{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T06:21:49Z","timestamp":1770704509091,"version":"3.49.0"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2014,6,11]],"date-time":"2014-06-11T00:00:00Z","timestamp":1402444800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper outlines and qualitatively compares the implementations of seven different methods for solving Poisson\u2019s equation on the disk. The methods include two classical finite elements, a cotan formula-based discrete differential geometry approach and four isogeometric constructions. The comparison reveals numerical convergence rates and, particularly for isogeometric constructions based on Catmull\u2013Clark elements, the need to carefully choose quadrature formulas. The seven methods include two that are new to isogeometric analysis. Both new methods yield O(h3) convergence in the L2 norm, also when points are included where n 6\u2260 4 pieces meet. One construction is based on a polar, singular parameterization; the other is a G1 tensor-product construction.<\/jats:p>","DOI":"10.3390\/axioms3020280","type":"journal-article","created":{"date-parts":[[2014,6,11]],"date-time":"2014-06-11T11:59:22Z","timestamp":1402487962000},"page":"280-299","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":80,"title":["A Comparative Study of Several Classical, Discrete Differential and Isogeometric Methods for Solving Poisson\u2019s Equation on the Disk"],"prefix":"10.3390","volume":"3","author":[{"given":"Thien","family":"Nguyen","sequence":"first","affiliation":[{"name":"Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32611, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ke\u00e7stutis","family":"Kar\u010diauskas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Vilnius University, LT-01513 Vilnius, Lithuania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00f6rg","family":"Peters","sequence":"additional","affiliation":[{"name":"Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32611, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2014,6,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1080\/10586458.1993.10504266","article-title":"Computing Discrete Minimal Surfaces and Their Conjugates","volume":"2","author":"Pinkall","year":"1993","journal-title":"Exp. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bobenko, A.I., and Suris, Y.B. (2008). Discrete Differential Geometry: Integrable Structure, AMS Bookstore.","DOI":"10.1090\/gsm\/098"},{"key":"ref_3","unstructured":"Kraft, R. (1997). Surface Fitting and Multiresolution Methods, Vanderbilt University Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1145\/1015706.1015715","article-title":"T-spline simplification and local refinement","volume":"23","author":"Sederberg","year":"2004","journal-title":"ACM Trans. Graph."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1016\/j.cagd.2012.12.005","article-title":"Polynomial splines over locally refined box-partitions","volume":"30","author":"Dokken","year":"2013","journal-title":"Comput. Aided Geom. Des."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1016\/j.cagd.2012.03.025","article-title":"THB\u2013splines: The truncated basis for hierarchical splines","volume":"29","author":"Giannelli","year":"2012","journal-title":"Comput. Aided Geom. Des."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1002\/nme.339","article-title":"(2001) Displacement and equilibrium models in the finite element method by B. Fraeijs de Veubeke, Chapter 9, Pages 145\u2013197, of Stress Analysis","volume":"52","author":"Zienkiewicz","year":"1965","journal-title":"Int. J. Numer. Meth. Engng."},{"key":"ref_8","unstructured":"Farin, G.E. (2001). Curves and Surfaces for CAGD: A Practical Guide, Morgan Kaufmann Publishers. [5th ed.]."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Braess, D. (2007). Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press.","DOI":"10.1017\/CBO9780511618635"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1023\/A:1014299228104","article-title":"Smooth macro-elements based on Powell\u2013Sabin triangle splits","volume":"16","author":"Alfeld","year":"2002","journal-title":"Adv. Comput. Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Lai, M.J., and Schumaker, L.L. (2007). Spline Functions on Triangulations, Cambridge University Press.","DOI":"10.1017\/CBO9780511721588"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.cma.2012.02.009","article-title":"Isogeometric analysis with Powell\u2013Sabin splines for advection\u2013diffusion\u2013reaction problems","volume":"221","author":"Speleers","year":"2012","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.cad.2013.08.017","article-title":"Isogeometric analysis on triangulations","volume":"46","author":"Jaxon","year":"2014","journal-title":"Comput.-Aided Des."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Wardetzky, M., Mathur, S., K\u00e4lberer, F., and Grinspun, E. (2007, January 4\u20136). Discrete Laplace operators: no free lunch. Proceedings of the Fifth Eurographics Symposium on Geometry Processing, Barcelona, Spain.","DOI":"10.1145\/1508044.1508063"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Desbrun, M., Meyer, M., Schr\u00f6der, P., and Barr, A.H. (1999, January 8\u201313). Implicit fairing of irregular meshes using diffusion and curvature flow. Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, Los Angeles, CA, USA.","DOI":"10.1145\/311535.311576"},{"key":"ref_16","first-page":"65","article-title":"Computational aspects of discrete minimal surfaces","volume":"2","author":"Polthier","year":"2005","journal-title":"Glob. Theory Minimal Surf."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/j.camwa.2004.05.001","article-title":"Convergence of discrete Laplace-Beltrami operators over surfaces","volume":"48","author":"Xu","year":"2004","journal-title":"Comput. Math. Appl."},{"key":"ref_18","unstructured":"Wardetzky, M. (2008). Discrete Differential Geometry, Birkh\u00e4user."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Crane, K., de Goes, F., Desbrun, M., and Schr\u00f6der, P. (2013, January 21\u201325). Digital Geometry Processing with Discrete Exterior Calculus. Proceedings of the SIGGRAPH \u201913 Special Interest Group on Computer Graphics and Interactive Techniques Conference, Anaheim, CA, USA.","DOI":"10.1145\/2504435.2504442"},{"key":"ref_20","unstructured":"Breen, D., and Lin, M. (2003, January 26\u201327). Discrete Shells. Proceedings of the Eurographics\/SIGGRAPH Symposium on Computer Animation, San Diego, CA, USA."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Pottmann, H., Liu, Y., Wallner, J., Bobenko, A., and Wang, W. (2007). Geometry of multi-layer freeform structures for architecture. ACM Trans. Graph., 26.","DOI":"10.1145\/1239451.1239516"},{"key":"ref_22","unstructured":"Hirani, A.N. (2003). Discrete Exterior Calculus. [Ph.D. Thesis, California Institute of Technology]."},{"key":"ref_23","unstructured":"Grinspun, E., Desbrun, M., Polthier, K., Schr\u00f6der, P., and Stern, A. (August, January 30). Discrete differential geometry: An applied introduction. Proceedings of the ACM SIGGRAPH Course, Boston, MA, USA."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4135","DOI":"10.1016\/j.cma.2004.10.008","article-title":"Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement","volume":"194","author":"Hughes","year":"2005","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"De Boor, C. (1978). A Practical Guide to Splines, Springer.","DOI":"10.1007\/978-1-4612-6333-3"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2039","DOI":"10.1002\/(SICI)1097-0207(20000430)47:12<2039::AID-NME872>3.0.CO;2-1","article-title":"Subdivision surfaces: A new paradigm for thin-shell finite-element analysis","volume":"47","author":"Cirak","year":"2000","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/S0010-4485(01)00061-6","article-title":"Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision","volume":"34","author":"Cirak","year":"2002","journal-title":"Comput.-Aided Des."},{"key":"ref_28","unstructured":"Peters, J. (2002). Handbook of Computer Aided Geometric Design, Elsevier."},{"key":"ref_29","unstructured":"Kar\u010diauskas, K., and Peters, J. (2013, January 11\u201313). Biquintic \n            G2\n           surfaces. Proceedings of the 14th IMA Conference on Mathematics of Surfaces, University of Birmingham, West Midlands, UK."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Peters, J., and Reif, U. (2008). Subdivision Surfaces, Geometry and Computing; Springer-Verlag.","DOI":"10.1007\/978-3-540-76406-9"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1145\/566654.566578","article-title":"CHARMS: A simple framework for adaptive simulation","volume":"21","author":"Grinspun","year":"2002","journal-title":"ACM Trans. Graph."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1575","DOI":"10.1111\/j.1467-8659.2010.01766.x","article-title":"Iso-geometric Finite Element Analysis Based on Catmull-Clark : Subdivision Solids","volume":"29","author":"Burkhart","year":"2010","journal-title":"Comput. Graph. Forum"},{"key":"ref_33","unstructured":"Burkhart, D. (2011). Subdivision for Volumetric Finite Elements. [Ph.D. Thesis, University of Kaiserslautern]."},{"key":"ref_34","unstructured":"Wawrzinek, A., Hildebrandt, K., and Polthier, K. (2011, January 4\u20136). Koiter\u2019s Thin Shells on Catmull-Clark Limit Surfaces. Proceedings of the Vision, Modeling, and Visualization Workshop 2011, Berlin, Germany."},{"key":"ref_35","unstructured":"Dikici, E., Snare, S.R., and Orderud, F. (2012, January 28\u201330). Isoparametric finite element analysis for Doo-Sabin subdivision models. Proceedings of the 2012 Graphics Interace Conference, Toronto, ON, Canada."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Vetter, R., Stoop, N., Jenni, T., Wittel, F.K., and Herrmann, H.J. (2012). Subdivision shell elements with anisotropic growth.","DOI":"10.1002\/nme.4536"},{"key":"ref_37","unstructured":"Barendrecht, P.J. (2013). IsoGeometric Analysis with Subdivision Surfaces, Eindhoven University of Technology."},{"key":"ref_38","unstructured":"Kar\u010diauskas, K., and Peters, J. (2014). Bi-5 Quad-Mesh Smoothing, Department CISE, University of Florida. Technical Report REP-2014-571."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1016\/j.cagd.2008.06.002","article-title":"Pairs of bi-cubic surface constructions supporting polar connectivity","volume":"25","author":"Myles","year":"2008","journal-title":"Comput. Aided Geom. Des."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/j.gmod.2012.05.006","article-title":"H2 regularity properties of singular parameterizations in isogeometric analysis","volume":"74","author":"Takacs","year":"2012","journal-title":"Graph. Models"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Halstead, M., Kass, M., and DeRose, T. (1993, January 2\u20136). Efficient, Fair Interpolation Using Catmull-Clark Surfaces. Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, Anaheim, CA, USA.","DOI":"10.1145\/166117.166121"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1142\/S0218202506001455","article-title":"Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes","volume":"16","author":"Bazilevs","year":"2006","journal-title":"Math. Models Methods Appl. Sci."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/3\/2\/280\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:12:22Z","timestamp":1760217142000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/3\/2\/280"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,11]]},"references-count":42,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2014,6]]}},"alternative-id":["axioms3020280"],"URL":"https:\/\/doi.org\/10.3390\/axioms3020280","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,6,11]]}}}