View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repositorio Universidad de Zaragoza SIAM J. SCI.COMPUT. c 2019 Society for Industrial and Applied Mathematics Vol. 41, No. 5, pp. S321{S345 A ROBUST MULTIGRID SOLVER FOR ISOGEOMETRIC ANALYSIS BASED ON MULTIPLICATIVE SCHWARZ SMOOTHERS∗ y y z ALVARO PE DE LA RIVA , CARMEN RODRIGO , AND FRANCISCO J. GASPAR Abstract. The design of fast solvers for isogeometric analysis is receiving a lot of attention due to the challenge that offers to find an algorithm with a robust convergence with respect to the spline degree. Here, we analyze the application of geometric multigrid methods to this type of discretization, and we propose a multigrid approach based on overlapping multiplicative Schwarz methods as smoothers. The size of the blocks considered within these relaxation procedures is adapted to the spline degree. A simple multigrid V-cycle with only one step of presmoothing results in a very efficient algorithm, whose convergence is independent on the spline degree and the spatial discretization parameter. Local Fourier analysis is shown to be very useful for the understanding of the problems encountered in the design of a robust multigrid method for IGA, and it is performed to support the good convergence properties of the proposed solver. In fact, an analysis for any spline degree and an arbitrary size of the blocks within the Schwarz smoother is presented for the one-dimensional case. The efficiency of the solver is also demonstrated through several numerical experiments, including a two-dimensional problem on a nontrivial computational domain. Key words. isogeometric analysis, multigrid methods, local Fourier analysis, overlapping multiplicative Schwarz methods AMS subject classifications. 65F10, 65M22, 65M55 DOI. 10.1137/18M1194407 1. Introduction. Isogeometric analysis (IGA) is a computational technique for the numerical solution of partial differential equations (PDEs), which was introduced by Hughes, Cottrell, and Bazilevs in the seminal paper [22]. Since then, this approach has been widely applied within different frameworks, and a detailed presentation of IGA together with a number of engineering applications can be found in the book [8]. IGA is based on the idea of using spline-type functions, which are exploited in com- puter aided design (CAD) software for the parametrization of the computational do- main, in order to approximate the unknown solution of the PDE. B-splines or nonuni- form rational B-splines (NURBS) are the most commonly used functions. There are several issues that make this approach advantageous over classical finite element meth- ods (FEMs). First, it allows us to represent exactlysome geometries like conic sec- tions, and also more complicated geometries are represented more accurately by this technique than by traditional polynomial based approaches. In addition, this precise description of the geometry is incorporated exactly at the coarsest grid level, making unnecessary further communication with the CAD system in order to do a mesh refine- ment procedure, which moreover does not modify the geometry. Another important ∗ Received by the editors June 15, 2018; accepted for publication (in revised form) June 5, 2019; published electronically October 29, 2019. https://doi.org/10.1137/18M1194407 Funding: The work of the first author was supported by the grant program of the Diputacion General de Aragon. The work of the second author was partially supported by the Spanish project FEDER/MCYT MTM2016-75139-R and the Diputacion General de Aragon (Grupo de referencia APEDIF, E24 17R). The work of the third author was supported by the European Union's Hori- zon 2020 research and innovation program under Marie Sklodowska-Curie grant agreement 705402, POROSOS. yIUMA and Applied Mathematics Department, University of Zaragoza, Zaragoza, Spain (apedela Downloaded 11/04/19 to 129.187.254.47. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php [email protected], [email protected]). zCWI, Centrum Wiskunde and Informatica, Amsterdam, The Netherlands ([email protected], http://www.unizar.es/pde/fjgaspar/). S321 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. S322 A. PE DE LA RIVA, C. RODRIGO, AND F. J. GASPAR p−1 advantage is the higher continuity, since IGA provides up to C interelement conti- nuity, denoting p the polynomial order (see [7, 10]). This corresponds to the so-called isogeometric k-method, which is one of the three refinement strategies for IGA pro- posed in [22], together with the h-refinement (reducing the mesh size by knot insertion) and p-refinement (order elevation, i.e., increase of the spline degree). The k-refinement is unique to IGA and its main advantage is that it maintains the maximum possible smoothness Cp−1 for the spline space of degree p. Due to its high performance, this is the most popular refinement strategy in the IGA community, and this is the one studied in this work. From the computational point of view, the efficient solution of the linear systems arising from the discretization of a PDE problem is a crucial point for its numerical simulation. When a discretization with high spline degrees is considered, this issue is even more challenging since the condition number of the stiffness matrices grows exponentially with the spline degree. The study of the computational efficiency for direct and iterative solvers was initiated in the papers [5, 6], respectively, and recently the design of iterative solvers has attracted much attention in the isogeometric com- munity. Many efforts have been devoted to develop efficient solvers for this type of discretizations. For example, in [4] a multilevel BPX-preconditioner is developed in the framework of IGA. Beir~aoda Veiga et al. analyze overlapping Schwarz methods for IGA in [11], whereas in [12] they study BDDC preconditioners by introducing appropriate discrete norms. In [16] algebraic multilevel iteration methods are ap- plied for the isogeometric discretization of scalar second order elliptic problems. The new isogeometric tearing and interconnecting method, which consists of a domain decomposition solver based on the ideas of the finite element tearing/interconnecting method, was proposed in [23]. In [27], the authors propose preconditioners based on fast solvers for the Sylvester equation. In all these works, the difficulty in achieving both robustness and computational efficiency for high-order isogeometric discretiza- tions is reported. For classical finite element, finite difference, or finite volume discretizations, multi- grid methods [2, 17, 29] are well known to be among the fastest solvers showing op- timal computational cost and convergence behavior. Thus, it seems natural to try to extend these methods to IGA, and in fact, in the early IGA literature, multigrid solvers for FEMs have been directly transferred to isogeometric discretizations with only minimal adaptations. However, a naive application of these multigrid methods to the isogeometric case results in an important deterioration of the convergence of the algorithms when the spline degree is increased. In particular, multigrid methods based on standard smoothers, like the Gauss{Seidel smoother, are not robust with respect to the spline degree (see, e.g., [15]). It was observed in [14] that the spectral radius of the multigrid iteration matrices based on standard smoothers tends to one exponentially as p increases. As it was pointed out in [13], this bad behaviour is due to the presence of many small eigenvalues associated with high-frequency eigenvectors. This deterioration of the convergence of standard multigrid algorithms has motivated advances toward robust multigrid methods with respect to the spline degree. In [14] a multigrid method was constructed based on a preconditioned Krylov smoother at the finest level and in [20] a mass matrix was proposed as a smoother within a multigrid framework. For both methods, an increase in the number of smoothing steps was needed in order to obtain robustness with respect tothe spline degree. To avoid the lack of robustness of the mass smoother, due to boundary effects, in [19] the authors Downloaded 11/04/19 to 129.187.254.47. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php introduce a boundary correction to that relaxation. In that work, it was not clear, however, how to extend this approach to three dimensions. To overcome this, in [18], Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. A ROBUST MG FOR IGA BASED ON SCHWARZ SMOOTHERS S323 the authors propose a multigrid smoother based on an additive subspace correction technique, applying a different smoother to each of the subspaces. In the regular interior subspace they use a mass smoother, whereas in the other subspaces they con- sider relaxations which exploit the particular structure of the subspaces. The authors observe a dependence of the convergence of this method on the space dimension and on the geometry transformation. We would like to remark that in all previous works the multigrid method is designed on the parametric domain and it is applied as a pre- conditioner for solving the problem in general geometries. In this work, however, we aim to propose a robust and efficient geometric multigrid algorithm for directly solv- ing isogeometric discretizations on a general domain. This multigrid method is based on a family of overlapping multiplicative Schwarz-type methods as smoothers. We will show that by choosing an appropriate Schwarz-type smoother we can efficiently remove those high-frequency components of the error associated with the small eigen- values. This makes it possible to obtain a very simple and efficient solver by using a multigrid V (1; 0)-cycle. It is well known that many details are open for discussion and decision in the design of a multigrid method for a target problem, since the performance of multigrid algorithms strongly depends on the choice of their components. There are no rules to facilitate this challenging task, but the local Fourier analysis (LFA) appears as a very useful tool for the design of the algorithm.