Local Approximation from Spline Spaces on Box Meshes

This paper analyzes the approximation properties of spaces of piecewise tensor product polynomials over box meshes with a focus on application to isogeometric analysis. Local and global error bounds with respect to Sobolev or reduced seminorms are provided. Attention is also paid to the dependence on the degree, and exponential convergence is proved for the approximation of analytic functions in the absence of non-convex extended supports.

[1]  Tom Lyche,et al.  Polynomial splines over locally refined box-partitions , 2013, Comput. Aided Geom. Des..

[2]  Giancarlo Sangalli,et al.  Exponential Convergence of the hp Version of Isogeometric Analysis in 1D , 2012 .

[3]  Andrea Bressan,et al.  A Versatile Strategy for the Implementation of Adaptive Splines , 2016, MMCS.

[4]  Giancarlo Sangalli,et al.  Mathematical analysis of variational isogeometric methods* , 2014, Acta Numerica.

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

[6]  T. Dupont,et al.  Polynomial approximation of functions in Sobolev spaces , 1980 .

[7]  Giancarlo Sangalli,et al.  Characterization of analysis-suitable T-splines , 2015, Comput. Aided Geom. Des..

[8]  Ulrich Reif.  Polynomial approximation on domains bounded by diffeomorphic images of graphs , 2012, J. Approx. Theory.

[9]  Rüdiger Verfürth,et al.  A note on polynomial approximation in Sobolev spaces , 1999 .

[10]  Giancarlo Sangalli,et al.  On Quasi-Interpolation Operators in Spline Spaces , 2015 .

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

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

[13]  Daniel Peterseim,et al.  Analysis-suitable adaptive T-mesh refinement with linear complexity , 2014, Comput. Aided Geom. Des..

[14]  David R. Forsey,et al.  Hierarchical B-spline refinement , 1988, SIGGRAPH.

[15]  Giancarlo Sangalli,et al.  Analysis-Suitable T-splines are Dual-Compatible , 2012 .

[16]  Philipp Morgenstern,et al.  Globally Structured Three-Dimensional Analysis-Suitable T-Splines: Definition, Linear Independence and m-graded local refinement , 2015, SIAM J. Numer. Anal..

[17]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[18]  L. Schumaker,et al.  Local Spline Approximation Methods , 1975 .

[19]  Nicholas S. North,et al.  T-spline simplification and local refinement , 2004, SIGGRAPH 2004.

[20]  Bert Jüttler,et al.  A hierarchical construction of LR meshes in 2D , 2015, Comput. Aided Geom. Des..

[21]  Dominik Schötzau,et al.  Exponential convergence for hp-version and spectral finite element methods for elliptic problems in polyhedra , 2015 .

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

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

[24]  Andrea Bressan,et al.  Some properties of LR-splines , 2013, Comput. Aided Geom. Des..

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

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

[27]  R. Range.  Holomorphic Functions and Integral Representations in Several Complex Variables , 1998 .

[28]  Ahmad H. Nasri,et al.  T-splines and T-NURCCs , 2003, ACM Trans. Graph..

[29]  Ricardo G. Durán,et al.  On Polynomial Approximation in Sobolev Spaces , 1983 .

[30]  Giancarlo Sangalli,et al.  Anisotropic NURBS approximation in isogeometric analysis , 2012 .

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

[32]  R ForseyDavid,et al.  Hierarchical B-spline refinement , 1988 .

[33]  Thomas J. R. Hughes,et al.  On linear independence of T-spline blending functions , 2012, Comput. Aided Geom. Des..

[34]  P. Müller.  E. F. Beckenbach and R. Bellman, Inequalities. (Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 30.) XII + 198 S. m. 6 Abb. Berlin/Göttingen/Heidelberg 1961. Springer‐Verlag. Preis Brosch. DM 48,60 , 1962 .

[35]  Ulrich Reif,et al.  Approximation with diversified B-splines , 2014, Comput. Aided Geom. Des..

[36]  Bert Jüttler,et al.  TDHB-splines: The truncated decoupled basis of hierarchical tensor-product splines , 2014, Comput. Aided Geom. Des..

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

[38]  Giancarlo Sangalli,et al.  ANALYSIS-SUITABLE T-SPLINES OF ARBITRARY DEGREE: DEFINITION, LINEAR INDEPENDENCE AND APPROXIMATION PROPERTIES , 2013 .