finises te proof. P T u 2 A k,t + 2 M k,t p û 2 Q( Ω) p u 2 A k + 2 M k, Lemma 28. For all grid sizes and spline degrees p N, te inequality (33) olds. Proof. Let k be arbitrary but fixed. Observe tat T k = K k E k V k and tat K k = {Ω k }. Certainly, te number of edges and te number of vertices do not exceed 4 (tey are smaller if te patc Ω k contributes to te (Diriclet) boundary), so E k V k 8 olds. Analogously to te proof of Lemma 24, we obtain u 2 A k,ωk + 2 M k,ωk u 2 A k + 2 M k + and, as T k = {Ω k } E k V k, T T k u 2 A k,t + 2 M k,t u 2 A k + 2 M k + T E k V k u 2 A k,t + 2 M k,t Using Lemmas 26 and 27 and E k V k 8, we obtain also T T k u 2 A k,t + 2 M k,t p u 2 A k + 2 M k. T E k V k u 2 A k,t + 2 M k,t. By adding tis up over all patces, we obtain using (28) tat K u 2 A T + 2 M T = u 2 A k,t + 2 M k,t T T k T T k=1 = p u 2 A+ 2 M, K p u 2 A k + 2 M k k=1 wic finises te proof. 20

22 Lemma 29. For all grid sizes, and spline degrees p N, te inequality (30) olds. Proof. Tis is just te combination of te Lemmas 24, 23 and 28. Based on tis, we can sow tat te multigrid solver converges robustly if O(p) smooting steps are applied. Teorem 30. Tere are constants c 1 and c 2 tat do not depend on te grid size, te spline degree p, and te number of patces K (but may depend on C G, C N, or C R ) suc tat τl 1 A 1 (45) for all τ (0, c 1 ] and te proposed two-grid metod converges for any τ satisfying (45) and any coice of te number of smooting steps ν > ν 0 := pτ 1 c 2 wit a convergence rate q = ν 0 /ν, i.e., u (1) A 1 f A + 2 M ν 0 ν u(0) A 1 f A + 2 M. Proof. We use Teorem 21, wose condition is sown by Lemma 29. Due to [17, Teorem 4], we know tat also te W-cycle multigrid metod converges. Remark 1. Because te computational costs for te (exact) solvers for te edges and te vertices are negligible, we obtain tat te overall computational complexity coincides wit tat of te subspace corrected mass smooter, as computed in [16, Section 5.4], multiplied wit te number of patces. So, we obtain as follows: setup costs: O(Np + Kp 6 ) application costs: O(Np + Kp 4 ), were N = Kn 2 is te number of unknowns, K is te number of patces and p is te spline degree. We obtain for p n tat te smooter is asymptotically not more expensive tan te computation of te residual. Te remaining parts of te multigrid solver (restriction, prolongation, solving on te coarsest grid) can also be done in optimal time, cf. [16, Section 5.4]. As we can prove convergence only if O(p) smooting steps are applied, tis does not sow tat te overall metod as optimal complexity. However, in Section 5, we will see tat te metod works well for fixed ν, so in practice te metod seems to be optimal. In te next section, we construct a multigrid solver were we can prove optimal complexity An optimal variant of te additive smooter First note tat te smooter L T is a robust preconditioner for A T + 2 M T. 21

23 Teorem 31. For all grid sizes and spline degrees p N, we obtain te relation L T A T + 2 M T for all T T. Proof. First note tat Lemma 23 states A T L T A T + 2 M T. So, it remains to sow tat 2 M T L T olds for all T T. (46) For T V, observe tat from (36) and (37), it follows tat p 2 2 M k,t A k,t. By summing up, we obtain p 2 2 M T A T, wic sows (46) as L T = A T and p 1. For T E, observe tat te combination of (43) and (44) yields PT (1) u Ak,T û û (0, 0) B p, û (m) (0, 1) B p, 2 (1) B L 2( Γ) p, 2 H 1 (0,1) (1) B p, 2 (1) H 1 (0,1) B p, 2 L P 2(0,1) T u Mk,T. Again, using (37), we obtain p 2 2 M k,t A k,t and by summing up, we obtain p 2 2 M T A T, wic sows (46) as L T = A T and p 1. For T K, te proof follows an idea by C. Hofreiter [15]. Note tat, in [16, Section 4.2], we ave constructed te smooter L T on subspaces of te spline space obtained by a stable splitting of te wole spline space S = S p, (for te particular patc) into subspaces S α. In two dimensions, we ave defined σ := 12 2 and L 00 = (1 + 2σ)M 0 M 0, L 01 = M 0 ((1 + σ)m 1 + K 1 ), L 10 = ((1 + σ)m 1 + K 1 ) M 0, L 11 = M 1 M 1 + K 1 M 1 + M 1 K 1, were M 0, M 1, K 0 and K 1 are te univariate mass and stiffness matrices corresponding to te spaces S 0 and S 1. Obviously, we ave L 00 2 M 0 M 0, L 10 2 M 1 M 0, and L 01 2 M 0 M 1. It remains to sow tat L 11 2 M 1 M 1 also olds. Note tat [16, Teorem 3] states (I Q 0 )u 2 L 2(0,1) 2 u 2 H 1 (0,1), wic yields also (I Q 0)u 2 L 2(0,1) 2 (I Q 0 )u 2 H 1 (0,1) and moreover 2 (I Q 0 ) M(I Q 0 ) (I Q 0 ) K(I Q 0 ), were M and K are te univariate mass and stiffness matrices corresponding to te wole spline space S. Note tat in [16, Section 3.2], we ave defined M 1 = (I Q 0 ) M(I Q 0 ) and K 1 = (I Q 0 ) K(I Q 0 ), so we ave 2 M 1 K 1. Using te definition of L 11, we obtain L 11 M 1 K 1 2 M 1 M 1. Now, we ave sown L αβ 2 M α M β for α, β {0, 1}. Using tis, te fact tat te spaces S α are by construction L 2 -ortogonal, we immediately obtain M T L T. (Note tat tis is completely analogous to [16, Lem. 8 and 9]). Using Lemma 22, we obtain (46). 22

24 Corollary 32. For all grid sizes and spline degrees p N, we obtain Proof. We can sow A + 2 M L p(a + 2 M ). A + 2 M T T P T (A T + 2 M T )P T analogously to te proof of Lemma 24. Using tis, Teorem 31 and te definition of L, we obtain A + 2 M L. Lemma 29 states L p(a + 2 M ). Based on tese results, we can construct a smooter tat can be applied wit optimal complexity and wic yields provably robust convergence rates. Te smooter is given by L 1 ( := ϱ 1 I ( p ( p = i i=1 ( I ϱl 1 ) p ) ( + 2 M ) ( + 2 M ) 1 ) ) i 1 L 1, ) ( ϱl 1 ( + 2 M ) were ϱ > 0 is cosen independent of te grid size, te spline degree p and te number of patces K suc tat ϱ(â + 2 M ) L. Tis is possible due to Corollary 32. Note tat L represents noting but p steps of a preconditioned Ricardson metod; so te smooting step (22) is to be realized by r (0,µ) p (0,µ,1) p (0,µ,i) u (0,µ) := f A u (0,µ 1) := ϱl 1 r(0,µ) := p (0,µ,i 1) := u (0,µ 1) + ϱl 1 ( r (0,µ) + τϱ 1 p (0,µ,p). ) ( + 2 M )p (0,µ,i 1) i = 2,..., p First observe tat tis metod can be realized wit optimal complexity. Remark 2. For applying te preconditioned Ricardson metod, we need (besides simple vector manipulations tat can be provided wit a complexity of O(N)) to apply te smooter L and to apply te matrix  + 2 M. Te latter can be done by applying it patc-wise, i.e., by computing K k=1 ((Âk + 2 Mk )p ). Note tat Âk and M k, stiffness and mass matrix on te parameter domain, ave tensor product structure. So, multiplication wit tem can be realized wit a computational complexity of O(N p), wic is not more tan te application costs of L 1, cf. Remark 1. 23

25 Te wole smooter L consists of p steps, so we ave to multiply te application costs wit p and obtain: setup costs: O(Np + Kp 6 ) application costs: O(Np 2 + Kp 5 ). In a multigrid setting, assuming O(log m) levels, were eac patc as m = m, m 2, m 4, m 8... intervals in eac dimension, we obtain by adding up te overall costs for smooting: in te V-cycle: O(Np 2 + K(log m)p 6 ), in te W-cycle: O(Np 2 + Kmp 5 + K(log m)p 6 ), were N Km 2 p 2 is te number of unknowns on te finest grid. Te full complexity including te costs for te exact coarse-grid solver and te intergrid transfers is asymptotically te same. Under mild assumptions on te relation between p and N, te overall complexity is asymptotically not more tan O(Np 2 ), wic is te cost for one application of te stiffness matrix. Tis sows tat te multigrid cycle as optimal complexity. Now, we sow tat tis approac leads to optimal convergence. Lemma 33. For all grid sizes and spline degrees p N, we ave L A + 2 M and L L. (47) Proof. Define X := ϱl 1 (Â + 2 M ) and note tat ϱ is cosen suc tat X I. Corollary 32 states tere is a constant C suc tat X C 1 p 1 I. So, we obtain 0 I X (1 C 1 p 1 )I and ( ρ((i X ) p ) 1 1 ) p e 1/C < 1, Cp were e is te Eulerian number. Tis implies I I (I X ) p = ϱ L 1 (Â + 2 M ), and L Â + 2 M, and using Lemma 22 finally te first relation in (47). As 0 I X I, we obtain (I X ) p I X, and consequently X I (I X ) p, wic implies L L, te second relation in (47). Using tis Lemma and Teorem 21, we obtain te following teorem. Teorem 34. Tere are constants c 1 and c 2 tat do not depend on te grid size, te spline degree p, and te number of patces K (but may depend on C G, C N, or C R ) suc tat τl 1 A I and ϱl 1 (Â + 2 M ) I (48) 24

26 for all τ (0, c 1 ] and all ϱ (0, c 2 ]. For any fixed coice of τ and ϱ satisfying (48), tere is some ν 0 tat does not depend on p,, or K suc tat te proposed two-grid metod converges for any coice of te number of smooting steps ν > ν 0 wit a convergence rate q = ν 0 /ν, i.e., u (1) A 1 f A + 2 M ν 0 ν u(0) A 1 f A + 2 M. Proof. Note tat τl 1 A I and Lemma 33 imply tat τa L. Using tis and Lemma 33, we obtain te conditions of Teorem 21, wic yields te desired statement. Due to [17, Teorem 4], we know tat also te W-cycle multigrid metod converges. 5. Numerical experiments In tis section, we present numerical experiments tat illustrate te efficiency of te proposed multigrid solver. Te multigrid solver was implemented in C++ based on te G+Smo library [24] Te unit square In tis section, we consider te domain Ω = ( 0.6, 1.4) 2, wic is decomposed into four patces Ω 1 = ( 0.6, 0.4) 2, Ω 2 = (0.4, 1.4) ( 0.6, 0.4), Ω 3 = ( 0.6, 0.4) (0.4, 1.4), and Ω 4 = (0.4, 1.4) 2 ; in all cases te geometry transformation is just a translation. We solve te problem u = 2π 2 sin(πx) sin(πy) u = g := sin(πx) sin(πy) in Ω on Ω (49) and note tat g is te exact solution of te problem. On te coarsest grid level l = 0, te wole patc is just one element. Te grid levels l = 1, 2,..., are obtained by uniform refinement. Te coarsest grid wic is actually used in te multigrid metod is cosen suc tat for all patces te condition m > p olds, i.e., tat te number of intervals is more tan p, cf. [16, Section 6.1]. l \ p Table 1: Multigrid for te unit square wit steps of smooter L as iterative metod 25

27 l \ p Table 2: Multigrid for te unit square wit steps of smooter L as preconditioner for conjugate gradient l \ p Table 3: Multigrid for te unit square wit steps of smooter L as iterative metod As first numerical example, we set up te W-cycle multigrid metod wit te proposed smooter L (cf. Section 4.1), were 1 pre- and 1 post-smooting step is applied. As damping parameter, we coose τ = Te parameter in te subspace-corrected mass smooter, cf. [16], is cosen as σ := Te iteration counts required to reduce te initial error by a factor of ɛ = 10 8 are given in Table 1. We observe tat te metod sows robustness bot in te grid size l := 2 l (wic was proven) and te spline degree p (were tis is only proven for O(p) smooting steps), were we observe as in [16] tat te convergence gets sligtly better if p is increased. We observe tat, as expected, te iteration counts are improved if we use te multigrid metod as a preconditioner for a conjugate gradient metod, cf. Table 2. Similar iteration numbers are obtained for te V-cycle. Finally, in Table 3, we consider te results for te smooter L (cf. Section 4.2). Here, we coose σ as above, ϱ = 0.95 and τ = 1. Again smooting steps are applied in a W-cycle multigrid iteration. We observe again tat te metod sows robustness in te grid size and te spline degree (wic was proven). We observe tat te iteration numbers decrease if te spline degree is increased. For large spline degrees p te iteration numbers are significantly smaller tan for te smooter L, owever te numerical experiments seem to indicate tat effect does not justify te additional effort required to realize te smooter L. 26

28 5.2. Te L-saped domain In tis section we consider te first non-trivial example. We extend te metod beyond te case covered by te convergence teory to te L-saped domain Ω = {(x, y) ( 0.6, 1.4) 2 : x < 0.4 y < 0.4}, were te regularity assumption does not old due to te reentrant corner. Te domain is decomposed into tree patces Ω 1 = ( 0.6, 0.4) 2, Ω 2 = (0.4, 1.4) ( 0.6, 0.4), and Ω 3 = ( 0.6, 0.4) (0.4, 1.4); in all cases te geometry transformation is just a translation. Again, we solve for te problem (49). l \ p Table 4: Multigrid for te L-saped domain wit steps of smooter L as iterative metod l \ p Table 5: Multigrid for te L-saped domain wit 1+1 steps of smooter L as preconditioner for conjugate gradient Again, we set up te W-cycle multigrid metod wit 1+1 smooting steps of te proposed smooter L. We coose τ = 0.95 and σ = Te iteration counts required to reduce te initial error by a factor of ɛ = 10 8 are given in Table 4. We observe tat te iteration counts are similar to tose for te unit square and tat te metod sows again robustness in te grid size and te spline degree. We observe tat, as expected, te iteration counts are improved if we use te multigrid metod as a preconditioner for a conjugate gradient metod, cf. Table Te Yeti footprint As tird domain, we consider te Yeti footprint, cf. Figure 1. Tis domain is a popular model problem for te IETI metod [20]. Tis domain as non-trivial geometry transformation functions. 27

29 Figure 1: Te Yeti footprint As te domain as a smoot boundary, it is covered by te teory presented witin te paper. Te domain is decomposed into 21 patces, wic can be seen in Figure 1. Again, we solve for te problem (49). For tis example, we ave to reduce te damping parameter. We coose τ = 0.25 and σ = If te multigrid metod is used as an iterative sceme, te metod suffers from te geometry transformation, so robust convergence is only obtained for 2+2 smooting steps, cf. Table 6. If te metod is used as a preconditioner for a conjugate gradient metod, again 1+1 smooting steps are sufficient for rater good convergence rates, cf. Table 7. Again we observe robustness bot in te grid size and te spline degree. Similar iteration counts are obtained for te V-cycle. l \ p Table 6: Multigrid for te Yeti footprint wit steps of smooter L as iterative metod l \ p Table 7: Multigrid for te Yeti footprint wit steps of smooter L as preconditioner for conjugate gradient In Table 8, we sow actual CPU times required for to execute te numerical tests from Table 7 on a standard personal computer 1 witout any parallelization. Te CPU times include te setup of te multigrid solver and te solution 1 12 core Intel(R) Xeon(R) CPU, 3.20GHz wit 15.6 GiB RAM 28

30 l \ p # of unknowns W-cycle 2.5 s 4.2 s 9.8 s 17.6 s V-cycle 1.8 s 3.1 s 8.0 s 15.0 s 5 # of unknowns W-cycle 10 s 17 s 30 s 48 s V-cycle 7 s 12 s 21 s 35 s 6 # of unknowns W-cycle 43 s 66 s 106 s 156 s V-cycle 30 s 47 s 74 s 112 s 7 # of unknowns W-cycle 185 s 284 s 465 s 712 s V-cycle 115 s 187 s 299 s 511 s Table 8: Multigrid for te Yeti footprint wit steps of smooter L as preconditioner for conjugate gradient of te problem (but it excludes te assembling of te stiffness matrix). We observe tat for -refinement, te CPU times grow linearly wit te number of unknowns. For te spline degree, we observe tat te complexity grows less tan quadratically wit te spline degree. Concluding, we observe tat te overall complexity does not exceed O(Np 2 ), te number of non-zero entries of te stiffness matrix. 6. Conclusions We ave introduced a multigrid smooter based on an additive domain decomposition approac and ave proven tat its convergence rates are robust bot in te grid size and te spline degree. Te proof only olds if O(p) smooting steps are applied, te experiments sow owever tat smooting steps are enoug. So, following te numerical experiments, te proposed smooter yields an optimal multigrid metod. Moreover, we ave given a variant of te smooter in Section 4.2, were we could actually prove optimal complexity. Te numerical experiments seem to indicate tat te original smooter is always superior to tat variant, so it is more of teoretical interest. Acknowledgments Te autor tanks C. Hofer and C. Hofreiter for fruitful discussions on topics related to tis publication. 29