Local multigrid solvers for adaptive isogeometric analysis in hierarchical spline spaces

We propose local multigrid solvers for adaptively refined isogeometric discretizations using (truncated) hierarchical B-splines ((T)HB-splines). Smoothing is only performed in or near the refinement areas on each level, leading to a computationally efficient solving strategy. We prove robust convergence of the proposed solvers with respect to the number of levels and the mesh sizes of the hierarchical discretization space under the assumption that the hierarchical mesh satisfies an admissibility condition, i.e., the number of interacting mesh levels is uniformly bounded. We also provide several numerical experiments. The main analytical tools are quasi-interpolators for THB-splines and the abstract convergence theory of subspace correction methods.

[1]  Karl Scherer,et al.  New Upper Bound for the B-Spline Basis Condition Number , 1999 .

[2]  C. D. Boor,et al.  Splines as linear combinations of B-splines. A Survey , 1976 .

[3]  R. Hiptmair,et al.  Local Multigrid in H(curl) , 2009, 0901.0764.

[4]  Bert Jüttler,et al.  Multigrid Methods for Isogeometric Analysis with THB-Splines , 2016 .

[5]  Guido Kanschat,et al.  Adaptive Multilevel Methods with Local Smoothing for H1- and Hcurl-Conforming High Order Finite Element Methods , 2011, SIAM J. Sci. Comput..

[6]  Dirk Praetorius,et al.  Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines , 2017, 1701.07764.

[7]  Zhiming Chen,et al.  Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems , 2006 .

[8]  Carlotta Giannelli,et al.  Complexity of hierarchical refinement for a class of admissible mesh configurations , 2015, Comput. Aided Geom. Des..

[9]  S. McCormick,et al.  The fast adaptive composite grid (FAC) method for elliptic equation , 1986 .

[10]  J. Zolésio,et al.  Springer series in Computational Mathematics , 1992 .

[11]  Tom Lyche,et al.  Foundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement , 2018 .

[12]  H. Yserentant.  Old and new convergence proofs for multigrid methods , 1993, Acta Numerica.

[13]  Hendrik Speleers,et al.  Effortless quasi-interpolation in hierarchical spaces , 2016, Numerische Mathematik.

[14]  T. Hughes,et al.  ISOGEOMETRIC ANALYSIS: APPROXIMATION, STABILITY AND ERROR ESTIMATES FOR h-REFINED MESHES , 2006 .

[15]  Hendrik Speleers,et al.  Robust and optimal multi-iterative techniques for IgA Galerkin linear systems This is a preprint of a paper published in Comput. Methods Appl. Mech. Engrg. 284 (2015) 230264. , 2015 .

[16]  W. Dahmen,et al.  Multilevel preconditioning , 1992 .

[17]  Joseph E. Pasciak,et al.  New estimates for multilevel algorithms including the V-cycle , 1993 .

[18]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[19]  Giancarlo Sangalli,et al.  BPX-preconditioning for isogeometric analysis , 2013 .

[20]  Achi Brandt,et al.  Local mesh refinement multilevel techniques , 1987 .

[21]  Hendrik Speleers,et al.  Symbol-Based Multigrid Methods for Galerkin B-Spline Isogeometric Analysis , 2017, SIAM J. Numer. Anal..

[22]  Rafael Vázquez Hernández,et al.  BPX preconditioners for isogeometric analysis using analysis-suitable T-splines , 2018, IMA Journal of Numerical Analysis.

[23]  Jinchao Xu,et al.  Convergence estimates for multigrid algorithms without regularity assumptions , 1991 .

[24]  Harry Yserentant,et al.  A basic norm equivalence for the theory of multilevel methods , 1993 .

[25]  Clemens Hofreither,et al.  Robust Multigrid for Isogeometric Analysis Based on Stable Splittings of Spline Spaces , 2016, SIAM J. Numer. Anal..

[26]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[27]  Clemens Hofreither,et al.  A robust multigrid method for Isogeometric Analysis in two dimensions using boundary correction , 2015, 1512.07091.

[28]  Clemens Hofreither,et al.  Spectral Analysis of Geometric Multigrid Methods for Isogeometric Analysis , 2014, NMA.

[29]  Hendrik Speleers,et al.  THB-splines: The truncated basis for hierarchical splines , 2012, Comput. Aided Geom. Des..

[30]  Carlotta Giannelli,et al.  Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates , 2017 .

[31]  Hendrik Speleers,et al.  Hierarchical spline spaces: quasi-interpolants and local approximation estimates , 2017, Adv. Comput. Math..

[32]  R. Bank,et al.  The hierarchical basis multigrid method , 1988 .

[33]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[34]  Hendrik Speleers,et al.  Strongly stable bases for adaptively refined multilevel spline spaces , 2014, Adv. Comput. Math..

[35]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[36]  Carlotta Giannelli,et al.  Adaptive isogeometric methods with hierarchical splines: error estimator and convergence , 2015, 1502.00565.

[37]  William F. Mitchell,et al.  Optimal Multilevel Iterative Methods for Adaptive Grids , 1992, SIAM J. Sci. Comput..

[38]  Ricardo H. Nochetto,et al.  Optimal multilevel methods for graded bisection grids , 2012, Numerische Mathematik.

[39]  Peter Oswald,et al.  Multilevel Finite Element Approximation , 1994 .