A Robust Multigrid Method for Isogeometric Analysis using Boundary Correction. C. Hofreither, S. Takacs, W. Zulehner. G+S Report No.

Transcription

1 A Robust Mutigrid Method for Isogeometric Anaysis using Boundary Correction C. Hofreither, S. Takacs, W. Zuehner G+S Report No. 33 Juy 2015

2 A Robust Mutigrid Method for Isogeometric Anaysis using Boundary Correction Cemens Hofreither a, Stefan Takacs b, Water Zuehner a a Department of Computationa Mathematics, Johannes Keper University Linz, Austria b Johann Radon Institute for Computationa and Appied Mathematics (RICAM), Austrian Academy of Sciences Abstract We consider geometric mutigrid methods for the soution of inear systems arising from isogeometric discretizations of eiptic partia differentia equations. For cassica finite eements, such methods are we known to be fast sovers showing optima convergence behavior. However, the naive appication of mutigrid to the isogeometric case resuts in significant deterioration of the convergence rates if the spine degree is increased. Recenty, a robust approximation error estimate and a corresponding inverse inequaity for B-spines of maximum smoothness have been shown, both with constants independent of the spine degree. We use these resuts to construct mutigrid sovers for discretizations based on B-spines with maximum smoothness which exhibit robust convergence rates. Keywords: Isogeometric Anaysis, geometric mutigrid, robustness 1. Introduction Isogeometric Anaysis (IgA), introduced by Hughes et a. [15], is an approach to the discretization of partia differentia equations (PDEs) which aims to bring geometric modeing and numerica simuation coser together. The fundamenta idea is to use spaces of B-spines or non-uniform rationa B-spines (NURBS) both for the geometric description of the computationa domain and as discretization spaces for the numerica soution of PDEs on such domains. As for cassica finite eement methods (FEM), this eads to inear systems with arge, sparse matrices. A good approximation of the soution of the PDE requires sufficient refinement, which causes both the dimension and the condition number of the stiffness matrix to grow. At east for probems on three-dimensiona Emai addresses: chofreither@numa.uni-inz.ac.at (Cemens Hofreither), stefan.takacs@ricam.oeaw.ac.at (Stefan Takacs), zuehner@numa.uni-inz.ac.at (Water Zuehner) December 22, 2015

3 domains or the space-time cyinder, the use of direct sovers does not seem feasibe. The deveopment of efficient inear sovers or preconditioners for such inear systems is therefore essentia. For cassica FEM, it is we-known that hierarchica methods, ike mutigrid and mutieve methods, are very efficient and show optima compexity, that is, the required number of iterations for reaching a fixed accuracy goa is independent of the grid size. In this case, the overa computationa compexity of the method grows ony ineary with the number of unknowns. It therefore seems natura to extend these methods to IgA, and severa resuts in this direction can be found in the iterature. Mutigrid methods for IgA based on cassica concepts have been considered in [10, 13, 14], and a cassica mutieve method in [2]. It has been shown eary on that a standard approach to constructing geometric mutigrid sovers for IgA eads to methods which are robust in the grid size [10]. However, it has been observed that the resuting convergence rates deteriorate significanty when the spine degree is increased. Even for moderate choices ike spines of degree four, too many iterations are required for practica purposes. Recenty, in [12] some progress has been made by using a Richardson method preconditioned with the mass matrix as a smoother (mass-richardson smoother). The idea is to carry over the concept of operator preconditioning to mutigrid smoothing: here the (inverse of) the mass matrix can be understood as a Riesz isomorphism representing the standard L 2 -norm, the Hibert space where the cassica mutigrid convergence anaysis is deveoped. Loca Fourier anaysis indicates that a mutigrid method equipped with such a smoother shoud show convergence rates that are independent of both the grid size and the spine degree. However, numerica resuts indicate that the proposed method is not robust in the spine degree in practice. This is due to boundary effects, which cannot be captured by oca Fourier anaysis. Simiar techniques have been used to construct symbo-based mutigrid approaches for IgA, see [8, 7, 9]. In the present paper, we take a coser ook at the origin of these boundary effects and introduce a boundary correction that deas with these effects. We prove that the mutigrid method equipped with a propery corrected mass- Richardson smoother converges robusty both in the grid size and the spine degree for one- and two-dimensiona probems. For the proof, we make use of the resuts of the recent paper [16], where robust approximation error estimates and robust inverse estimates for spines of maximum smoothness have been shown. We present numerica resuts that iustrate the theoretica resuts. Throughout the paper we restrict ourseves to the case of spines with maximum continuity, which is of particuar interest in IgA. The buk of our anaysis is first carried out in the one-dimensiona setting. Whie mutigrid sovers are typicay not interesting in this setting from a practica point of view, the tensor product structure of the spine spaces commony used in IgA ends itsef very we to first anayzing the one-dimensiona case and then extending the resuts into higher dimensions. In the present work, we extend the ideas from the one-dimensiona to the two-dimensiona case and obtain a robust and efficienty reaizabe smoother for that setting. 2

4 The paper is organized as foows. In Section 2, we introduce the spine spaces used for the discretization and the eiptic mode probem. A genera mutigrid framework is introduced in Section 3. This framework requires a proper choice of the smoother, which is discussed in detai for one-dimensiona domains in Section 4. In Section 5, we extend these resuts to the case of two-dimensiona domains. Some detais on the numerica reaization of the proposed smoother as we as numerica experiments iustrating the theory are given in Section 6. In Section 7, we cose with some concuding remarks. A few proofs are postponed to the Appendix. 2. Preiminaries 2.1. B-spines and tensor product B-spines First, we consider the case of one-dimensiona domains and assume, without oss of generaity, Ω = (0, 1). We introduce for any N 0 = {0, 1, 2,...} a uniform subdivision (grid) by spitting Ω into n = n 0 2 subintervas of ength h := 1 n = 1 n 0 2 by uniform dyadic refinement. On these grids we introduce spaces of spine functions with maximum smoothness as foows. Definition 1. For p N := {1, 2, 3,...}, S p, (0, 1) is the space of a functions in C p 1 (0, 1) which are poynomias of degree p on each subinterva ((i 1)h, ih ) for i = 1,..., n. Here C m (Ω) denotes the space of a continuous functions mapping Ω R that are m times continuousy differentiabe. We assume that the coarsest grid is chosen such that n 0 > p hods. Note that the spaces are nested for fixed spine degree p, that is, S p, 1 (0, 1) S p, (0, 1), and the number of degrees of freedom roughy doubes in each refinement step. The parameter wi pay the roe of the grid eve in the construction of our mutigrid agorithm. As a basis for S p, (Ω), we choose B-spines as described by, e.g., de Boor [6]. To this end, we introduce an open knot vector with the first and ast knot repeated p+1 times each, (0,..., 0, h, 2h,..., (n 1)h, 1,..., 1), and define the normaized B-spine basis over this knot vector in the standard way. We denote the B-spine basis functions by {ϕ (1) p,,..., ϕ(m ) p, }, where m = dim S p, (Ω) = n + p. They form a partition of unity, that is, m j=1 ϕ(j) p, (x) = 1 for a x Ω. In higher dimensions, we wi assume Ω = (0, 1) d with d > 1 and introduce tensor product B-spine basis functions of the form (x, y) ϕ (j1) p, (x)ϕ(j2) p, (y) for two dimensions and anaogous functions for higher dimensions. The space spanned by these basis functions wi again be denoted by S p, (Ω). The extension to the case where Ω R d is the tensor product of d arbitrary bounded and open intervas is straightforward. Simiary, it is no probem to have different spine degrees, grid sizes, and/or number of subintervas for each dimension, as ong as the grid sizes are approximatey equa in a directions. However, for the sake of simpicity of the notation, we wi aways assume that the discretization is identica in each coordinate direction. 3

5 2.2. Mode probem For the sake of simpicity, we restrict ourseves to the foowing mode probem. Let Ω = (0, 1) d and assume f L 2 (Ω) to be a given function. Find a function u : Ω R such that u + u = f in Ω, u = 0 on Ω. n In variationa form, this probem reads: find u V := H 1 (Ω) such that ( u, v) L 2 (Ω) + (u, v) L 2 (Ω) }{{} (u, v) A := = (f, v) L 2 (Ω) }{{} f, v := v V. Here and in what foows, L 2 (Ω) is the standard Lebesgue space of square integrabe functions and H 1 (Ω) denotes the standard Soboev space of weaky differentiabe functions with derivatives in L 2 (Ω). Appying a Gaerkin discretization using spine spaces, we obtain the foowing discrete probem: find u V := S p, (Ω) V such that (u, v ) A = f, v v V. Using the B-spine basis introduced in the previous subsection, the discretized probem can be rewritten in matrix-vector notation, A u = f, (1) where A is the B-spine stiffness matrix. Here and in what foows, underined symbos for prima variabes, ike u and v, are the coefficient vectors representing the corresponding functions u and v with respect to the B-spine basis. Remark 1. Our mode probem may seem rather restrictive in the sense that we ony aow for tensor product domains. In practica IgA probems, one often uses such domains as parameter domains and introduces a geometry mapping, usuay given in the same basis as used for the discretization, which maps the parameter domain to the actua physica domain of interest. As ong as the geometry mapping is we-behaved, a good sover on the parameter domain can be used as a preconditioner for the probem on the physica domain, and our mode probem therefore captures a essentia difficuties that arise in the construction of mutigrid methods in this more genera setting. Geometry mappings with singuarities as we as muti-patch domains are beyond the scope of this paper. 3. A mutigrid sover framework 3.1. Description of the mutigrid agorithm We now introduce a standard mutigrid agorithm for soving the discretized equation (1) on grid eve. Starting from an initia approximation u (0), the next iterate u (1) is obtained by the foowing two steps: 4

6 Smoothing procedure: For some fixed number ν of smoothing steps, compute ( ) u (0,m) := u (0,m 1) + τl 1 f A u (0,m 1) for m = 1,..., ν, (2) where u (0,0) := u (0). The choice of the matrix L and the damping parameter τ > 0 wi be discussed beow. Coarse-grid correction: Compute the defect and restrict it to grid eve 1 using a restriction matrix I 1 : r (1) 1 := I 1 ( f A u (0,ν) Compute the correction p (1) 1 by approximatey soving the coarsegrid probem A 1 p (1) = 1 r(1) 1. (3) Proongate p (1) 1 to the grid eve using a proongation matrix I 1 and add the resut to the previous iterate: u (1) := u (0,ν) + I 1 p (1) 1. As we have assumed nested spaces, the intergrid transfer matrices I 1 and I 1 can be chosen in a canonica way: I 1 is the canonica embedding and the restriction I 1 its transpose. In the IgA setting, the proongation matrix I 1 can be computed by means of knot insertion agorithms. If the probem (3) on the coarser grid is soved exacty (two-grid method), the coarse-grid correction is given by ( ) u (1) := u (0,ν) + I 1 A 1 1 I 1 f A u (0,ν). (4) In practice, the probem (3) is approximatey soved by recursivey appying one step (V-cyce) or two steps (W-cyce) of the mutigrid method. On the coarsest grid eve ( = 0), the probem (3) is soved exacty by means of a direct method. The crucia remaining task is the choice of the smoother. For mutigrid methods for eiptic probems with a FEM discretization, it is common to use a damped Jacobi iteration or a Gauss-Seide iteration as the smoother. This is aso possibe if an isogeometric discretization is used, but eads to significant deterioration in the convergence rates as p is increased ([10, 14]). This motivates to choose a non-standard smoother. In the seque of this section, we derive conditions on the matrix L, guaranteeing convergence, which we use in the foowing two sections to construct a smoother for the mode probem. ). 5

7 3.2. Abstract mutigrid convergence theory As the two-grid method is a inear iteration scheme, its action on the error is fuy described by the iteration matrix. Let u denote the exact soution of (1). Then the initia error u u(0) and the error after one two-grid cyce u u(1) are reated by the equation where u u (1) = (I T 1 )S ν (u u (0) ), S = I τl 1 A, T 1 = I 1 A 1 1 I 1 A. Observe that S and I T 1 are the iteration matrices of the smoother and of the coarse-grid correction, respectivey. Throughout the paper we use the foowing notations. For a symmetric and positive definite matrix Q and a vector u, we define u Q := (Q u, u ) 1/2 = Q 1/2 u, where and (, ) are the Eucidean norm and scaar product, respectivey. The associated matrix norms are denoted by the same symbos Q and. For ease of notation, we use the same symbos aso for the norm of functions u V and mean in this case their appication to the coefficient vector, i.e., u Q := u Q. As A is the matrix representation of the biinear form (, ) A, we have u A = u A, where A := (, ) 1/2 A is the energy norm. We assume that the matrix L appearing in (2) is symmetric and positive definite. Foowing the cassica ine of, e.g., [1], we show convergence in that norm, i.e., we show that q := (I T 1 )S ν L < 1, (5) which obviousy impies the q-inear convergence property u u (1) L q u u (0) L. To anayze (5), we use semi-mutipicity of norms and obtain (I T 1 )S ν L = L 1/2 (I T 1 )S ν L 1/2 L 1/2 (I T 1 )A 1 L 1/2 L 1/2 A S ν L 1/2. (6) Therefore, it suffices to prove the two conditions L 1/2 (I T 1 )A 1 L 1/2 C A, (approximation property) (7) L 1/2 A S ν L 1/2 C S ν 1, (smoothing property) (8) cf. [11, eq. (6.1.5)] for this spitting. Then (5) foows immediatey for ν > C A C S, that is, if sufficienty many smoothing steps are appied. As we are interested in robust convergence, the constants C A and C S shoud be independent of both the grid size and the spine degree. The approximation property (7) is a direct consequence of the foowing approximation error estimate. Observe that T 1 is the matrix representation 6

8 of the A-orthogona projector from V to V 1. For simpicity, we use the same symbo T 1 aso for the A-orthogona projector from V to V 1 itsef and, ater on, aso for the A-orthogona projector from the whoe space V = H 1 (Ω) to V 1. Lemma 1. The approximation error estimate (I T 1 )u 2 L C A u 2 A u V (9) is equivaent to the approximation property (7) with the same constant C A. Proof. The estimate (9) can be rewritten as X C 1/2 A with X := L 1/2 (I T 1 )A 1/2, which is equivaent to X X T C A. We have X X T = L 1/2 (I T 1 )A 1 (I T 1 ) T L 1/2 = L 1/2 (I T 1 ) 2 A 1 L 1/2, where we used ((I T 1 )A 1 ) T = (I T 1 )A 1, which foows from I T 1 being sef-adjoint with respect to A. Since I T 1 is a projector, the statement foows. The foowing emma states that the smoothing property is a direct consequence of an inverse inequaity. Lemma 2. Let A and L be symmetric and positive definite matrices. Assume that the inequaity u 2 A C I u 2 L u V (10) hods. Then for τ (0, C 1 I ], the smoother (2) satisfies the smoothing property (8) with C S = τ 1. Under the same assumptions, S A 1 hods. Proof. The proof is based on [11, Lemma 6.2.1]. From (10), we immediatey obtain τ 1 A 1/2 L 1/2 2 = ρ(l 1/2 A L 1/2 ), where ρ( ) is the spectra radius. Observe that L 1/2 A S ν L 1/2 = L 1/2 A (I τl 1 A ) ν L 1/2 = Ā(I τā) ν with Ā := L 1/2 A L 1/2 and that Ā(I τā) ν is symmetric. Furthermore, we have the spectra radius bound ρ(τā) = τρ(l 1/2 A L 1/2 ) 1. Thus, L 1/2 A S ν L 1/2 = ρ(ā(i τā) ν ) = sup λ σ(ā) ( ) ν ν = τ ν 1 + ν τ 1 ν, λ(1 τλ) ν τ 1 sup µ(1 µ) ν µ [0,1] which shows (8). Here σ( ) denotes the spectrum of a matrix. Simiary we have S A = ρ(i τā) 1. 7

9 Observe that estimate (10) is, indeed, of the typica form of an inverse inequaity, if L is a (propery scaed) mass matrix representing the L 2 inner product. In view of Lemmas 1 and 2, we can state sufficient conditions for the convergence of a two-grid method. Theorem 3. Assume that there are constants C A and C I, independent of the grid size and the spine degree, such that the approximation error estimate (9) and the inverse inequaity (10) hod. Then the two-grid method converges for the choice τ (0, C 1 I ] and ν > ν 0 := τ 1 C A with rate q = ν 0 /ν. Proof. The statement on the convergence of the two-grid method foows directy from (6), Lemma 1, and Lemma 2. The assumptions of Theorem 3 are aso sufficient to prove convergence of a symmetrica variant of the W-cyce mutigrid method. Here, in addition to the smoothing steps before the coarse-grid correction (pre-smoothing), we perform an equa number of smoothing steps after the coarse-grid correction (post-smoothing). Theorem 4. Under the assumptions of Theorem 3, the W-cyce mutigrid method with ν/2 pre- and ν/2 post-smoothing steps converges in the energy norm A for the choice τ (0, C 1 I ] and ν > 4ν 0 = 4τ 1 C A with rate q = 2ν 0 /ν. Proof. Theorem 3 states that (I T 1 )S ν L τ 1 C A. ν As S ν/2 (I T 1 )S ν/2 is sef-adjoint in the scaar product (, ) A, this impies S ν/2 (I T 1 )S ν/2 A = ρ((i T 1 )S ν ) (I T 1 )S ν L τ 1 C A. (11) ν The iteration matrix of the W-cyce mutigrid method is recursivey given by W = S ν/2 (I I 1(I W 1)A I 1 A )S ν/2 for > 0, and W 0 = 0. Using the triange inequaity and semi-mutipicativity of norms, we obtain for the convergence rate q = W A = S ν/2 (I I 1(I W 1)A I 1 A )S ν/2 A S ν/2 (I T 1 )S ν/2 A + S ν/2 I 1W 1A I 1 A S ν/2 A S ν/2 (I T 1 )S ν/2 A + S ν A A 1/2 I 1A 1/2 1 2 W 1 2 A 1. As the spaces V are nested, we obtain A 1 = I 1 A I 1. Thus it foows A 1/2 I 1 A 1/2 1 2 = ρ(i 1 A I 1 A 1 1 ) = 1. Lemma 2 states that S A 1. Using these two statements, (11) and q 1 = W 1 A 1, we obtain q ν 0 ν + q2 1. Using the assumption ν 4ν 0, one shows by induction that q 2ν 0 /ν. 8

10 4. A robust mutigrid method for one-dimensiona domains 4.1. Robust estimates for a subspace of the spine space For the mode probem, we have V = H 1 (Ω) and A = H 1 (Ω). (12) The standard mutigrid convergence anaysis as introduced by Hackbusch [11] gives convergence in a (propery scaed) L 2 -norm, L = h 1 L 2 (Ω), (13) and thus the choice L = h 2 M, where M is the mass matrix, consisting of pairwise L 2 -scaar products of the basis functions. To show robust convergence of the mutigrid sover with a Jacobi smoother, one woud show that the diagona of A and h 2 M are spectray equivaent. This equivaence hods robusty in the grid size, however the invoved constants deteriorate with increasing spine degree p. This issue is cosey reated to the so-caed condition number of the B-spine basis (see, e.g., [4]). The growth of the condition number with p expains why standard Jacobi iteration and Gauss-Seide iteration do not work we for B-spines. The aforementioned equivaence is not necessary for a direct appication of the choice L = h 2 M, which is the mass-richardson smoother aready studied in [12]. Loca Fourier anaysis indicates that this smoother shoud ead to robust convergence of the mutigrid sover. However, the numerica experiments in [12] show that this is not the case and iteration numbers sti deteriorate with p. The reason for this effect is motivated as foows. With the choice (12) and (13), the condition (10) reads u H1 (Ω) C 1/2 I h 1 u L2 (Ω) u V = S p, (Ω). (14) In other words, it is required that the spine space satisfies an inverse inequaity with a constant that is independent of the grid size and the spine degree. However, such a robust inverse inequaity does not hod, as the counterexampe given in [16] shows: for each p N and each N 0, there exists a spine w S p, (Ω) with w H1 (Ω) ph 1 w L2 (Ω). (15) As we have to choose τ C 1 I for the smoother not to diverge, the existence of such spines w impies that one has to choose τ = O(p 2 ), which causes the convergence rates to deteriorate. The spines w in the counterexampe are nonperiodic and therefore cannot be captured by oca Fourier anaysis and simiar toos which consider ony the periodic setting. In [16] it was shown that a robust inverse estimate of the form (14) does hod for the foowing arge subspace of S p, (Ω). 9

11 Definition 2. Let Ω = (0, 1). We denote by S p, (Ω) the space of a u S p, (Ω) whose odd derivatives of order ess than p vanish at the boundary, 2+1 x 2+1 u (0) = 2+1 x 2+1 u (1) = 0 for a N 0 with < p. The space S p, (Ω) is amost as arge as S p, (Ω): their dimensions differ by p (for p even) or p 1 (for p odd). Theorem 5 ([16]). For a spine degrees p N and a grid eves N 0, we have the inverse inequaity u H1 (Ω) 2 3h 1 u L2 (Ω) u S p, (Ω). This resut makes it cear that the counterexampe (15) describes the effect of ony a few outiers (see aso [3] on the topic of spectra outiers in IgA), connected to the boundary. We make use of this fact by constructing a mass-richardson smoother with a (ow-rank) boundary correction. Using u 2 A = u 2 H 1 (Ω) + u 2 L 2 (Ω) and h 1, the inverse inequaity impies u A c u h 2 M u S p, (Ω). (16) Here and in the seque, we use c to refer to a generic constant which is independent of both the grid eve and the spine degree p. In addition to the inverse estimate, aso an approximation property for the subspace S p, (Ω) was proved in [16]. We sighty refine the resut here as foows. Theorem 6. We have the approximation error estimate (I T )u L2 (Ω) 2 2h u H1 (Ω) u H 1 (Ω), where T : H 1 (Ω) S p, (Ω) is the A-orthogona projector. The proof for this theorem is given in the Appendix. Using (16) and Theorem 6, we immediatey obtain assumptions (10) and (9) of Theorem 3 with the choice V = S p, (Ω) and L = h 2 M. Thus, a two-grid method using the mass smoother woud be robust in this subspace. However, the probems we are interested in are typicay discretized in the fu spine space S p, (Ω). We extend the choice of L robusty to such probems in the foowing Extension of the inverse inequaity to the entire spine space In this section, we modify the mass matrix in such a way as to satisfy a robust inverse inequaity in the entire spine space. This is done by means of an additiona term which is essentiay a discrete harmonic extension. We set L := h 2 M + (I T ) T A(I T ). 10

12 Here, the second term corrects for the vioation of the inverse inequaity in the A-orthogona compement of S p, (Ω) by incorporating the origina operator A. This term satisfies the energy minimization property u (I T ) T A(I T ) = (I T )u A = inf u + v A. v S p, (Ω) We now prove the inverse inequaity for the modified norm. The L proof requires both the inverse inequaity and the approximation error estimate in S p, (Ω). Lemma 7. We have the inverse inequaity u A c u L u S p, (Ω) with a constant c which is independent of and p. Proof. Let u S p, (Ω). Using the triange inequaity, the inverse inequaity (16) and again the triange inequaity we obtain u A T u A + (I T )u A ch 1 T u M + (I T )u A ch 1 u M + ch 1 (I T )u M + (I T )u A. By Theorem 6, we have (I T )u M = (I T ) 2 u M ch (I T )u A, and it foows ( u A c h 1 u M + (I T ) )u A. The statement foows since (I T )u A = u (I T ) T A (I T ) A robust mutigrid method for the whoe spine space We first show that Theorem 6 can be easiy extended to the projection into the entire spine space. Lemma 8. The approximation error estimate (I T )u L2 (Ω) ch u H1 (Ω) u H 1 (Ω), hods, where T : H 1 (Ω) S p, (Ω) is the A-orthogona projector. Proof. As S p, (Ω) S p, (Ω), we have T = T T. This identity, the triange inequaity, Theorem 6, and the stabiity of the A-orthogona projector T yied (I T )u 2 L 2 (Ω) 2( (I T )u 2 L 2 (Ω) + (I T )T u 2 L 2 (Ω) ) 2ch ( u 2 H 1 (Ω) + T u 2 H 1 (Ω) ) 4ch u 2 H 1 (Ω) u H 1 (Ω). The robust inverse inequaity (Lemma 7) together with the approximation property (Lemma 8) aow us to prove robust convergence of the foowing mutigrid method for the space V := S p, (Ω). It turns out that we aso obtain robust convergence if we do not project into S p, (Ω), but into a subspace Sp, I (Ω), which may be easier to hande in practice. 11

13 Theorem 9. Let Sp, I (Ω) S p, (Ω) a subspace and T I : S p,(ω) Sp, I (Ω) the A-orthogona projector. Consider the smoother (2) with the choice L := h 2 M + (I T I ) T A (I T I ). (17) Then there exists a damping parameter τ > 0 and a choice of ν 0 > 0, both independent of and p, such that the two-grid method with ν > ν 0 smoothing steps converges with rate q = ν 0 /ν. With the same choice of τ and with ν > 4ν 0, aso the W-cyce mutigrid method with ν/2 pre- and ν/2 post-smoothing steps converges in the energy norm with convergence rate q = 2ν 0 /ν. Proof. We show the assumptions of Theorem 3. The statements then foow from Theorems 3 and 4. Proof of assumption (10). Let u S p, (Ω). Lemma 7 states u 2 A c( u 2 + (I T h 2 M )u 2 A). Using the energy-minimizing property of the projection operators, we obtain (I T )u A = inf u v A v S p, (Ω) inf v S I p, (Ω) u v A = (I T I )u A since S I p, (Ω) S p, (Ω). Combining these estimates, we get the desired statement u 2 A c u 2 L. (18) Proof of assumption (9). Let u S p, (Ω). Lemma 8 impies (I T 1 )u 2 h 2 M c u 2 A Since both I T I and I T 1 are A-orthogona projectors and thus are stabe in the A-norm, we obtain the approximation error estimate (9) via (I T 1 )u 2 L = (I T 1 )u 2 + (I T I h 2 M )(I T 1 )u 2 A (c + 1) u 2 A. The choice S I p, (Ω) = S p, (Ω) in the above theorem is admissibe and resuts in a robust mutigrid method. However, the space S p, (Ω) is somewhat difficut to work with in practice since it is not easy to represent in terms of the standard B-spine basis. Therefore, we choose a sighty smaer space for Sp, I (Ω) which is based on a simpe spitting of the degrees of freedom. To this end, we spit the spine space S p, (Ω) into boundary and inner functions, S p, (Ω) = S Γ p,(ω) + S I p,(ω), 12

14 where Sp, Γ (Ω) is spanned by the first p B-spine basis functions ϕ(1) p,,..., ϕ(p) p, and the ast p basis functions ϕ (m p+1) p,,..., ϕ (m ) p,, whereas the space Sp, I (Ω) is spanned by a the remaining basis functions. By construction, Sp, I (Ω) consists of a spines that vanish on the boundary together with a derivatives up to order p 1. Therefore, we have Sp, I (Ω) S p, (Ω). Due to this subspace reation, the inverse inequaity (16) remains vaid in Sp, I (Ω). However, the anaogue of the approximation property Theorem 6 does not in genera hod in this smaer space. The existence of the arger subspace S p, (Ω) in which both properties hod is crucia for the proof of assumption (10) above, even though we do not directy use the space S p, (Ω) in practice. Concerning the practica reaization of such a smoother, observe that we can reorder the degrees of freedom based on the spitting S p, (Ω) = Sp, Γ (Ω)+SI p, (Ω) and write the matrix A and the vector u in bock structure as ( AΓΓ, A T IΓ, A = A IΓ, A II, ), u = ( uγ, u I, ). (19) Using a matrix representation of the projector T I, direct computation yieds ( ) (I T I u )u = Γ, A 1 II, A, IΓ,u Γ, and therefore the matrix L can be represented using the Schur compement as ( L = h 2 M + C := h 2 AΓΓ, A M + T IΓ, A 1 II, A ) IΓ, 0. (20) 0 0 We remark that u T C u represents the (squared) energy of the discrete harmonic extension of u Γ, from S Γ p, (Ω) to S p,(ω). 5. A robust mutigrid method for two-dimensiona domains In this section, we extend the theory presented in the previous section to the (more reevant) case of two-dimensiona domains. As outined in Remark 1, we restrict ourseves to probems without geometry mapping, i.e., probems on the unit square ony. However, these approaches can be used as preconditioners for probems on genera geometries if there is a reguar geometry mapping. In the foowing, we are interested in soving the probem (1) with d = 2. Here, we assume to have a tensor product space. Without oss of generaity, we assume for sake of simpicity that Ω = (0, 1) 2 and S p, (Ω) := S p, (0, 1) S p, (0, 1). Beow, we wi use caigraphic etters to refer to matrices for the twodimensiona domain, whereas standard etters refer to matrices for the onedimensiona domain. Using the tensor product structure of the probem and the discretization, the mass matrix has the tensor product structure M = M M, 13

15 where denotes the Kronecker product. To keep the notation simpe, we assume that we have the same basis and thus the same matrix M for both directions of the two-dimensiona domain. However, this is not needed for the anaysis. The stiffness matrix in two dimensions has the representation A = K M + M K + M M, (21) where K represents the scaar product ( u, v ) L 2 (Ω) in one dimension. This decomposition refects that (u, v ) A = ( x u, x v ) L 2 (Ω) + ( y u, y v ) L 2 (Ω) + (u, v ) L 2 (Ω). The probem of interest is now A u = f. Here, the mutigrid framework from Section 3 appies unchanged, ony repacing standard etters by caigraphic etters. Again, we have to choose a suitabe smoother by determining the matrix L. To this end, we again have a ook at the approximation error estimate and the inverse inequaity. Let S p, (Ω) := S p, (0, 1) S p, (0, 1) and et A be the energy norm for the probem in two dimensions, i.e., such that u A = u A hods for a u S p, (Ω). An approximation error estimate for the space S p, (Ω) was shown in [16, Theorem 8]. However, there, not the A-orthogona projector was anayzed. The foowing statement is anaogous to Theorem 6 and can be proved based on the one-dimensiona resut. Theorem 10. The approximation error estimate (I T )u L2 (Ω) ch u A u S p, (Ω), where T : S p, (Ω) S p, (Ω) is the A-orthogona projector, hods with a constant c which is independent of and p. Proof. Theorem 6 impies that (I T )u M ch u A u S p, (0, 1), where T : S p, (0, 1) S p, (0, 1) is the A-orthogona projector. Stabiity of the A-orthogona projector means that (I T )u A u A u S p, (0, 1). Combining these statements, we obtain (I T T )u A M +M A ch u A A u S p, (Ω). 14

16 Note that A = K + M. Therefore, A = K M + M K + M M A M +M A hods, where is understood in the spectra sense. Moreover, A M 1 A = K M 1 K M + K K + K M + K K + M K M 1 K + M K + K M + M K + M M K K + K M + M K + M M = A A hods. Using these two estimates, we obtain further (I T T )u A ch u A M 1 A u S p, (Ω). As the A-orthogona projection minimizes the norm A = A, we obtain or in matrix form (I T )u A By transposing and using A 1 T T ch u A M 1 A u S p, (Ω), A 1/2 (I T )A 1 M 1/2 ch. M 1/2 = T A 1, it foows (I T )A 1/2 ch which can be equivaenty rewritten as the desired resut (I T )u M ch u A u S p, (Ω). Competey anaogousy to the proof of Lemma 8, we can extend this approximation error estimate to (I T 1 )u L 2 (Ω) ch u A u S p, (Ω), (22) where T : H 1 (Ω) S p, (Ω) is the A-orthogona projector. From [16, Theorem 9], the foowing robust inverse inequaity in S p, (Ω), anaogous to the one-dimensiona resut (16), foows: u A 2 6h 1 u L 2 (Ω) u S p, (Ω). As in Section 4 for the one-dimensiona case, we coud set up a smoother based on the A-orthogona projection to an interior space Sp, I (0, 1) SI p, (0, 1) and prove, by competey anaogous arguments, that the resuting mutigrid method is robust. However, the reaization of this smoother requires the discrete harmonic extension from a boundary ayer of width p to the interior, which is too computationay expensive. Instead we propose a sighty modified smoother which better expoits the tensor product structure of the underying spaces. Using A = K + M, we obtain A A M + M A. 15

17 As we have seen that A cl for the choice of L introduced in Section 4, the inverse inequaity woud be satisfied for the choice L M + M L, which can be expressed using L = h 2 M + C from (20) as foows: 2h 2 M M + C M + M C Instead we choose L to be a sight perturbation of that matrix, namey, L := h 2 M M + C M + M C = h 2 (L L C C ), Thus, L is the sum of a tensor product matrix h 2 L L and a ow-rank correction h 2 C C. The foowing two emmas show the assumptions for mutigrid convergence when using the smoother based on this L. Lemma 11. There is a constant c, independent of and p, such that A c L. Proof. The one-dimensiona inverse inequaity (18) in matrix formuation reads A cl. Using L = h 2 M + C, we obtain A A M + M A c(l M + M L ) = c(2h 2 M M + C M + M C ) 2c(h 2 M M + C M + M C ) = 2cL. Lemma 12. The approximation property (I T 1 )u L c u A u S p, (Ω) hods with a constant c which is independent of and p. Proof. By an energy minimization argument, we know that C A and therefore L h 2 M + M A + A M. Furthermore, since M A + A M = A + M and h 1, we obtain L 2h 2 M + A. Thus, using (22) and the stabiity of the A-orthogona projector I T 1, we have (I T 1 )u 2 L 2 (I T 1 )u 2 + (I T h 2 M 1 )u 2 A (2c + 1) u 2 A for a u S p, (Ω). We now immediatey obtain the foowing convergence resut. 16

18 Theorem 13. Consider the smoother (2) with L = h 2 (L L C C ), where L = h 2 M + C and C = (I T I)T A(I T I ) as in Section 4, and T I : S p,(0, 1) Sp, I (0, 1) is the A-orthogona projector. Then, there is some damping parameter τ > 0 and some choice ν 0 > 0, both independent of and p, such that the two-grid method with ν > ν 0 smoothing steps converges with rate q = ν 0 /ν. With the same choice of τ and with ν > 4ν 0, aso the W-cyce mutigrid method with ν/2 pre- and ν/2 post-smoothing steps converges in the energy norm with convergence rate q = 2ν 0 /ν. Proof. Lemma 11 and Lemma 12 show the assumptions (10) and (9) of Theorem 3, respectivey. The statements then foow from Theorems 3 and 4. To obtain a quasi-optima method, we need a way of inverting the matrix L efficienty. Grouping the degrees of freedom as in (19), we obtain L = h 2 L L h 2 [P T P T ][Q Q ][P P ] (23) with ( I P := 0 ) and Q := A ΓΓ, A T IΓ,A 1 II, A IΓ,. (24) Expoiting the Kronecker product structure, L L can be efficienty inverted using the ideas from [5]. The second term, h 2 [P T P T ][Q Q ][P P ], is a ow-rank correction that ives ony near the four corners of the domain. We can thus invert the smoother using the Sherman-Morrison-Woodbury formua and obtain L 1 = h 2 (I + ([L 1 P ] [L 1 P ])R 1 (P T P T ))(L 1 where P and Q are defined as in (24), R := Q 1 Q 1 W 1 L 1 ), W 1, (25) W := Q + h 2 (M ΓΓ, MIΓ,M T 1 II, M IΓ,), (26) and M II,, M IΓ, and M ΓΓ, are the bocks of the mass matrix M if the degrees of freedom are grouped as in (19). 6. Numerica reaization and resuts 6.1. Numerica reaization As the proposed smoothing procedure appears rather compex, we give in Listing 6.1 pseudo code describing how the two-dimensiona smoother based on 17

19 function smoother() given matrices: one-dimensiona mass and stiffness matrices M, A R m m input: function vaue u (0,n) Preparatory steps, ony performed once: R m2, corresponding residua r (0,n) 1. Compute the Choesky factorization of A II,. 2. Determine Q R 2p 2p as defined in (24). 3. Compute the Choesky factorization of M II,. 4. Determine W R 2p 2p as defined in (26). 5. Determine R R 4p2 4p 2 as defined in (25). 6. Compute the Choesky factorization of R. 7. Determine the sparse matrix L R m m as defined in (20). 8. Compute the Choesky factorization of L. For each smoothing step do: 1. Determine q (0) := h 2 (L 1 L 1 )r. 2. Determine q (1) := (P T P T )q(0). 3. Determine q (2) 4. Determine q := ([L 1 := R 1 q (1) using the Choesky factorization of R. P ] [L 1 P ])q (2) using the Choesky factorization of L. 5. Set p = q (0) + q. 6. Update the function u (0,n+1) 7. Update the residua r (0,n+1) R m2 := u (0,n) + τp. := r (0,n) τa p. output: function vaue u (0,n+1), corresponding residua r (0,n+1) Listing 1: Pseudo-code impementation for the smoother in two dimensions 18

20 the matrix L as described in (23) can be impemented efficienty such that the overa mutigrid method achieves optima compexity. The overa costs of the preparatory steps are O(m p 2 + p 6 ) foating point operations, where m > p is the number of degrees of freedom in one dimension and m 2 is the overa number of degrees of freedom. Here, for the preparatory steps 1 and 3, O(m p 2 ) operations are required, as the dimension of the matrices A II, and M II, is m 2p = O(m ) and the bandwidth is O(p). For the preparatory steps 2 and 4, it is required to sove O(p) inear systems invoving this factorization, which requires again O(m p 2 ) operations. The preparatory steps 5 and 6 just ive on the 4 vertices of the square and require O(p 6 ) operations. The preparatory step 7 costs O(m p) operations just for adding A and L. The Choesky factorization to be performed in the preparatory step 8 has as in the steps 1 and 3 a computationa compexity of O(m p 2 ) operations. The steps 1 and 4 of the smoother itsef require O(m 2 p) operations each, if the tensor product structure L 1 L 1 = (I L 1 )(L 1 I) is used (see [5] for the agorithmic idea). Step 2 of the smoother requires O(m p 2 ) operations. In step 3, ony O(p 4 ) operations are required as R is a dense 4p 2 4p 2 -matrix. The steps 5 and 6, which are ony adding up, can be competed with O(m 2 ) operations. Step 7 can be computed using the decomposition A = (I M )(K I) + (M I)(I K ) + (M I)(I M ) (27) with O(m 2 p) operations. In summary, the preparatory steps require O(m p 2 + p 6 ) operations and the smoother itsef requires O(m 2 p) operations. Furthermore, aso the coarsegrid correction can be performed with O(m 2 p) operations, if the tensor product structure of the intergrid transfer matrices I 1 and I 1 is used. The overa costs are therefore O(m 2 p +p6 ) or, assuming p 5 m 2, O(m2 p) operations. The method can therefore be caed asymptoticay optima, since the mutigrid sover requires O(m 2 p) operations in tota, which is the same effort as for mutipication with A. In the case of variabe coefficients or geometry transformations, A is no onger a sum of tensor-product matrices. In this case, the computation of the residua of the origina probem requires O(m 2 p2 ) operations, whie the appication of the mutigrid method as a preconditioner can sti be performed with O(m 2 p) operations Experimenta resuts As a numerica exampe, we sove the mode probem from Section 2.2 with the right-hand side d f(x) = dπ 2 sin(π(x j )) j=1 and homogeneous Neumann boundary conditions on the domain Ω = (0, 1) d, d = 1, 2. We perform a (tensor product) B-spine discretization using uniformy 19

21 sized knot spans and maximum-continuity spines for varying spine degrees p. We refer to the coarse discretization with ony a singe interva as eve = 0 and perform uniform, dyadic refinement to obtain the finer discretization eves with 2 d eements (intervas or quadriateras). We set up a V-cyce mutigrid method according to the framework estabished in Section 3 and using the proposed smoother (20) for the one-dimensiona domain and (23) for the two dimensiona domain. We aways use one pre- and one post-smoothing step. The damping parameter τ was chosen as τ = 0.14 for d = 1 and τ = 0.08 for d = 2. In the one-dimensiona case, we appy the damping factor ony to the mass component of the smoother. This sighty improves the iteration numbers but is non-essentia. Due to the requirement that the number of intervas exceed p so that the space Sp, I (0, 1) is nonempty and the construction of the smoother is vaid, we cannot aways use = 0 as the coarse grid for the mutigrid method. We perform two-grid iteration unti the Eucidean norm of the initia residua is reduced by a factor of The iteration numbers using varying spine degrees p as we as varying fine grid eves for the one-dimensiona domain are given in Tabe 1. Here we aways used = 5 as the coarse grid since this is an admissibe choice for a tested p. We observe that the iteration numbers are robust with respect to the grid size, the number of eves, and the spine degree p. In the two-dimensiona experiments, for every degree p, we determine the coarsest eve on which Sp, I (0, 1) is nonempty and use the next coarser eve as the coarsest grid for the mutigrid method. The resuting iteration numbers are dispayed in Tabe 2. Here, the bottom-most iteration number in each coumn corresponds to a two-grid method, the next higher one to a three-grid method, and so on. Again, the iteration numbers remain uniformy bounded with respect to a discretization parameters, athough they are much higher than in the onedimensiona case. To mitigate this, we can sove the probem with Conjugate Gradient (CG) iteration preconditioned with one mutigrid V-cyce. As the resuting iteration numbers in Tabe 3, now ony for = 7, show, this eads to a significant speedup whie maintaining robustness. p Tabe 1: Mutigrid iteration numbers in 1D for varying fine-grid eves and spine degrees. 7. Concusions and outook We have anayzed in detai the convergence properties of a geometric mutigrid method for an isogeometric mode probem using B-spines. In a first step, 20

22 p Tabe 2: Mutigrid iteration numbers in 2D for varying fine-grid eves and spine degrees. p = Tabe 3: Iteration numbers in 2D for CG preconditioned with V-cyce, varying p. we discussed one-dimensiona domains. Based on the obtained insights, in particuar on the importance of boundary effects on the convergence rate, we then proposed a boundary-corrected mass-richardson smoother. We have proved that this new smoother yieds convergence rates which are robust with respect to the spine degree, a resut which is not attainabe with cassica [14] or purey mass-based smoothers [12]. Expoiting the tensor product structure of spine spaces commony used in IgA, we extended the construction of the smoother to the two-dimensiona setting and again proved robust convergence for this case. We have shown how the proposed smoother can be efficienty reaized such that the overa mutigrid method has quasi-optima compexity. Athough we have restricted ourseves to simpe tensor product domains, our technique is easiy extended to probems with non-singuar geometry mappings as discussed in Remark 1. The extension of this approach to three or more dimensions remains open. In particuar, the representation of the smoother as the Kronecker product of one-dimensiona smoothers pus some ow-rank correction as in (23) encounters some difficuties in this case. Therefore, the construction of robust and efficient smoothers for the three- and higher-dimensiona cases is eft as future work. Acknowedgments The work of the first and third author was supported by the Nationa Research Network Geometry + Simuation (NFN S117, ), funded by the Austrian Science Fund (FWF). The work of the second author was funded by the Austrian Science Fund (FWF): J3362-N25. 21

23 Appendix Our aim in this section is to prove Theorem 6. First, however, we show a refined version of the main approximation resut from [16]. Theorem 14. Let N 0 and p N. Then (I Π )u L 2 (Ω) 2h u H 1 (Ω) u H 1 (Ω), where Π : H 1 (Ω) S p, (Ω) is the H (Ω)-orthogona 1 projector and (u, v) H 1 (Ω) := ( u, v) L2 (Ω) + 1 ( ) ( ) u(x)dx v(x)dx. Ω Ω Ω Proof. The proof is a sighty improved version of the proof of [16, Theorem 1]. There, for any fixed u H 1 (Ω), a function u S p, (Ω) was constructed which satisfies the approximation error estimate u u L2 (Ω) 2h u H1 (Ω). (The space S p, (Ω) coincides with the space S p,h (Ω) in [16].) Now, it remains to show that u coincides with Π u, i.e., that u is the H 1 (Ω)-orthogona projection of u to the space S p, (Ω). By definition, this means that we have to show that (u u, ũ ) H 1 (Ω) = 0 ũ S p, (Ω). (28) Now, et Ŝp,( 1, 1) be the space of periodic spines on ( 1, 1) with degree p and grid size h (cf. [16, Definition 2]). Note that, by construction in [16, Theorem 1], u is the restriction of w to (0, 1), where w is the H 1 -orthogona projection of w(x) := u( x ) to Ŝp,( 1, 1). Define w Ŝp,( 1, 1) by w (x) := ũ ( x ) and observe (w w, w ) H 1 ( 1,1) = 2(u u, ũ ) H 1 (0,1). As w is by construction the H 1 -orthogona projection of w, we have (w w, w ) H 1 (0,1) = 0. This shows (28) and therefore u = Π u. The error estimate for the A-orthogona, i.e., the H 1 (Ω)-orthogona, projector T now foows immediatey by a perturbation argument. Proof (of Theorem 6). Define, for u, v H 1 (Ω), the scaar product (u, v) B := (u, v) A (u, v) H 1 (Ω) = (u, v) L 2 (Ω) 1 ( ) ( ) udx vdx Ω Ω Ω and observe that the Cauchy-Schwarz inequaity impies 0 (v, v) B (v, v) L 2 (Ω) for a v H 1 (Ω). Let u H 1 (Ω) arbitrary. Orthogonaity impies ((I Π )u, w ) H 1 (Ω) = 0 w S p, (Ω), ((I T )u, w ) A = 0 w S p, (Ω). (29) 22

24 The first of these two equations impies ((I Π )u, w ) A = ((I Π )u, w ) B w S p, (Ω), and subtracting (29) yieds (( T Π )u, w ) A = ((I Π )u, w ) B w S p, (Ω). With the choice w := ( T Π )u as we as A = H1 (Ω) and the Cauchy- Schwarz inequaity, we obtain ( T Π )u 2 H 1 (Ω) (I Π )u B ( T Π )u B, where B := (, ) 1/2 B. Using B L 2 (Ω) H 1 (Ω), it foows that ( T Π )u L 2 (Ω) (I Π )u L 2 (Ω). Thus, using the triange inequaity and Theorem 14 we obtain (I T )u L 2 (Ω) (I Π )u L 2 (Ω) + ( T Π )u L 2 (Ω) 2 (I Π )u L2 (Ω) 2 2h u H1 (Ω). Bibiography [1] S.C. Brenner, Mutigrid methods for parameter dependent probems, RAIRO, Modéisation Math. Ana. Numér 30 (1996), [2] A. Buffa, H. Harbrecht, A. Kunoth, and G. Sangai, BPX-preconditioning for isogeometric anaysis, Computer Methods in Appied Mechanics and Engineering 265 (2013), [3] J.A. Cottre, A. Reai, Y. Bazievs, and T.J.R. Hughes, Isogeometric anaysis of structura vibrations, Computer Methods in Appied Mechanics and Engineering 195 (2006), no , , John H. Argyris Memoria Issue. Part II. [4] C. de Boor, On cacuating with B-spines, Journa of Approximation Theory 6 (1972), no. 1, [5], Efficient computer manipuation of tensor products, ACM Transactions on Mathematica Software (TOMS) 5 (1979), no. 2, [6], A practica guide to spines (revised edition), Appied Mathematica Sciences, vo. 27, Springer, [7] M. Donatei, C. Garoni, C. Manni, S. Serra-Capizzano, and H. Speeers, Robust and optima muti-iterative techniques for IgA Gaerkin inear systems, Computer Methods in Appied Mechanics and Engineering 284 (2014),

25 [8], Symbo-based mutigrid methods for Gaerkin B-spine isogeometric anaysis, TW Reports (2014), no [9], Robust and optima muti-iterative techniques for IgA coocation inear systems, Computer Methods in Appied Mechanics and Engineering 284 (2015), [10] K. P. S. Gahaaut, J. K. Kraus, and S. K. Tomar, Mutigrid methods for isogeometric discretization, Computer Methods in Appied Mechanics and Engineering 253 (2013), [11] W. Hackbusch, Muti-Grid Methods and Appications, Springer, Berin, [12] C. Hofreither and W. Zuehner, Mass smoothers in geometric mutigrid for isogeometric anaysis, NuMa Report , Institute of Computationa Mathematics, Johannes Keper University Linz, Austria, 2014, Accepted for pubication in the proceedings of the 8th Internationa Conference Curves and Surfaces. [13], On fu mutigrid schemes for isogeometric anaysis, NuMa Report , Institute of Computationa Mathematics, Johannes Keper University Linz, Austria, 2014, Accepted for pubication in the proceedings of the 22nd Internationa Conference on Domain Decomposition Methods. [14], Spectra anaysis of geometric mutigrid methods for isogeometric anaysis, Numerica Methods and Appications (I. Dimov, S. Fidanova, and I. Lirkov, eds.), Lecture Notes in Computer Science, vo. 8962, Springer Internationa Pubishing, 2015, pp [15] T. J. R. Hughes, J. A. Cottre, and Y. Bazievs, Isogeometric anaysis: CAD, finite eements, NURBS, exact geometry and mesh refinement, Computer Methods in Appied Mechanics and Engineering 194 (2005), no , [16] S. Takacs and T. Takacs, Approximation error estimates and inverse inequaities for B-spines of maximum smoothness, Preprint arxiv: , Institute of Computationa Mathematics, Johannes Keper University Linz, Austria, 2015, Submitted. 24