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January  2019, 39(1): 241-261. doi: 10.3934/dcds.2019010

Adaptive isogeometric methods with hierarchical splines: An overview

Cesare Bracco 1, , Annalisa Buffa 2,3,, , Carlotta Giannelli 1, and  Rafael Vázquez 2,3,

1. 

Dipartimento di Matematica e Informatica 'U. Dini', Università degli Studi di Firenze, viale Morgagni 67a, 50134 Firenze, Italy
2. 

Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne, Switzerland
3. 

Istituto di Matematica Applicata e Tecnologie Informatiche "E. Magenes" del CNR, via Ferrata 5, 27100 Pavia, Italy

* Corresponding author: Annalisa Buffa

Received  December 2017 Revised  June 2018 Published  October 2018

We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and present the study of its numerical properties. By following [10,12,11], optimal convergence rates of the AIGM can be proved when suitable approximation classes are considered. This is in line with the theory of adaptive methods developed for finite elements, recently well reviewed in [43]. The important output of our analysis is the definition of classes of admissibility for meshes underlying hierarchical splines and the design of an optimal adaptive strategy based on these classes of meshes. The adaptivity analysis is validated on a selection of numerical tests. We also compare the results obtained with suitably graded meshes related to different classes of admissibility for 2D and 3D configurations.

Keywords: Isogeometric analysis, adaptive methods, error estimate, hierarchical splines, THB-splines.
Mathematics Subject Classification: Primary: 41A15, 65N30; Secondary: 65D07, 65N12.
Citation: Cesare Bracco, Annalisa Buffa, Carlotta Giannelli, Rafael Vázquez. Adaptive isogeometric methods with hierarchical splines: An overview. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 241-261. doi: 10.3934/dcds.2019010
References:
[1]

M. Actis, P. Morin and M. S. Pauletti, A new perspective on hierarchical spline spaces for adaptivity, arXiv: 1808.02053 Google Scholar

[2]

Y. Bazilevs, V. M. Calo, J. A. Cottrell, J. Evans, T. J. R. Hughes, S. Lipton, M. A. Scott and T. W. Sederberg, Isogeometric analysis using T-Splines, Comput. Methods Appl. Mech. Engrg., 199 (2010), 229-263.  doi: 10.1016/j.cma.2009.02.036.  Google Scholar

[3]

L. Beirão da Veiga, A. Buffa, D. Cho and G. Sangalli, Analysis-Suitable T-splines are Dual-Compatible, Comput. Methods Appl. Mech. Engrg., 249/252 (2012), 42-51.  doi: 10.1016/j.cma.2012.02.025.  Google Scholar

[4]

L. Beirão da Veiga, A. Buffa, G. Sangalli and R. Vázquez, Analysis-suitable T-splines of arbitrary degree: Definition, linear independence and approximation properties, Math. Models Methods Appl. Sci., 23 (2013), 1979-2003.  doi: 10.1142/S0218202513500231.  Google Scholar

[5]

A. Bonito and R. H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal., 48 (2010), 734-771.  doi: 10.1137/08072838X.  Google Scholar

[6]

C. Bracco, C. Giannelli and R. Vázquez, Refinement algorithms for adaptive isogeometric methods with hierarchical splines, Axioms, 7 (2018), 43. doi: 10.3390/axioms7030043.  Google Scholar

[7]

A. Buffa and E. M. Garau, Refinable spaces and local approximation estimates for hierarchical splines, IMA J. Numer. Anal., 37 (2017), 1125-1149.  doi: 10.1093/imanum/drw035.  Google Scholar

[8]

A. Buffa and E. M. Garau, A posteriori error estimators for hierarchical B-spline discretizations, Math. Models Methods Appl. Sci., 28 (2018), 1453-1480.  doi: 10.1142/S0218202518500392.  Google Scholar

[9]

A. Buffa, E. M. Garau, C. Giannelli and G. Sangalli, On quasi-interpolation operators in spline spaces, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations (eds. G. R. Barrenechea et al.), vol. 114, Lecture Notes in Computational Science and Engineering, 2016, 73-91. doi: 10.1007/978-3-319-41640-3_3.  Google Scholar

[10]

A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence, Math. Models Methods Appl. Sci., 26 (2016), 1-25.  doi: 10.1142/S0218202516500019.  Google Scholar

[11]

A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates, Math. Models Methods Appl. Sci., 27 (2017), 2781-2802.  doi: 10.1142/S0218202517500580.  Google Scholar

[12]

A. Buffa, C. Giannelli, P. Morgenstern and D. Peterseim, Complexity of hierarchical refinement for a class of admissible mesh configurations, Comput. Aided Geom. Design, 47 (2016), 83-92.  doi: 10.1016/j.cagd.2016.04.003.  Google Scholar

[13]

C. de Boor, A Practical Guide to Splines, Springer, revised ed., 2001.  Google Scholar

[14]

T. Dokken, T. Lyche and K. F. Pettersen, Polynomial splines over locally refined box-partitions, Comput. Aided Geom. Design, 30 (2013), 331-356.  doi: 10.1016/j.cagd.2012.12.005.  Google Scholar

[15]

M. R. Dörfel, B. Jüttler and B. Simeon, Adaptive isogeometric analysis by local h-refinement with T-splines, Comput. Methods Appl. Mech. Engrg., 199 (2010), 264-275.  doi: 10.1016/j.cma.2008.07.012.  Google Scholar

[16]

W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.  doi: 10.1137/0733054.  Google Scholar

[17]

E. J. Evans, M. A. Scott, X. Li and D. C. Thomas, Hierarchical T-splines: Analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 284 (2015), 1-20.  doi: 10.1016/j.cma.2014.05.019.  Google Scholar

[18]

M. Feischl, G. Gantner, A. Haberl and D. Praetorius, Adaptive 2D IGA boundary element methods, Engineering Analysis with Boundary Elements, 62 (2016), 141-153.  doi: 10.1016/j.enganabound.2015.10.003.  Google Scholar

[19]

M. Feischl, G. Gantner, A. Haberl and D. Praetorius, Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations, Numer. Math., 136 (2017), 147-182.  doi: 10.1007/s00211-016-0836-8.  Google Scholar

[20]

G. Gantner, D. Haberlik and D. Praetorius, Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines, Math. Models Methods Appl. Sci., 27 (2017), 2631-2674.  doi: 10.1142/S0218202517500543.  Google Scholar

[21]

E. Garau and R. Vázquez, Algorithms for the implementation of adaptive isogeometric methods using hierarchical B-splines, Appl. Numer. Math., 123 (2018), 58-87.  doi: 10.1016/j.apnum.2017.08.006.  Google Scholar

[22]

C. Giannelli and B. Jüttler, Bases and dimensions of bivariate hierarchical tensor-product splines, J. Comput. Appl. Math., 239 (2013), 162-178.  doi: 10.1016/j.cam.2012.09.031.  Google Scholar

[23]

C. Giannelli, B. Jüttler, S. K. Kleiss, A. Mantzaflaris, B. Simeon and J. Špeh, THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 299 (2016), 337-365.  doi: 10.1016/j.cma.2015.11.002.  Google Scholar

[24]

C. Giannelli, B. Jüttler and H. Speleers, THB-splines: The truncated basis for hierarchical splines, Comput. Aided Geom. Design, 29 (2012), 485-498.  doi: 10.1016/j.cagd.2012.03.025.  Google Scholar

[25]

C. Giannelli, B. Jüttler and H. Speleers, Strongly stable bases for adaptively refined multilevel spline spaces, Adv. Comput. Math., 40 (2014), 459-490.  doi: 10.1007/s10444-013-9315-2.  Google Scholar

[26]

P. Hennig, M. Kästner, P. Morgenstern and D. Peterseim, Adaptive mesh refinement strategies in isogeometric analysis— A computational comparison, Comput. Methods Appl. Mech. Engrg., 316 (2017), 424-448.  doi: 10.1016/j.cma.2016.07.029.  Google Scholar

[27]

T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194 (2005), 4135-4195.  doi: 10.1016/j.cma.2004.10.008.  Google Scholar

[28]

K. A. Johannessen, T. Kvamsdal and T. Dokken, Isogeometric analysis using LR B-splines, Comput. Methods Appl. Mech. Engrg., 269 (2014), 471-514.  doi: 10.1016/j.cma.2013.09.014.  Google Scholar

[29]

K. A. Johannessen, F. Remonato and T. Kvamsdal, On the similarities and differences between Classical Hierarchical, Truncated Hierarchical and LR B-splines, Comput. Methods Appl. Mech. Engrg., 291 (2015), 64-101.  doi: 10.1016/j.cma.2015.02.031.  Google Scholar

[30]

G. Kiss, C. Giannelli, U. Zore, B. Jüttler, D. Großmann and J. Barner, Adaptive CAD model (re-)construction with THB-splines, Graphical models, 76 (2014), 273-288.  doi: 10.1016/j.gmod.2014.03.017.  Google Scholar

[31]

R. Kraft, Adaptive and linearly independent multilevel B-splines, in Surface Fitting and Multiresolution Methods (eds. A. Le Méhauté, C. Rabut and L. L. Schumaker), Vanderbilt University Press, Nashville, 1997,209-218.  Google Scholar

[32]

M. Kumar, T. Kvamsdal and K. A. Johannessen, Simple a posteriori error estimators in adaptive isogeometric analysis, Comput. Math. Appl., 70 (2015), 1555-1582.  doi: 10.1016/j.camwa.2015.05.031.  Google Scholar

[33]

M. Kumar, T. Kvamsdal and K. A. Johannessen, Superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 316 (2017), 1086-1156.  doi: 10.1016/j.cma.2016.11.014.  Google Scholar

[34]

G. Kuru, C. V. Verhoosel, K. G. van der Zeeb and E. H. van Brummelen, Goal-adaptive isogeometric analysis with hierarchical splines, Comput. Methods Appl. Mech. Engrg., 270 (2014), 270-292.  doi: 10.1016/j.cma.2013.11.026.  Google Scholar

[35]

X. Li, J. Zheng, T. W. Sederberg, T. J. R. Hughes and M. A. Scott, On linear independence of T-spline blending functions, Comput. Aided Geom. Design, 29 (2012), 63-76.  doi: 10.1016/j.cagd.2011.08.005.  Google Scholar

[36]

G. Lorenzo, M. A. Scott, K. Tew, T. J. R. Hughes and H. Gomez, Hierarchically refined and coarsened splines for moving interface problems, with particular application to phase-field models of prostate tumor growth, Comput. Methods Appl. Mech. Engrg., 319 (2017), 515-548.  doi: 10.1016/j.cma.2017.03.009.  Google Scholar

[37]

D. Mokriš, B. Jüttler and C. Giannelli, On the completeness of hierarchical tensor-product B-splines, J. Comput. Appl. Math., 271 (2014), 53-70.  doi: 10.1016/j.cam.2014.04.001.  Google Scholar

[38]

P. Morgenstern, Globally structured three-dimensional analysis-suitable T-splines: definition, linear independence and m-graded local refinement, SIAM J. Numer. Anal., 54 (2016), 2163-2186.  doi: 10.1137/15M102229X.  Google Scholar

[39]

P. Morgenstern, Mesh Refinement Strategies for the Adaptive Isogeometric Method, PhD thesis, Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2017. Google Scholar

[40]

P. Morgenstern and D. Peterseim, Analysis-suitable adaptive T-mesh refinement with linear complexity, Comput. Aided Geom. Design, 34 (2015), 50-66.  doi: 10.1016/j.cagd.2015.02.003.  Google Scholar

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P. Morin, R. H. Nochetto and M. S. Pauletti, An adaptive method for hierarchical splines. A posteriori estimation via local problems, convergence and optimality, In preparation. Google Scholar

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R. H. Nochetto, K. G. Siebert and A. Veeser, Theory of adaptive finite element methods: An introduction, in Multiscale, Nonlinear and Adaptive Approximation (eds. R. DeVore and A. Kunoth), Springer Berlin Heidelberg, 2009,409-542. doi: 10.1007/978-3-642-03413-8_12.  Google Scholar

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show all references

References:
[1]

M. Actis, P. Morin and M. S. Pauletti, A new perspective on hierarchical spline spaces for adaptivity, arXiv: 1808.02053 Google Scholar

[2]

Y. Bazilevs, V. M. Calo, J. A. Cottrell, J. Evans, T. J. R. Hughes, S. Lipton, M. A. Scott and T. W. Sederberg, Isogeometric analysis using T-Splines, Comput. Methods Appl. Mech. Engrg., 199 (2010), 229-263.  doi: 10.1016/j.cma.2009.02.036.  Google Scholar

[3]

L. Beirão da Veiga, A. Buffa, D. Cho and G. Sangalli, Analysis-Suitable T-splines are Dual-Compatible, Comput. Methods Appl. Mech. Engrg., 249/252 (2012), 42-51.  doi: 10.1016/j.cma.2012.02.025.  Google Scholar

[4]

L. Beirão da Veiga, A. Buffa, G. Sangalli and R. Vázquez, Analysis-suitable T-splines of arbitrary degree: Definition, linear independence and approximation properties, Math. Models Methods Appl. Sci., 23 (2013), 1979-2003.  doi: 10.1142/S0218202513500231.  Google Scholar

[5]

A. Bonito and R. H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal., 48 (2010), 734-771.  doi: 10.1137/08072838X.  Google Scholar

[6]

C. Bracco, C. Giannelli and R. Vázquez, Refinement algorithms for adaptive isogeometric methods with hierarchical splines, Axioms, 7 (2018), 43. doi: 10.3390/axioms7030043.  Google Scholar

[7]

A. Buffa and E. M. Garau, Refinable spaces and local approximation estimates for hierarchical splines, IMA J. Numer. Anal., 37 (2017), 1125-1149.  doi: 10.1093/imanum/drw035.  Google Scholar

[8]

A. Buffa and E. M. Garau, A posteriori error estimators for hierarchical B-spline discretizations, Math. Models Methods Appl. Sci., 28 (2018), 1453-1480.  doi: 10.1142/S0218202518500392.  Google Scholar

[9]

A. Buffa, E. M. Garau, C. Giannelli and G. Sangalli, On quasi-interpolation operators in spline spaces, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations (eds. G. R. Barrenechea et al.), vol. 114, Lecture Notes in Computational Science and Engineering, 2016, 73-91. doi: 10.1007/978-3-319-41640-3_3.  Google Scholar

[10]

A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence, Math. Models Methods Appl. Sci., 26 (2016), 1-25.  doi: 10.1142/S0218202516500019.  Google Scholar

[11]

A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates, Math. Models Methods Appl. Sci., 27 (2017), 2781-2802.  doi: 10.1142/S0218202517500580.  Google Scholar

[12]

A. Buffa, C. Giannelli, P. Morgenstern and D. Peterseim, Complexity of hierarchical refinement for a class of admissible mesh configurations, Comput. Aided Geom. Design, 47 (2016), 83-92.  doi: 10.1016/j.cagd.2016.04.003.  Google Scholar

[13]

C. de Boor, A Practical Guide to Splines, Springer, revised ed., 2001.  Google Scholar

[14]

T. Dokken, T. Lyche and K. F. Pettersen, Polynomial splines over locally refined box-partitions, Comput. Aided Geom. Design, 30 (2013), 331-356.  doi: 10.1016/j.cagd.2012.12.005.  Google Scholar

[15]

M. R. Dörfel, B. Jüttler and B. Simeon, Adaptive isogeometric analysis by local h-refinement with T-splines, Comput. Methods Appl. Mech. Engrg., 199 (2010), 264-275.  doi: 10.1016/j.cma.2008.07.012.  Google Scholar

[16]

W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.  doi: 10.1137/0733054.  Google Scholar

[17]

E. J. Evans, M. A. Scott, X. Li and D. C. Thomas, Hierarchical T-splines: Analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 284 (2015), 1-20.  doi: 10.1016/j.cma.2014.05.019.  Google Scholar

[18]

M. Feischl, G. Gantner, A. Haberl and D. Praetorius, Adaptive 2D IGA boundary element methods, Engineering Analysis with Boundary Elements, 62 (2016), 141-153.  doi: 10.1016/j.enganabound.2015.10.003.  Google Scholar

[19]

M. Feischl, G. Gantner, A. Haberl and D. Praetorius, Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations, Numer. Math., 136 (2017), 147-182.  doi: 10.1007/s00211-016-0836-8.  Google Scholar

[20]

G. Gantner, D. Haberlik and D. Praetorius, Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines, Math. Models Methods Appl. Sci., 27 (2017), 2631-2674.  doi: 10.1142/S0218202517500543.  Google Scholar

[21]

E. Garau and R. Vázquez, Algorithms for the implementation of adaptive isogeometric methods using hierarchical B-splines, Appl. Numer. Math., 123 (2018), 58-87.  doi: 10.1016/j.apnum.2017.08.006.  Google Scholar

[22]

C. Giannelli and B. Jüttler, Bases and dimensions of bivariate hierarchical tensor-product splines, J. Comput. Appl. Math., 239 (2013), 162-178.  doi: 10.1016/j.cam.2012.09.031.  Google Scholar

[23]

C. Giannelli, B. Jüttler, S. K. Kleiss, A. Mantzaflaris, B. Simeon and J. Špeh, THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 299 (2016), 337-365.  doi: 10.1016/j.cma.2015.11.002.  Google Scholar

[24]

C. Giannelli, B. Jüttler and H. Speleers, THB-splines: The truncated basis for hierarchical splines, Comput. Aided Geom. Design, 29 (2012), 485-498.  doi: 10.1016/j.cagd.2012.03.025.  Google Scholar

[25]

C. Giannelli, B. Jüttler and H. Speleers, Strongly stable bases for adaptively refined multilevel spline spaces, Adv. Comput. Math., 40 (2014), 459-490.  doi: 10.1007/s10444-013-9315-2.  Google Scholar

[26]

P. Hennig, M. Kästner, P. Morgenstern and D. Peterseim, Adaptive mesh refinement strategies in isogeometric analysis— A computational comparison, Comput. Methods Appl. Mech. Engrg., 316 (2017), 424-448.  doi: 10.1016/j.cma.2016.07.029.  Google Scholar

[27]

T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194 (2005), 4135-4195.  doi: 10.1016/j.cma.2004.10.008.  Google Scholar

[28]

K. A. Johannessen, T. Kvamsdal and T. Dokken, Isogeometric analysis using LR B-splines, Comput. Methods Appl. Mech. Engrg., 269 (2014), 471-514.  doi: 10.1016/j.cma.2013.09.014.  Google Scholar

[29]

K. A. Johannessen, F. Remonato and T. Kvamsdal, On the similarities and differences between Classical Hierarchical, Truncated Hierarchical and LR B-splines, Comput. Methods Appl. Mech. Engrg., 291 (2015), 64-101.  doi: 10.1016/j.cma.2015.02.031.  Google Scholar

[30]

G. Kiss, C. Giannelli, U. Zore, B. Jüttler, D. Großmann and J. Barner, Adaptive CAD model (re-)construction with THB-splines, Graphical models, 76 (2014), 273-288.  doi: 10.1016/j.gmod.2014.03.017.  Google Scholar

[31]

R. Kraft, Adaptive and linearly independent multilevel B-splines, in Surface Fitting and Multiresolution Methods (eds. A. Le Méhauté, C. Rabut and L. L. Schumaker), Vanderbilt University Press, Nashville, 1997,209-218.  Google Scholar

[32]

M. Kumar, T. Kvamsdal and K. A. Johannessen, Simple a posteriori error estimators in adaptive isogeometric analysis, Comput. Math. Appl., 70 (2015), 1555-1582.  doi: 10.1016/j.camwa.2015.05.031.  Google Scholar

[33]

M. Kumar, T. Kvamsdal and K. A. Johannessen, Superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 316 (2017), 1086-1156.  doi: 10.1016/j.cma.2016.11.014.  Google Scholar

[34]

G. Kuru, C. V. Verhoosel, K. G. van der Zeeb and E. H. van Brummelen, Goal-adaptive isogeometric analysis with hierarchical splines, Comput. Methods Appl. Mech. Engrg., 270 (2014), 270-292.  doi: 10.1016/j.cma.2013.11.026.  Google Scholar

[35]

X. Li, J. Zheng, T. W. Sederberg, T. J. R. Hughes and M. A. Scott, On linear independence of T-spline blending functions, Comput. Aided Geom. Design, 29 (2012), 63-76.  doi: 10.1016/j.cagd.2011.08.005.  Google Scholar

[36]

G. Lorenzo, M. A. Scott, K. Tew, T. J. R. Hughes and H. Gomez, Hierarchically refined and coarsened splines for moving interface problems, with particular application to phase-field models of prostate tumor growth, Comput. Methods Appl. Mech. Engrg., 319 (2017), 515-548.  doi: 10.1016/j.cma.2017.03.009.  Google Scholar

[37]

D. Mokriš, B. Jüttler and C. Giannelli, On the completeness of hierarchical tensor-product B-splines, J. Comput. Appl. Math., 271 (2014), 53-70.  doi: 10.1016/j.cam.2014.04.001.  Google Scholar

[38]

P. Morgenstern, Globally structured three-dimensional analysis-suitable T-splines: definition, linear independence and m-graded local refinement, SIAM J. Numer. Anal., 54 (2016), 2163-2186.  doi: 10.1137/15M102229X.  Google Scholar

[39]

P. Morgenstern, Mesh Refinement Strategies for the Adaptive Isogeometric Method, PhD thesis, Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2017. Google Scholar

[40]

P. Morgenstern and D. Peterseim, Analysis-suitable adaptive T-mesh refinement with linear complexity, Comput. Aided Geom. Design, 34 (2015), 50-66.  doi: 10.1016/j.cagd.2015.02.003.  Google Scholar

[41]

P. Morin, R. H. Nochetto and M. S. Pauletti, An adaptive method for hierarchical splines. A posteriori estimation via local problems, convergence and optimality, In preparation. Google Scholar

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Figure 1.  A univariate quadratic B-spline of level $\ell$ (left) and its truncation obtained by considering $\Omega^{\ell+1} = [1,3.0]$ (right)
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Figure 2.  The REFINE and REFINE_RECURSIVE modules
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Figure 3.  Exact solutions for the smooth function with a peak (left) and the singular function (right)
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Figure 4.  Numerical error and estimator (left) and effectivity index (right) for the smooth function with a peak. On the left, the error (solid lines) and the estimator (dashed lines) are plotted for different degrees. The red, green, blue and black lines (star, diamond, cross and circle markers, respectively) represent admissibility classes $m = 2,3,4,\infty$, respectively, while the magenta line (square marker) corresponds to uniform refinement. The same coloring and marking is used for the effectivity index (divided by 10) on the right figure
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Figure 5.  Hierarchical meshes obtained for the smooth solution with a peak
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Figure 6.  Numerical error and estimator (left) and effectivity index (right) for the example with singular solution. On the left, the error (solid lines) and the estimator (dashed lines) are plotted for different degrees. The red, green, blue and black lines (star, diamond, cross and circle markers, respectively) represent admissibility classes $m = 2,3,4,\infty$, respectively, while the magenta line (square marker) corresponds to uniform refinement. The same coloring and marking is used for the effectivity index (divided by 10) on the right figure
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Figure 7.  Hierarchical meshes obtained for the example with singular solution
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Figure 8.  Geometry and solution for the 3D example
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Figure 9.  Convergence of the estimator for the 3D example. The red, green, blue and black lines (star, diamond, cross and circle markers, respectively) represent admissibility classes $m = 2,3,4,\infty$, respectively, while the magenta line (square marker) corresponds to uniform refinement
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Figure 10.  Meshes obtained for the 3D example degree $\mathbf{p} = (4,4)$ and different values of the admissibility class $m$
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Table 1.  Number of elements ($\# {\cal Q}$) and degrees of freedom ($\# {\cal T}({\cal Q})$) for Example 1
$\mathbf{p}=(2,2)$ $\mathbf{p}=(3,3)$ $\mathbf{p}=(4,4)$
$\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$
$m=2$ $3232$ $2876$ $3880 $ $3313$ $3736 $ $3400$
$m=3$ $2500$ $2244$ $3868 $ $3317$ $2824 $ $2224$
$m=4$ $3628$ $3272$ $3064 $ $2614$ $2224 $ $1672$
$m=\infty$ $3616$ $3244$ $2980 $ $2446$ $2284 $ $1756$
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$\mathbf{p}=(2,2)$ $\mathbf{p}=(3,3)$ $\mathbf{p}=(4,4)$
$\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$
$m=2$ $3232$ $2876$ $3880 $ $3313$ $3736 $ $3400$
$m=3$ $2500$ $2244$ $3868 $ $3317$ $2824 $ $2224$
$m=4$ $3628$ $3272$ $3064 $ $2614$ $2224 $ $1672$
$m=\infty$ $3616$ $3244$ $2980 $ $2446$ $2284 $ $1756$
Table 2.  Number of elements ($\# {\cal Q}$) and degrees of freedom ($\# {\cal T}({\cal Q})$) for Example 2
$\mathbf{p}=(2,2)$ $\mathbf{p}=(3,3)$ $\mathbf{p}=(4,4)$
$\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$
$m=2$ $68677$ $68860$ $8776 $ $8938$ $3700 $ $3822$
$m=3$ $68014$ $68201$ $6346 $ $6631$ $2875 $ $2966$
$m=4$ $68008$ $68195$ $6208 $ $6484$ $3010 $ $3138$
$m=\infty$ $68008$ $68195$ $6208 $ $6484$ $2827 $ $2895$
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$\mathbf{p}=(2,2)$ $\mathbf{p}=(3,3)$ $\mathbf{p}=(4,4)$
$\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$
$m=2$ $68677$ $68860$ $8776 $ $8938$ $3700 $ $3822$
$m=3$ $68014$ $68201$ $6346 $ $6631$ $2875 $ $2966$
$m=4$ $68008$ $68195$ $6208 $ $6484$ $3010 $ $3138$
$m=\infty$ $68008$ $68195$ $6208 $ $6484$ $2827 $ $2895$
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