We present a robust and efficient multigrid method for single-patch isogeometric discretizations using tensor product B-splines of maximum smoothness. Our method is based on a stable splitting of the spline space into a large subspace of interior splines which satisfy a robust inverse inequality, as well as one or several smaller subspaces which capture the boundary effects responsible for the spectral outliers which occur in Isogeometric Analysis. We then construct a multigrid smoother based on an additive subspace correction approach, applying a different smoother to each of the subspaces. For the interior splines, we use a mass smoother, whereas the remaining components are treated with suitably chosen Kronecker product smoothers or direct solvers.
We prove that the resulting multigrid method exhibits iteration numbers which are robust with respect to the spline degree and the mesh size. Furthermore, it can be efficiently realized for discretizations of problems in arbitrarily high geometric dimension. Some numerical examples illustrate the theoretical results and show that the iteration numbers also scale relatively mildly with the problem dimension.
Scalable multigrid methods for immersed finite element methods and immersed isogeometric analysis
Eindhoven University of Technology（埃因霍温理工大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）；German Aerospace Center (DLR)（德国航空航天中心）；Walt Disney Animation Studios（华特迪士尼动画工作室）
Brigham Young University（杨百翰大学）；Coreform LLC（Coreform有限责任公司）；Technische Universität Kaiserslautern（凯泽斯劳滕工业大学）；Istituto di Matematica Applicata e Tecnologie Informatiche del CNR（意大利国家研究委员会应用数学与信息技术研究所）
Bézier projection: a unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis
Brigham Young University（杨百翰大学）；University of Colorado, Boulder（科罗拉多大学博尔德分校）
Isogeometric analysis with $C^1$ cubic Powell-Sabin splines
Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；Department of Mathematics, University of Rome Tor Vergata（罗马托维加塔大学数学系）
Robust Preconditioning for Space-Time Isogeometric Analysis of Parabolic Evolution Problems
Johannes Kepler University（约翰内斯·开普勒大学）；Institute of Computational Mathematics, Johannes Kepler University（约翰内斯·开普勒大学计算数学研究所）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉顿计算与应用数学研究所）
Adaptive isogeometric analysis with hierarchical box splines
Istituto Nazionale di Alta Matematica, Unità di Ricerca di Firenze c/o DiMaI “U. Dini”, Università di Firenze（意大利高等数学研究所佛罗伦萨研究单元）；Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze（佛罗伦萨大学数学与信息学系）；Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”（罗马托维加塔大学数学系）
Element-based B-spline basis function spaces: construction and application in isogeometric analysis
University of Electronic Science and Technology of China（电子科技大学）；University of Science and Technology of China（中国科学技术大学）；Wayne State University（韦恩州立大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub- and super- geometric analysis to Geometry Independent Field approximaTion (GIFT)
University of Chile（智利大学）；Hangzhou Dianzi University（杭州电子科技大学）；Institute of Computational Engineering, University of Luxembourg（卢森堡大学计算工程研究所）；School of Engineering, Cardiff University（卡迪夫大学工程学院）
An isogeometric boundary element method for electromagnetic scattering with compatible B-spline discretizations
University of Glasgow（格拉斯哥大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” del CNR（CNR应用数学与信息技术研究所“E. Magenes”）；University of Colorado Boulder（科罗拉多大学博尔德分校）
Adaptive isogeometric phase-field modeling of the Cahn-Hilliard equation: Suitably graded hierarchical refinement and coarsening on multi-patch geometries
Suitably graded THB-spline refinement and coarsening: Towards an adaptive isogeometric analysis of additive manufacturing processes
University of Pavia（帕维亚大学）；Technische Universität München（慕尼黑工业大学）；Università degli Studi di Firenze（佛罗伦萨大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” del CNR（意大利国家研究委员会应用数学与信息技术研究所）
