Robust multigrid solvers for the biharmonic problem in isogeometric analysis

Transcription

1 Robust multigrid solvers for te biarmonic problem in isogeometric analysis J. Sogn, S. Takacs RICAM-Report

2 Robust multigrid solvers for te biarmonic problem in isogeometric analysis Jarle Sogn a,, Stefan Takacs b a Institute of Computational Matematics, Joannes Kepler University Linz, Altenberger Str. 69, 4040 Linz, Austria b Joann Radon Institute for Computational and Applied Matematics (RICAM), Austrian Academy of Sciences, Altenberger Str. 69, 4040 Linz, Austria Abstract In tis paper, we develop multigrid solvers for te biarmonic problem in te framework of isogeometric analysis (IgA). In tis framework, one typically sets up B-splines on te unit square or cube and transforms tem to te domain of interest by a global smoot geometry function. Wit tis approac, it is feasible to set up H 2 -conforming discretizations. We propose two multigrid metods for suc a discretization, one based on Gauss Seidel smooting and one based on mass smooting. We prove tat bot are robust in te grid size, te latter is also robust in te spline degree. Numerical experiments illustrate te convergence teory and indicate te efficiency of te proposed multigrid approaces, particularly of a ybrid approac combining bot smooters. Keywords: Biarmonic problem, Isogeometric analysis, Robust multigrid 2010 MSC: 35J30, 65D07, 65N55 1. Introduction Isogeometric analysis (IgA) was introduced around a decade ago as a new paradigm to te discretization of partial differential equations (PDEs) and as gained increasing attention (cf. [1] for te original paper and [2] for a survey paper). Te idea of IgA from te tecnical point of view is to use B-spline spaces or similar spaces, like NURBS spaces, to discretize te problem. In contrast to standard C 0 -smoot ig-order finite elements, te introduction of discretizations wit iger smootness on general computational domains is not straigt forward. In IgA, splines are first set up on te unit square or te unit cube, wic is usually called te parameter domain. Ten, a global smoot geometry transformation mapping from te parameter domain to te pysical domain, i.e., te domain of interest, is used to define te ansatz functions on te pysical domain. Corresponding autor

3 Suc an approac allows to construct arbitrarily smoot ansatz functions. So, we easily obtain H 2 -conforming discretizations wic can be used as conforming discretizations of te biarmonic problem, wic is for example of interest in plate teory (cf. [3]), Stokes streamline equations (cf. [4]), or Scur complement preconditioners (cf. [5, 6]). For te latter, also te tree dimensional version of te biarmonic problem is of interest. Suc H 2 -conforming discretizations are ard to realize in a standard finite element sceme. One option is te Bogner-Fox-Scmit element, wic requires a rectangular mes, anoter option is te Argyris elements for triangular meses. For suc H 2 -conforming elements, besides various kinds of oter preconditioners (cf. [7] and references terein), also multigrid solvers ave been proposed (cf. [8]). As alternative, multigrid solvers for various kinds of mixed or non-conforming formulations ave been developed (cf. [9, 10, 11] and references terein). In tis paper, we develop iterative solvers for conforming Galerkin discretizations of te biarmonic problem in an isogeometric setting. Multigrid metods are known to solve linear systems arising from te discretization of partial differential equations wit optimal complexity, i.e., teir computational complexity grows typically only linearly wit te number of unknowns. In an isogeometric setting, multigrid and multilevel metods ave been discussed witin te last years (cf. [12, 13, 14, 15, 16]). It was observed tat multigrid metods based on standard smooters, like te Gauss Seidel smooter, sow robustness in te grid size witin te isogeometric setting, teir convergence rates owever deteriorate significantly if te spline degree is increased. Tis motivated te recent publications [15, 16]. In te latter, a subspace corrected mass smooter was introduced, based on te approximation error estimates and inverse inequalities from [17]. Te present paper is a continuation of [17] and [16]. We propose two multigrid metods for te linear system resulting from te discretization of te biarmonic problem, one based on Gauss Seidel smooting and one based on a subspace corrected mass smooter. We prove tat bot are robust in te grid size, te latter is also robust in te spline degree. For tis purpose, non-trivial extensions to bot previous papers are required. [17] covers te approximation wit functions wose odd derivatives vanis on te boundary; an extension to functions wose even derivatives (including te function value itself) vanis on te boundary migt be straigt-forward, owever, for te first biarmonic problem we need a combination of bot. A straigt-forward extension of [16] would require full H 4 regularity, wic cannot even be assumed on te unit square (cf. [18]). So, we only require partial regularity (Assumption 2) and derive te convergence results using Hilbert space interpolation. We give numerical experiments bot for domains described by trivial and non-trivial geometry transformation in two and tree dimensions. We observe tat te subspace corrected mass smooter outperforms te Gauss Seidel smooter for significant large spline degrees. Te negative effects of te geometry transformation to te subspace corrected mass smooter, wic ave also been observed for te Poisson problem, are amplified in case of te biarmonic problem. Approaces to master tese effects are of particular interest for te bi- 2

4 armonic problem. We propose a ybrid smooter wic combines te strengts of bot proposed smooters and works well in our numerical experiments (cf. Section 6.3). Te remainder of te paper is organized as follows. We introduce te model problem and its discretization in Section 2. Ten, in Section 3, we develop te required approximation error estimates. In Section 4, we set up a stable splitting of te spline spaces. In Section 5, we introduce te multigrid algoritms and prove teir convergence. Finally, in Section 6, we give results from te numerical experiments and draw conclusions. 2. Preliminaries 2.1. Model problem In tis paper, we consider te first biarmonic problem as model problem, wic reads as follows. For a given domain Ω R d wit piecewise C 2 -smoot Lipscitz boundary Γ = Ω and a given source function f, find te unknown function u suc tat 2 u = f in Ω, u = 0 on Γ, (1) u n = 0 on Γ, were n is te outer normal vector; for simplicity, we restrict ourselves to omogenous boundary conditions. Our proposed solver can be extended to oter boundary conditions, namely to te second and te tird biarmonic problem, cf. Remarks 2 and 3. Following te principle of IgA, we assume tat te computational domain Ω is represented by a bijective geometry transformation G : Ω Ω (2) mapping from te parameter domain Ω := (0, 1) d to te pysical domain Ω. Te variational formulation of model problem (1) is as follows. Problem 1. Given f L 2 (Ω), find u V := H 2 0 (Ω) suc tat ( u, v) L2 (Ω) = (f, v) L2 (Ω) }{{} v V. (3) (u, v) B(Ω) := Here and in wat follows, L 2 and H r denote te standard Lebesgue and Sobolev spaces wit standard inner products (, ) L 2, (, ) H r, norms L 2, H r and seminorms H r = (, ) 1/2 H. r H2 0 (Ω) is te standard subspace of H 2, containing te functions were te values and te derivatives vanis on te boundary, i.e., H 2 0 (Ω) = { v H 2 (Ω) v = v n = 0 on Γ }. 3

5 Note tat te inner products (, ) H2 (Ω) and (, ) B(Ω) coincide on H0 2 (Ω) (cf. [19]), i.e., (u, v) B(Ω) = (u, v) H2 (Ω) u, v H 2 0 (Ω). (4) Let (, ) B(Ω) be te inner product obtained by removing te cross terms from te inner product (, ) B(Ω), i.e., (u, v) B(Ω) := d ( xk x k u, xk x k u) L2 (Ω). (5) k=1 Here and in wat follows, x := partial derivatives. x and xy := x y and r x := r x r denote Lemma 1. Te inner products defined in (3) and (5) are spectrally equivalent, i.e., (u, u) B(Ω) (u, u) B(Ω) d (u, u) B(Ω) u H0 2 (Ω). Proof. Using te Caucy-Scwarz inequality and ab 1 2 (a2 + b 2 ), we obtain u 2 B(Ω) = 1 2 d k=1 l=1 d d ( xk x k u, xl x l u) L 2 (Ω) d k=1 l=1 ( ) xk x k u 2 L 2 (Ω) + x l x l u 2 L 2 (Ω) = d u 2 B(Ω), wic sows one direction. Using te boundary conditions and (4), we obtain u 2 B(Ω) = u 2 H 2 (Ω) = u 2 B(Ω) + wic sows te oter direction Spline space d k=1 l {1,...,d}\{k} ( xk x l u, xk x l u) L 2 (Ω), }{{} 0 We consider standard tensor product B-spines wit maximum continuity (see, e.g., [20]). Let te interval (0, 1) be subdivided into m N elements of lengt = 1/m. Te space of splines of degree p N := {1, 2, 3,...} wit maximum continuity is defined by S p, (0, 1) := { u C p 1 (0, 1) : u ((i 1),i) P p j = 1,... m }, were C p 1 (0, 1) is te space of all p 1 times continuously differentiable functions on (0, 1) and P p is te space of all polynomials wit degree at most p. We use te standard B-splines wit open knot vector as basis for S p, (0, 1). Te dimension of S p, (0, 1) is n := dim S p, (0, 1) = m + p. We will from time to 4

6 time omit te subscripts p and of a spline space S p, (0, 1) and write S(0, 1) or just S. For iger dimensions d > 1, we use te tensor product splines S p, ( Ω) = S p, (0, 1)... S p, (0, 1), defined over Ω = (0, 1) d. For notational convenience, we assume tat all of tose univariate spline spaces S p, ave te same spline degree p and te same number of elements m, owever, tis in not necessary and te results in tis paper can easily be generalized to te case wit different p and m. Based on te spline space on te parameter space, we define te spline space on te pysical space using te standard pull-back principle as S p, (Ω) = {u : u G S p, ( Ω)}, were G is te geometry transformation (2). We assume tat te geometry transformation is sufficiently smoot suc tat te following estimate olds. Assumption 1. Assume tat tere exist constants α > 0 and α suc tat α u Hq (Ω) u G Hq ( Ω) α u H q (Ω) u H q (Ω), q {2, 3}. We discretize te Problem 1 using te Galerkin principle as follows. Problem 2. Given f L 2 (Ω), find u V := S 0 p, (Ω) := H2 0 (Ω) S p, (Ω) suc tat (u, v) B(Ω) = (f, v) L2 (Ω) v V. (6) By fixing a basis for te space Sp, 0 (Ω), we can rewrite te Problem 2 in matrix-vector notation as B u = f, (7) were B is a standard stiffness matrix, u is te representation of te corresponding function u wit respect to te cosen basis and te vector f is obtained by testing te rigt and side functional (f, ) L2 (Ω) wit te basis functions. For convenience, we use te following notation. Notation 1. Trougout tis paper, c is a generic positive constant independent of and p, but may depend on d and G Regularity In te following sections, we use Aubin-Nitsce duality arguments for sowing te desired error estimates. Tis requires tat te following assumption olds. Assumption 2. For a given f H 1 (Ω), te solution u H 2 0 (Ω) of te first biarmonic problem (1) satisfies u H 3 (Ω) and u H3 (Ω) c f H 1 (Ω). 5

7 Suc a result is satisfied for convex polygonal domains (cf. [18]). It is wort noting tat tis implies tat te result also olds for te parameter domain Ω = (0, 1) 2. As we only rely on a partial regularity result, we use Hilbert space interpolation (cf. [21, 22]) to derive our estimates. defined, e.g., wit te K-metod, is a Hilbert space wit norm [A1,A 2] θ. Applied to Sobolev spaces H m (Ω) and H n (Ω), we obtain 2 [H m (Ω),H n (Ω)] θ = 2 H (1 θ)m+θn (Ω), (8) see [21, Teorem 6.4.5], applied to scaled Hilbert spaces A 1 and γa 2 wit a scaling parameter γ > 0, we obtain 2 [A 1,γA 2] θ = γ θ 2 [A 1,A 2] θ (9) and applied to te intersections of two Hilbert spaces A 1 A 2 2 A 1 A 2 := 2 A A 2, we obtain wit norm 2 [A 1,A 1 A 2] θ c 2 A 1 [A 1,A 2] θ, (10) see [23, Lemma 6.1], and applied to dual norms, we obtain 2 ([A 1,A 2] θ ) = 2 [A 1,A 2 ] θ, (11) see [21, Teorem 3.7.1]. As te interpolation defined by te K-metod is an exact interpolation function, see [21, Teorem 3.1.2], we know tat any bounded operator Ψ, wic maps from a Hilbert space A 1 to a Hilbert space B 1 and from a Hilbert space A 2 to a Hilbert space B 2, maps also from [A 1, A 2 ] θ to [B 1, B 2 ] θ and satisfies Ψa [B1,B 2] θ cm 1 θ 1 M θ 2 a [A1,A 2] θ wit M i := sup a i A i Ψa i Bi a i Ai (12) for all θ (0, 1), were c only depends on θ. 3. Approximation error estimates One vital component needed to prove multigrid convergence is an approximation error estimate. Approximation error estimates between te spaces L 2 (Ω) and H 1 (Ω) are given in [17, 24] and used in [15, 16] to prove convergence for a multigrid solver for te Poisson problem. For te biarmonic problem we need similar estimates for H 2 (Ω) Approximation error estimates for te periodic case We start te analysis for te periodic case. We define for eac q N te periodic Sobolev space H q per( 1, 1) := { u H q ( 1, 1) : u (l) ( 1) = u (l) (1), } l N 0 wit l < q 6

8 and for eac p N te periodic spline space } S per p, {u ( 1, 1) := S( 1, 1) : u (l) ( 1) = u (l) (1) l N 0 wit l < p. Let T q,per p, be te H q, -ortogonal projection into S per p, ( 1, 1), were te underlying scalar product (, ) H q, ( 1,1) is given by were 1 2 (u, v) H r, ( 1,1) := { (u, v) H q ( 1,1) u dx 1 v dx for q > 0, 1 (u, v) L 2 ( 1,1) for q = 0, 1 1 u dx 1 v dx is added to enforce uniqueness. 1 Teorem 1. Let p N 0, q N 0 wit p q and p < 1. Ten, (I T q,per p, )u H q ( 1,1) 2 u H q+1 ( 1,1) u H q+1 per ( 1, 1). Proof. We use induction wit respect to q. Proof for q = 0. [17, Lemma 4.1] gives an approximation error estimate for te H 1, -ortogonal projection of u into S per 0,per p, for p 1. Because Tp, minimizes te L 2 -norm, we obtain (I T 0,per p, )u 1,per L2 (I Tp, )u L 2 2 u H 1 u Hper( 1, 1 1), i.e., te desired result. For p = 0, we observe tat tere are no periodicity conditions for te space S per p,. Te desired result on approximation by piecewise constants is standard and can be found, e.g, in [25, Teorem 6.1]. Proof for q > 0. We already know tat te induction ypotesis olds true for q 1, i.e., we ave u T q 1,per p 1, u H q 1 ( 1,1) 2 u Hq ( 1,1) u H q per( 1, 1). (13) As a next step we sow tat for all u H q+1 per ( 1, 1), tere is a u S per p, suc tat u u H q ( 1,1) 2 u H q+1 ( 1,1). (14) By plugging u into (13), we immediately obtain u T q 1,per p 1, u H q 1 ( 1,1) 2 u H q+1 ( 1,1) u H q+1 per ( 1, 1). Let v := T q 1,per p 1, u and define u (x) := x 1 v (ξ)dξ + γ, were γ R suc tat 1 1 u (x)dx = 0. For tis coice, we obtain te desired estimate (14). It remains to sow u S per p,. As we ave v S per p 1,, we obtain tat u is a spline of degree p. Te continuity estimates u (l) ( 1) = u(l) (1) for l = 1,..., p 1 7

9 follow directly from v (l) ( 1) = v(l) (1) for l = 0,..., p 2. So, it remains to sow u ( 1) = u (1). Note tat, as u is periodic, we ave u ( 1) u (1) = (u(1) u( 1)) (u (1) u ( 1)) = = v(x) v (x)dx = ((I T q 1,per p 1, )v, 1) L 2 ( 1,1). Note tat (, 1) H r, ( 1,1) = (, 1) L 2 ( 1,1) for any r, so we obtain u ( 1) u (1) = ((I T q 1,per p 1, )v, 1) H q 1, ( 1,1) u (x) u (x)dx and finally, as 1 S per p 1,, Galerkin ortogonality sows tat tis term is 0. So, we ave sown u S per q,per p, and (14). As te projector Tp, minimizes te H q -seminorm, we obtain (I T q,per p, )u H q ( 1,1) u u H q ( 1,1) 2 u H q+1 ( 1,1), i.e., te desired result Approximation error estimates for te univariate case Now, we derive approximation error estimates for univariate splines tat satisfy te desired boundary conditions. First, we consider te approximation of functions in te Sobolev space of functions wit vanising even derivatives (and function values) on te boundary, given by } H q D {u (0, 1) := H q (0, 1) : u (2l) (0) = u (2l) (1) = 0, l N 0 wit 2l < q, by functions in a corresponding spline space, given by { S D,0 (0, 1) := u S(0, 1) : u (2l) (0) = u (2l) (1) = 0 } l N 0 wit 2l < p. Now, we define Π D,0 to be te H 2 -ortogonal projection from HD 2 (0, 1) into S D,0 (0, 1). Tis projector satisfies te following error estimate. Teorem 2. Let p N wit p 3 and p < 1. Ten, (I Π D,0 )u H 2 (0,1) 2 2 u H 4 (0,1) u H 4 D(0, 1) Proof. Assume u HD 4 (0, 1) to be arbitrary but fixed. Define w on ( 1, 1) by w(x) := sign(x) u( x ) and observe tat w H 4 per( 1, 1). Using Teorem 1, we obtain (I T 2,per p, )w H 2 ( 1,1) = (I T 2,per p, )(I T 3,per p, )w H 2 ( 1,1) 2 (I T 3,per p, )w H 3 ( 1,1) 2 2 w H 4 ( 1,1). 8

10 First observe tat w H4 ( 1,1) = 2 u H4 (0,1). Define w := T 2,per p, w and observe tat we obtain w (x) = w ( x) using a standard symmetry argument. Tis implies tat u, te restriction of w to (0, 1), satisfies u S D,0. Moreover, we ave w w H 2 ( 1,1) = 2 u u H 2 (0,1) and, as a consequence, u u H 2 (0,1) 2 2 u H 4 (0,1). As te projector Π D,0 minimizes te H 2 -seminorm, te desired result follows. Now, we consider te boundary conditions of interest for te first biarmonic problem. Here, te continuous space is H 2 0 (0, 1) and te discretized space is S 0 (0, 1), given by S 0 (0, 1) := {u S(0, 1) : u(0) = u (0) = u(1) = u (1) = 0} = S(0, 1) H 2 0 (0, 1). Now, we define Π 0 to be te H 2 -ortogonal projection from H0 2 (0, 1) into S 0 (0, 1). Tis projector satisfies te following error estimate. Teorem 3. Let p N wit p 3 and p < 1. Ten, (I Π 0 )u H2 (0,1) 2 2 u H4 (0,1) u H 4 (0, 1) H 2 0 (0, 1). Proof. First, define S (0, 1) := S H 1 0 (0, 1) and Π : H 2 D (0, 1) S to be te H 2 -ortogonal projector into te corresponding space. Since S D,0 S, Teorem 2 directly implies (I Π ) w H 2 (0,1) 2 2 w H 4 (0,1) w H 4 D(0, 1). Now let u H 4 (0, 1) H 1 0 (0, 1) be arbitrary but fixed. Observe tat for w(x) := u(x) (x3 3x 2 + 2x) u (0) 1 }{{} 6 (x3 x) u (1), }{{} φ 1 (x) := φ 2 (x) := we obtain w H 4 D (0, 1). Note tat φ 1, φ 2 S and φ 1 H 4 (0,1) = φ 2 H 4 (0,1) = 0. So, (I Π ) u H 2 (0,1) = inf u S u u H 2 (0,1) = inf w S w w H2 (0,1) 2 2 w H4 (0,1) = 2 2 u H4 (0,1). Now, consider te function ψ 1 (x) := 1 2 (x2 x) and observe ψ 1 S and 0 = ((I Π ) u, ψ 1 ) H2 (0,1) = [(I Π ) u] (1) [(I Π ) u] (0). (15) As (I Π ) u H 1 0, we obtain 0 = [(I Π ) u](1) [(I Π ) u](0) = ((I Π ) u, ψ 2 ) H1 (0,1), 9

11 were ψ 2 (x) := 1 6 (x3 x). Integration by parts and ψ 2 (0) = 0 yields 0 = ((I Π ) u, ψ 2 ) H 1 (0,1) = ((I Π ) u, ψ 2 ) H 2 (0,1) + [[(I Π ) u] ψ 2 ](1). As ψ 2 S, Galerkin ortogonality yields ((I Π ) u, ψ 2 ) H2 (0,1) = 0, so we ave [(I Π ) u] (1) = 0. Tis implies, in combination wit (15), tat u (1) = (Π u) (1) and u (0) = (Π u) (0) olds. So, for any u H 2 0 (0, 1), we ave Π u S H 2 0 = S 0. As Π 0 minimizes te same norm, we obtain for any u H 2 0 (0, 1) tat Π u = Π 0 u, so also te projector Π 0 satisfies te desired error estimate. Teorem 4. Let p N wit p 3 and p < 1. Ten, (I Π 0 )u L 2 (0,1) 2 2 u H 2 (0,1) u H 2 0 (0, 1). Proof. Tis is sown using a classical Aubin Nitsce duality trick. Let u H0 2 (0, 1) be arbitrary but fixed and coose v H 4 (0, 1) H0 2 (0, 1) suc tat v = u Π 0 u. Using integration by parts and Teorem 3, we obtain u Π 0 u L2 (0,1) = ( u Π 0 u, u Π 0 u ) L 2 (0,1) u Π 0 u L 2 (0,1) = ( u Π 0 u, v ) L 2 (0,1) v H 4 (0,1) = ( u Π 0 u, v ) H 2 (0,1) v H4 (0,1) ( u Π 0 u, v ) 2 2 H 2 (0,1) v Π 0. v H2 (0,1) Galerkin ortogonality gives ( u Π 0 u, Π 0 v ) = 0. Using tis, te Caucy- H 2 (0,1) Scwarz inequality and tis H 2 -stability of Π 0, we finally obtain u Π 0 u L 2 (0,1) 2 2 ( u Π 0 u, v Π 0 v ) H 2 (0,1) v Π 0 v H2 (0,1) wic finises te proof. 2 2 u Π 0 u H 2 (0,1) 2 2 u H 2 (0,1), 3.3. Approximation error estimates for te parameter domain In tis subsection, we derive robust approximation error estimates for te space S 0 ( Ω). For tis purpose, we define te following projectors on u H 2 ( Ω): (Π x k )u(x 1,..., x k 1,, x k+1,..., x d ) := Π 0 u(x 1,..., x k 1,, x k+1,..., x d ) (x 1,..., x k 1, x k+1,..., x d ) (0, 1) d 1 for k = 1,..., d. Lemma 2. Te projectors Π x k are commutative; tat is, Π xi Π xj = Π xj Π xi for i = 1,..., d and j = 1,..., d. Proof. Te proof is completely analogous to tat of [26, Lemma 12]. 10

12 Let Π := Π p, be te H 2 -ortogonal projection from H 2 0 ( Ω) into S 0 ( Ω) = S 0 p, ( Ω). Teorem 5. Let p N wit p 3 and p < 1. Ten, (I Π)u H 2 ( Ω) c 2 u H 4 ( Ω) u H 4 ( Ω) H 2 0 ( Ω). Proof. For sake of simplicity, we restrict te proof to te two dimensional case. Using te triangle inequality and te H 2 -stability of Π x, we obtain xx (u Π x Π y u) L 2 ( Ω) xx(u Π x u) L 2 ( Ω) + xxπ x (u Π y u) L 2 ( Ω) xx (u Π x u) L 2 ( Ω) + xx(u Π y u) L 2 ( Ω). Using Teorems 3 and 4 and a + b c(a 2 + b 2 ) 1/2, we obtain ( 1/2 xx (u Π x Π y u) L 2 ( Ω) c 2 xxxx u 2 + L 2 ( Ω) xxyyu 2 L ( Ω)). 2 Using Lemma 2 and te same arguments as above, we obtain ( 1/2 yy (u Π x Π y u) L 2 ( Ω) c 2 yyxx u 2 + L 2 ( Ω) yyyyu 2 L ( Ω)). 2 Using tis and Lemma 1, we finally obtain (I Π)u 2 (I H 2 ( Ω) Πx Π y ) u 2 = (I H 2 ( Ω) Πx Π y ) u 2 B( Ω) 2 (I Π x Π y ) u 2 B( Ω) = 2 xx (u Π x Π y u) L 2 ( Ω) yy(u Π x Π y u) 2 ( L 2 ( Ω) ) c 4 xxxx u 2 + L 2 ( Ω) xxyyu 2 + L 2 ( Ω) yyxxu 2 + L 2 ( Ω) yyyyu 2 L 2 ( Ω) = c 4 u 2. H 4 ( Ω) Teorem 6. Let p N wit p 3 and p < 1. Ten, (I Π)u H 2 ( Ω) c u H 3 ( Ω) u H 3 ( Ω) H 2 0 ( Ω). Proof. Teorem 5 states (I Π)u H 2 ( Ω) c 2 u H 4 ( Ω) u H 4 ( Ω) H 2 0 ( Ω), and, as Π is stable in H 2, we ave (I Π)u H 2 ( Ω) u H 2 ( Ω) u H 2 0 ( Ω), Using (12) for θ = 1/2 and (8), we obtain te desired result. Teorem 7. Let p N wit p 3 and p < 1. Ten, (I Π)u H1 ( Ω) c u H 2 ( Ω) u H 2 0 ( Ω). 11

13 Proof. Tis proof is a variant of te classical Aubin Nitsce duality trick. Let u H 2 0 ( Ω) be arbitrary but fixed. Define f H 1 ( Ω) by f, := (u Πu, ) H 1 ( Ω) and define w H2 0 (Ω) to be suc tat ( w, w) L 2 ( Ω) = f, w w H2 0 ( Ω). Lax Milgram lemma yields w H2 ( Ω) = f H 2 ( Ω). Assumption 2 (applied to te parameter domain) implies w H 3 ( Ω) and w H3 ( Ω) c f H 1 ( Ω) = c u Πu H1 ( Ω). We obtain u Πu H 1 ( Ω) = (u Πu, u Πu) H 1 ( Ω) u Πu H 1 ( Ω) c (u Πu, u Πu) H 1 ( Ω). w H3 ( Ω) Using Teorem 6, we furter obtain u Πu H 1 ( Ω) c(u Πu, u Πu) H1 ( Ω) w Πw. H2 ( Ω) Te definitions of f and w, Galerkin ortogonality, Caucy-Scwarz inequality and te H 2 -stability of Π yield u Πu H1 ( Ω) f, u Πu c w Πw = c ( w, (u Πu)) L 2 ( Ω) H 2 ( Ω) w Πw H 2 ( Ω) c (w, u Πu) H2 ( Ω) w Πw H2 ( Ω) c u H2 ( Ω), wic finises te proof. = c (w Πw, u Πu) H2 ( Ω) w Πw H2 ( Ω) c u Πu H2 ( Ω) 3.4. Approximation error estimates for te pysical domain In tis subsection, we extend te robust approximation error estimates for te space S 0 ( Ω) to te space S 0 (Ω) = S(Ω) H 2 0 (Ω). For tis purpose, we define Π = Π p, to be te H 2 -ortogonal projection from H 2 0 (Ω) into S 0 (Ω) = S 0 p, (Ω). Here and in wat follows, always refers to te grid size on te parameter domain. All estimates directly carry over to te grid size Ω on te pysical domain because we ave c 1 Ω c. Teorem 8. Let p N wit p 3 and p < 1. Ten, (I Π) u H2 (Ω) c u H3 (Ω) u H 3 (Ω) H 2 0 (Ω). 12

14 Proof. Let u H 3 (Ω) H0 2 (Ω) and û := u G. inequality, Teorem 6 and Assumption 1, we obtain By combining Friedrics α u [ Πû] G 1 H2 (Ω) (I Π)û H 2 ( Ω) c (I Π)û H 2 ( Ω) c û H 3 ( Ω) α c u H3 (Ω) α c u H3 (Ω), were [ Πû] G 1 S 0 (Ω). As Π minimizes te H 2 -seminorm, we obtain (I Π) u H2 (Ω) u [ Πû] G 1 H2 (Ω). Using α/α c, te desired result follows. Teorem 9. Let p N wit p 3 and p < 1 and assume tat Ω is suc tat Assumption 2 olds. Ten, (I Π) u H 1 (Ω) c u H 2 (Ω) u H 2 0 (Ω). Proof. Te proof is analogous to te proof of Teorem 7. In te proof, we use Teorem 8 instead of Teorem Stable splitting of te spline space In tis section, we introduce an L 2 -ortogonal splitting of te spline space S 0 and sow tat te splitting is stable in H 2 analogously to [16]. To do tis, we need some more approximation error estimates and inverse inequalities Approximation error estimates and inverse inequalities First, we give an estimate for te periodic case. Teorem 10. Let p N wit p 3 and p < 1. Ten, (I T 2,per p, )u L 2 ( 1,1) 2 2 u H 2 ( 1,1) u H 2 per( 1, 1). Proof. Teorem 1 for q = 2 and q = 3 can be combined to (I T 2,per p, )u H 2 ( 1,1) = (I T 2,per p, 3,per )(I T )u H 2 ( 1,1) 2 2 u H 4 ( 1,1) for all u H 4 per( 1, 1). Te desired estimate is sown by an Aubin Nitsce duality trick, wic is completely analogous to Teorem 4. Now, we extend te approximation error estimate to non-periodic splines. Teorem 11. Let p N wit p 3 and p < 1. Ten, (I Π D,0 )u L 2 (0,1) 2 2 u H 2 (0,1) u H 2 D(0, 1). p, 13

15 Proof. Assume u HD 2 (0, 1) to be arbitrary but fixed. Define w on ( 1, 1) by w(x) := sign(x) u( x ) and observe tat w H 2 per( 1, 1). Using Teorem 10, we obtain (I T 2,per p, )w L 2 ( 1,1) 2 2 w H 2 ( 1,1). First, observe tat w H 2 ( 1,1) = 2 u H 2 (0,1). Define w := T 2,per p, w and u as te restriction of w. Observe tat we obtain w (x) = w ( x) using a standard symmetry argument. Tis implies u S D,0. It follows tat w w L2 ( 1,1) = 2 u u L2 (0,1). Using tis, we obtain u u L2 (0,1) 2 2 u H2 (0,1). It remains to sow tat u coincides wit Π D,0 u, i.e., tat u u is H 2 - ortogonal to S D,0. By definition, tis means tat we ave to sow (u u, v ) H 2 (0,1) = 0 v S D,0. (16) Let w S per be defined as w := sign(x) v ( x ) and observe tat (w w, w ) H 2 ( 1,1) = 2(u u, v ) H 2 (0,1) since u, u, v are restrictions of w, w, w, respectively. Furtermore, (w w, w ) H2 ( 1,1) = 0 by construction since w := T 2,per p, w. Tis sows (16) and finises te proof. Next, we need an inverse inequality. We extend te H 1 L 2 -inverse inequality from [17] to te pair H 2 L 2 and te space S D,0 Teorem 12. For all grid sizes and eac p N, u H2 (0,1) 12 2 u L2 (0,1) u S D,0. (17) Proof. We extend u to ( 1, 1) by defining w (x) = sign(x) u ( x ). Observe tat w H 2,per ( 1, 1). Analogously to te proof of [17, Teorem 6.1], we obtain w H1 ( 1,1) w L2 ( 1,1) and w H1 ( 1,1) w L2 ( 1,1). Te combination of tese two results yields w H2 ( 1,1) 12 2 w L 2 ( 1,1). As w H2 ( 1,1) = 2 u H2 (0,1) and w L2 ( 1,1) = 2 u L2 (0,1), te desired result immediately follows. 14

16 4.2. Stable splitting in te univariate case In te previous section, we ave introduced te projectors Π D,0 : H 2 D S D,0. Now, we introduce te L 2 -ortogonal projectors wic split S into te direct sum Q D,0 : S S D,0 and Q D,1 := I Q D,0, S = S D,0 S D,1 u = Q D,0 u + Q D,1 u, were S D,1 is te L 2 -ortogonal complement of S D,0 in S. Because te splitting is L 2 -ortogonal, we obtain u 2 L 2 (0,1) = Q D,0 u 2 L 2 (0,1) + Q D,1 u 2 L 2 (0,1) u S. (18) We sow tat te splitting is stable in te H 2 -norm. Teorem 13. Let p N wit p 3 and p < 1. Ten, c 1 u 2 H 2 (0,1) QD,0 u 2 H 2 (0,1) + QD,1 u 2 H 2 (0,1) c u 2 H 2 (0,1) u S. Proof. Te proof is analogous to [16, Teorem 4]. Te left inequality follows from Caucy-Scwarz inequality wit c = 2. For te rigt inequality, we ave Q D,0 u H 2 (0,1) Π D,0 u H 2 (0,1) + (Π D,0 Q D,0 )u H 2 (0,1) u H2 (0,1) + c 2 (Π D,0 Q D,0 )u L 2 (0,1), using te triangle inequality and te inverse inequality Teorem 12. Using te triangle inequality and te approximation error estimate Teorem 11, we get Q D,0 u H2 (0,1) u H2 (0,1) + c 2 ( (I Π D,0 )u L2 (0,1) + (I Q D,0 )u L2 (0,1) c u H 2 (0,1). Using te inequality above togeter wit Q D,0 u 2 H 2 (0,1) + QD,1 u 2 H 2 (0,1) 2 u 2 H 2 (0,1) + 3 QD,0 u 2 H 2 (0,1), completes te proof Stable splitting in te multivariate case Te generalization to two and more dimensions is straigt forward. Let Ω = (0, 1) d and let α {0, 1} d be a multiindices. Te space S( Ω) is split into te direct sum of 2 d subspaces S( Ω) = S D,α ( Ω) were S D,α ( Ω) = S D,α1... S D,αd. α {0,1} d Te L 2 ( Ω)-ortogonal projectors are given by Q D,α := Q D,α1... Q D,α d : S( Ω) S D,α ( Ω). As in te univariate case, te splitting is stable. ) 15

17 Teorem 14. Let p N wit p 3 and p < 1. Ten, u 2 = Q D,α u 2 u S( Ω), (19) L 2 ( Ω) L 2 ( Ω) α {0,1} d c 1 u 2 B( Ω) Q D,α u 2 B( Ω) c u 2 B( Ω) u S( Ω). (20) α {0,1} d Proof. Te equation (19) follows immediately from te equality in te one dimensional case. Te left inequality in (20) follows immediately from te Caucy- Scwarz inequality. It remains to sow te rigt inequality in (20). Let α and u be arbitrary but fixed. We ave Q D,α u 2 B( Ω) = We obtain d x 2 k Q D,α u 2 = L 2 ( Ω) k=1 d x 2 k Q D,α1 Q D,α d u 2. L 2 ( Ω) k=1 2 x k Q D,α1 Q D,α d u 2 L 2 ( Ω) c 2 x k u 2 L 2 ( Ω) by applying (18) for all Q D,α l wit l k and by applying Teorem 13 for Q D,α k. Combining tese two inequalities yields Q D,α u 2 B( Ω) c u 2 B( Ω). Summing over all multi-indices α yields te desired estimate. 5. Constructing a robust multigrid metod In tis section, we develop a robust multigrid metod for solving te linear system (7). We assume tat we ave constructed a ierarcy of grids by uniform refinement. We obtain V H V for two consecutive grids wit grid sizes and H := 2. For tese spaces, we define P : V H V to be te canonical embedding. We denote te its matrix representation wit te same symbol, te restriction is realized as its transpose P. For a given initial iterate u (k), we obtain te next iterate u(k+1) by applying te following steps. First, we perform ν N smooting steps, given by u (k,i) := u (k,i 1) + τl 1 ( f B u (k,i 1) ), for i = 1,..., ν, were u (k,0) := u (k), L represents te cosen smooter and τ is an appropriately cosen damping parameter. Te coice of L and τ is discussed below. Second, we perform a coarse-grid correction step, wic is for te two-grid metod given by ( ) u (k+1) := u (k,ν) + P B 1 H P f B u (k,ν). 16

18 Given a sequence of spaces, we replace te application of B 1 H by one or two steps of te metod on te next coarser level. Tis results in te V-cycle or W-cycle multigrid metod, respectively. Te application of B 1 H is realized by means of a direct solver only on te coarsest grid level. In te sequel, we discuss two possibilities for te smooter, te Gauss Seidel smooter and a subspace corrected mass smooter. Wile only te latter is robust in te spline degree, te Gauss Seidel smooter is superior for small spline degrees and for cases were a non-trivial geometry transformation is involved. First, we introduce te framework for te convergence analysis and give common results for bot smooters. We sow te convergence of te multigrid metod based on te splitting of te analysis into approximation property and smooting property (cf. [27]). As we do not assume full H 4 -regularity, we coose to sow convergence in te norm B + 2 K, were K is te matrix obtained by discretizing (, ) H 1 (Ω). Te approximation property (21) and te smooting property (22) read as follows: (B + 2 K ) 1/2 (I P B 1 H P B )B 1 (B + 2 K ) 1/2 C A, (21) (B + 2 K ) 1/2 B (I τl 1 B ) ν (B + 2 K ) 1/2 ν 1/2 C S. (22) Te combination of tese two properties yields q := (I P B 1 H P B )(I τl 1 B ) ν B + 2 K C AC S ν, i.e., te two-grid metod convergences if sufficiently many smooting steps are applied. Te convergence of te W-cycle multigrid metod follows under weak assumptions (cf. [27]). Te approximation property follows from te approximation error estimates we ave sown in Section 3. Teorem 15. Let p N wit p 3 and Hp < 1. Ten, te approximation property (21) is satisfied wit a constant C A being independent of and p (cf. Notation 1). Proof. Teorem 9 states tat te H 2 -ortogonal projector Π 0 p,h V H = Sp,H 0 (Ω) satisfies te approximation error estimate : H2 0 (Ω) ( I Π 0 p,h) u H 1 (Ω) c H u H 2 (Ω) u H 2 0 (Ω). As te considered functions are in H0 2 (Ω), Lemma 1 implies te same for te B-ortogonal projector. For u V = Sp, 0 (Ω), we can rewrite tis is matrixvector notation: ( I P B 1 H P B ) u K c H u B. Using te stability of projectors, we also obtain ( I P B 1 H P B ) u B u B. 17

19 By combining tese two results, we obtain using H = 2 c tat ( I P B 1 H P B ) u B c u + 2 K B. Tis reads in matrix-notation as (B + 2 K ) 1/2 ( I P B 1 H P B ) 1/2 B c. As T T T 2, we obtain tat (B + 2 K ) 1/2 ( I P B 1 H P B ) ( I P B 1 H P B ) B 1 (B + 2 K ) 1/2 is bounded by some constant c and, as we ave (I Q)(I Q) = I Q for any projector Q, te desired statement (21). In te two subsequent subsections we sow te smooting estimate (B + 4 M ) 1/2 B (I τl 1 B ) ν (B + 4 M ) 1/2 ν 1 CS (23) and te stability estimate B 1/2 (I τl 1 B ) ν B 1/2 1. (24) Estimate (23), togeter wit an L 2 H 2 -approximation error estimate for te B-ortogonal projector would allow to prove a convergence result in te norm B + 4 M, were M denotes te mass matrix. However, te proof of suc an error estimate requires a full H 4 -regularity assumption, wic is not satisfied in te cases of interest. Using Hilbert space interpolation, we obtain te following lemma. Lemma 3. Te combination of (23) and (24) yields (22), were C S only depends on C S. Proof. First observe tat Lemma 1, (23) and (24) yield B (I τl 1 B ) ν u [H 2 (Ω) 4 L 2 (Ω)] ν 1 CS u H 2 (Ω) 4 L 2 (Ω), B (I τl 1 B ) ν u [H2 (Ω)] u H 2 (Ω) u V, were B and L : V V denote te operator interpretations of te corresponding matrices. Using (12) for θ = 1/2, (11), (10), (9) and (8), we obtain B (I τl 1 B ) ν 1/2 u [H2 (Ω) 2 H 1 (Ω)] c C S ν 1/2 u H2 (Ω) 2 H 1 (Ω), 1/2 were C S := c C S only depends on C S. Tis directly implies (22). 18

20 5.1. Gauss-Seidel smooter Te most obvious coice of a multigrid smooter is te (symmetric) Gauss- Seidel metod. For simplicity, we restrict ourselves to te symmetric Gauss- Seidel smooter, consisting of one forward Gauss-Seidel sweep and one backward Gauss-Seidel sweep. Let B be composed into B = D C C, were C is a (strict) left-lower triangular matrix and D is a diagonal matrix. Ten, te symmetric Gauss-Seidel metod is represented by L := (D C )D 1 (D C ) = B + C D 1 C, see, e.g., [27, Note ]. Using standard arguments, we can sow as follows. Lemma 4. Te matrix L satisfies B L B + c(p) 4 M, (25) were c(p) is independent of te grid size, but depends on te spline degree p and te geometry transformation G. Proof. As C D 1 C 0, te first part of te inequality is obvious. Now, observe tat B as not more tan O(p d ) non-zero entries per row, so also te matrix D 1/2 C D 1/2 as not more tan O(p d ) non-zero entries per row. Te absolute value of eac of tem is bounded by 1 due to te Caucy- Scwarz inequality. So, we obtain using Gerscorin s teorem tat te eigenvalues of D 1/2 C D 1/2 are bounded by cp d, wic implies L B + cp 2d D. A standard inverse estimate (cf. [28, Teorem 3.91]) yields L B + cp 2d+8 4 diag (M ) were diag (M ) is te diagonal of te mass matrix M. Note tat te condition number of te B-splines of degree p is bounded by p2 p (cf. [29]), so we obtain L B + c2 p p 2d+9 4 M, wic finises te proof. Now, we can sow te convergence of te multigrid metod. Teorem 16. Let p N wit p 3 and Hp < 1. Ten, tere exists a constant c(p), wic is indepedent of but depends on p and G, suc tat te two-grid metod wit te symmetric Gauss-Seidel smooter (wit τ = 1) satisfies q c(p) ν 1/2, i.e., it converges if sufficiently many smooting steps ν are applied. 19

21 Proof. From (25), we obtain τl 1 B I for τ = 1. [15, Lemma 2] implies L 1/2 B (I τl 1 B ) ν L 1/2 cν 1, from wic te smooting statement (23) follows using (25). Te stability statement (24) can be sown analogously. Lemma 3 yields te smooting property (22) wit C S = c(p)ν 1/2. Teorem 15 yields te approximation property (21) wit C A = c. Te combination of smooting property and approximation property yields convergence Subspace corrected mass smooter We now construct a smooter tat satisfies c 1 B L c(b + 4 M ), (26) were te constant c is independent of p and (Notation 1). To reduce te complexity of te smooter, we construct te local smooters not around te original stiffness matrix B, representing (, ) B(Ω), but around te spectrally equivalent matrix B, representing (, ) B( Ω). Moreover, we observe tat te original mass matrix M is spectrally equivalent to M, representing (, ) L 2 ( Ω). Using te spectral equivalence, we obtain tat te condition c 1 B L c( B + 4 M ), (27) is equivalent to (26). We follow te ideas of te paper [16] and construct local smooters L α for any of te spaces V,α := S D,α S 0, were α = (α 1,..., α d ) {0, 1} d is a multi-index. Tese local contributions are cosen suc tat tey satisfy te corresponding local condition were c 1 Bα L α c( B α + 4 Mα ), (28) B α := Q,α B (Q,α ) and Mα := Q,α M (Q,α ) and Q,α is te matrix representation of te canonical embedding V,α V. Te canonical embedding as tensor product structure, i.e., Q,α1 Q,αd, were te Q,αi are te matrix representations of te corresponding univariate embeddings. In te two-dimensional case, B and M ave te representation B = B M + M B and M = M M, were M and B are te corresponding univariate mass and stiffness matrices. Restricting B to te subspace V (α1,α2) gives B α1,α 2 = B α1 M α2 + M α1 B α2, 20

22 were B αi = Q,αi B(Q,αi ) and M αi = Q,αi M(Q,αi ). Te inverse inequality for S D,0 (Teorem 12), allows us to estimate B 0 σm 0, were σ = Using tis, we define te smooters L α1,α 2 as follows and obtain estimates for tem as follows: B 00 2σM 0 M 0 =: L 00 c( B M 00 ), B 01 M 0 (σm 1 + B 1 ) =: L 01 c( B M 01 ), B 10 (B 1 + σm 1 ) M 0 =: L 10 c( B M 10 ), B 11 B 1 M 1 + M 1 B 1 =: L 11 c( B M 11 ). (29) Te extension to tree and more dimensions is completely straigt-forward (cf. [16]). For eac of te subspaces V,α, we ave defined a symmetric and positive definite smooter L α. Te overall smooter is given by L := (Q D,α ) L α Q D,α, α {0,1} d were Q D,α 1 = M α (Q,α ) M is te matrix representation of te L 2 projection from V to V,α. Completely analogous to [16, Section 5.2], we obtain L 1 = α (Q,α ). α {0,1} d Q,α L 1 Remark 1. How to realize te smooter computationally efficient, is discussed in [16, Section 5]. Te local estimates from (28) can be carried over to te wole smooter L analogous to te results from [16]. Teorem 17. Let p N wit p 3 and p < 1. Assume tat c 1 Bα L α c( B α + 4 Mα ) α {0, 1} d. (30) Ten, te subspace corrected mass smooter satisfies (27). Proof. Using Teorem 14 and (30), we obtain B c (Q D,α ) B Q D,α c (Q D,α ) L Q D,α = cl α {0,1} d α {0,1} d and L c α {0,1} d (Q D,α ) ( B + 4 M )Q D,α c( B + 4 M ), wic finises te proof. 21

23 Now, we can sow te robust convergence of te multigrid metod. Teorem 18. Let p N wit p 3 and Hp < 1. Ten, tere exist two constants τ 0 and c independent of and p (cf. Notation 1) suc tat for any τ (0, τ 0 ] te two-grid metod wit te subspace corrected mass smoter satisfies q cτ 1/2 ν 1/2, i.e., it converges if sufficiently many smooting steps ν are applied. Proof. (29) and Teorem 17 sow (27), te spectral equivalence of B and B ten sows (26). From tat estimate, we obtain L c 1 B, wic implies tat tere is some constant τ 0 suc tat τl 1 B I for all τ (0, τ 0 ]. [15, Lemma 2] implies L 1/2 B (I τl 1 B ) ν L 1/2 cτ 1 ν 1, from wic te smooting statement (23) follows using (26). Te stability statement (24) can be sown analogously. Lemma 3 yields te smooting property (22) wit C S = cτ 1/2 ν 1/2. Teorem 15 yields te approximation property (21) wit C A = c. Te combination of smooting property and approximation property yields convergence. Remark 2. Te multigrid metods discussed in tis paper can be applied also to te second biarmonic problem 2 u = f in Ω wit u = u = 0 on Γ. Remark 3. Te multigrid metods discussed in tis paper can be applied also to te tird biarmonic problem 2 u = f in Ω wit u n = u n = 0 on Γ on te parameter domain. In tis case, te subspace corrected mass smooter as to be based on te splitting of S into te space of functions in S wose odd derivatives vanis on te boundary and its ortogonal complement. Tis is te same splitting wic was used in [16]. How to transform a strong formulation of te boundary condition to te pysical domain, is not obvious. 6. Numerical results In tis section, we compare multigrid solvers based on te two smooters introduced in Section 5, te symmetric Gauss-Seidel smooter and te subspace corrected mass smooter. Tis is done first for a problem wit a trivial geometry transformation, ten for a problem wit a nontrivial geometry transformation. All numerical experiments are implemented using te G+Smo library [30]. 22

24 6.1. Experiments on te parameter domain We solve te model problem on te unit square and te unit cube; tat is, 2 u = f in Ω := (0, 1) d wit u = u n = 0 on Γ, for d = 2, 3 wit te rigt-and side f(x 1,..., x d ) := d 2 π 4 d sin (πx j ). Te problem is discretized using tensor product B-splines wit equidistant knot spans and maximum continuity. j=1 l \ p Symmetric Gauss-Seidel Subspace corrected mass smooter Table 1: Iteration counts for te unit square. l \ p Symmetric Gauss-Seidel Subspace corrected mass smooter Table 2: Iteration counts for te unit cube. We solve te resulting system using a conjugate gradient (CG) solver, preconditioned wit one multigrid V-cycle wit 1 pre and 1 post smooting step. 23

25 Wen using te W-cycle, wic is covered by te convergence teory, one obtains comparable iteration counts; as te V-cycle is more efficient, we present our results for tat case. Wen using te Gauss-Seidel smooter, we perform te multigrid metod directly on te system matrix B. Wen using te subspace corrected mass smooter, we perform te multigrid metod on te auxiliary operator B, representing te reduced inner product (, ) B(Ω). Here, we use tat te matrices B and B are spectrally equivalent wit constants independent of p and. For te subspace corrected mass smooter, we coose σ 1 := for d = 2 and σ 1 := for d = 3. In all cases, we coose τ := 1. Te initial guess is a random vector. Tables 1 and 2 sow te number of iterations needed to reduce te initial residual by a factor of 10 8 for te unit square and te unit cube. We do te experiments for several coices of te spline degree p and several uniform refinement levels l. (Te refinement level l = 0 corresponds to te domain consisting only of one element.) On te finest considered grid, te number of degrees of freedom ranges for d = 2 between around 65 and 69 tousand and for d = 3 between 250 and 301 tousand. Te number of non-zero entries of te stiffness matrix ranges for d = 2 between around 3 and 29 million and for d = 3 between 79 and 855 million. As predicted, te iteration counts of te multigrid solver wit Gauss-Seidel smooter eavily depend on p. Tis effect is amplified in te tree dimensional case. Te mass smooter (wic is proven to be p-robust) outperforms te Gauss-Seidel smooter for p 7 in te two dimensional case and for p 5 in te tree dimensional case Experiments on nontrivial computational domains In tis subsection, we present te results for te same model problem as in te previous subsection, but on te nontrivial geometries sown in Figures 1 and Figure 1: Te two-dimensional domain Figure 2: Te tree-dimensional domain Wen using te Gauss-Seidel smooter, we again perform te multigrid metod directly on te system matrix B. Wen using te subspace corrected mass smooter, we perform te multigrid metod on te auxiliary operator B, representing te reduced inner product (, ) B( Ω) on te parameter domain. Again, we use tat te matrices B and B are spectrally equivalent 24

26 wit constants independent of p and, but wic certainly depend on te geometry transformation. For te subspace corrected mass smooter, we coose σ 1 := for d = 2 and σ := for d = 3. Again, we coose τ = 1 in all cases. l \ p Symmetric Gauss-Seidel Subspace corrected mass smooter Table 3: Iteration counts for 2D pysical domain given in Figure 1 l \ p Symmetric Gauss-Seidel OoM Subspace corrected mass smooter OoM Table 4: Iteration counts for 3D pysical domain given in Figure 2 Tables 3 and 4 sow te number of iterations 1 required to reduce te initial residual by a factor of Again, we obtain very nice results for te Gauss Seidel smooter wic as for te case of trivial computational domains deteriorate if p is increased. For te mass smooter, we ave proven robustness in p and. Here, te results migt look like te mass smooter is not robust in. Te reason is tat a sufficiency small grid size is needed to capture te full effect of te geometry 1 Te entry OoM indicates tat we ran out of memory wen assembling te stffness matrix. 25

27 transformation. A similar observation can also be made for te Possion problem (cf. [16, Table 4]). Te effects of te geometry transformation can be measured by te condition number of B 1 B. For te Poisson problem, tis condition number was estimated, e.g., in [31]. For te biarmonic problem, te condition number is typically te square of te condition number for te Poisson problem, wic explains tat te dependence on te geometry transformation is more severe for te biarmonic problem A ybrid smooter Te numerical experiments ave sown tat te Gauss-Seidel smooter captures te effects of te geometry transformation quite well and tat it is superior to te mass smooter for nontrivial domains, unless p is particularly ig. Te mass smooter is robust in p, but does not perform well for non-trivial geometries. So, it seems to be a good idea to set up a ybrid smooter wic combines te strengts of bot proposed smooters. We set up again a conjugate gradient solver, preconditioned wit one multigrid V-cycle wit 1 pre and 1 post smooting step. Here, in order to represent te geometry well, te multigrid solver is set up on te original system matrix B. Te ybrid smooter consists of one forward Gauss-Seidel sweep, followed by one step of te subspace corrected mass smooter, finally followed by one backward Gauss-Seidel sweep. As always, te subspace corrected mass smooter wic requires a tensor-product matrix is constructed based on te reduced matrix B on te parameter domain. For te Gauss-Seidel sweeps, we coose τ = 1; and for te subspace corrected mass smooter, we coose τ = and σ 1 = for d = 2 and τ = 0.09 and σ 1 = for d = 3. Tables 5 and 6 sow te iteration numbers for te ybrid smooter. We see tat te iteration counts are quite robust bot in te spline degree p and in te grid level l. For small spline degrees, te iteration counts are comparable to te multigrid preconditioner wit Gauss Seidel smooter. For ig spline degrees, te ybrid smooter outperforms bot oter approaces, even if one considers tat te overall costs for one step te ybrid smooter are comparable to te overall costs of two smooting steps of one of te oter smooters. l \ p Hybrid smooter Table 5: Iteration counts for 2D pysical domain given in Figure 1 26

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