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Adaptive IGAFEM with optimal convergence rates: hierarchical B-splines. (English) Zbl 1376.41006

Summary: We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension \(d \geq 2\). We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (respectively, the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.

MSC:

41A15 Spline approximation
65D07 Numerical computation using splines
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

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[1] Ainsworth, M. and Oden, J. T., A posteriori Error Estimation in Finite Element Analysis, (John Wiley & Sons, 2000). · Zbl 1008.65076
[2] Aurada, M., Ferraz-Leite, S. and Praetorius, D., Estimator reduction and convergence of adaptive BEM, Appl. Numer. Math.62 (2012) 787-801. · Zbl 1237.65131
[3] Bazilevs, Y., da Veiga, L. Beirão, Cottrell, J. A., Hughes, T. J. R. and Sangalli, G., Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes, Math. Models Methods Appl. Sci.16 (2006) 1031-1090. · Zbl 1103.65113
[4] da Veiga, L. Beirão, Brezzi, F., Cangiani, A., Manzini, G., Marini, L. Donatella and Russo, A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci.23 (2013) 199-214. · Zbl 1416.65433
[5] da Veiga, L. Beirão, Buffa, A., Sangalli, G. and Vázquez, R., Analysis-suitable T-splines of arbitrary degree: Definition, linear independence and approximation properties, Math. Models Methods Appl. Sci.23 (2013) 1979-2003. · Zbl 1270.65009
[6] da Veiga, L. Beirão, Buffa, A., Sangalli, G. and Vázquez, R., Mathematical analysis of variational isogeometric methods, Acta Numerica23 (2014) 157-287. · Zbl 1398.65287
[7] Bespalov, A., Haberl, A. and Praetorius, D., Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems, Comput. Methods Appl. Mech. Eng.317 (2017) 318-340. · Zbl 1439.65148
[8] Binev, P., Dahmen, W. and DeVore, R., Adaptive finite element methods with convergence rates, Numer. Math.97 (2004) 219-268. · Zbl 1063.65120
[9] A. Buffa and E. M. Garau, A posteriori error estimators for hierarchical B-spline discretizations, preprint (2016), arXiv:1611.07816. · Zbl 1398.65291
[10] Buffa, A. and Garau, E. M., Refinable spaces and local approximation estimates for hierarchical splines, IMA J. Numer. Anal.37 (2016) 1125-1149. · Zbl 1433.41005
[11] Buffa, A. and Giannelli, C., Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence, Math. Models Methods Appl. Sci.26 (2016) 1-25. · Zbl 1336.65181
[12] A. Buffa and C. Giannelli, Adaptive Isogeometric Methods with Hierarchical Splines: Optimality and Convergence Rates, Technical Report (École Polytechnique Fédérale de Lausanne, March, 2017). · Zbl 1376.41004
[13] Buffa, A., Giannelli, C., Mørgenstern, P. and Peterseim, D., Complexity of hierarchical refinement for a class of admissible mesh configurations, Comput.-Aided Geom. Design47 (2016) 83-92. · Zbl 1418.65011
[14] Carstensen, C., Feischl, M., Page, M. and Praetorius, D., Axioms of adaptivity, Comput. Math. Appl.67 (2014) 1195-1253. · Zbl 1350.65119
[15] Cascon, J. M., Kreuzer, C., Nochetto, R. H. and Siebert, K. G., Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal.46 (2008) 2524-2550. · Zbl 1176.65122
[16] Cottrell, J. A., Hughes, T. J. R. and Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (John Wiley & Sons, 2009). · Zbl 1378.65009
[17] De Boor, C., A Practical Guide to Splines, rev. edn. (Springer, 2001). · Zbl 0987.65015
[18] Dokken, T., Lyche, T. and Pettersen, K. F., Polynomial splines over locally refined box-partitions, Comput.-Aided Geom. Design30 (2013) 331-356. · Zbl 1264.41011
[19] Dörfler, W., A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal.33 (1996) 1106-1124. · Zbl 0854.65090
[20] C. Erath, Adaptive Finite Volumen Methode, Master’s Thesis (Institute for Analysis and Scientific Computing, TU Wien, 2005).
[21] Feischl, M., Führer, T. and Praetorius, D., Adaptive FEM with optimal convergence rates for a certain class of nonsymmetric and possibly nonlinear problems, SIAM J. Numer. Anal.52 (2014) 601-625. · Zbl 1300.65086
[22] Feischl, M., Gantner, G., Haberl, A. and Praetorius, D., Adaptive 2D IGA boundary element methods, Eng. Anal. Bound. Element62 (2016) 141-153. · Zbl 1403.65267
[23] Feischl, M., Gantner, G., Haberl, A. and Praetorius, D., Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations, Numer. Math.136 (2017) 147-182. · Zbl 1362.65131
[24] Feischl, M., Gantner, G. and Praetorius, D., Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations, Comput. Methods Appl. Mech. Eng.290 (2015) 362-386. · Zbl 1425.65200
[25] Gaspoz, F. D. and Morin, P., Convergence rates for adaptive finite elements, IMA J. Numer. Anal.29 (2008) 917-936. · Zbl 1183.65134
[26] Giannelli, C., Jüttler, B. and Speleers, H., THB-splines: The truncated basis for hierarchical splines, Comput.-Aided Geom. Design29 (2012) 485-498. · Zbl 1252.65030
[27] Giannelli, C., Jüttler, B. and Speleers, H., Strongly stable bases for adaptively refined multilevel spline spaces, Adv. Comput. Math.40 (2014) 459-490. · Zbl 1298.41010
[28] Hennig, P., Kästner, M., Mørgenstern, P. and Peterseim, D., Adaptive mesh refinement strategies in isogeometric analysis — A computational comparison, Comput. Methods Appl. Mech. Eng.316 (2017) 424-448. · Zbl 1439.65169
[29] Hughes, T. J. R., Cottrell, J. A. and Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng.194 (2005) 4135-4195. · Zbl 1151.74419
[30] Johannessen, K. A., Kvamsdal, T. and Dokken, T., Isogeometric analysis using LR B-splines, Comput. Methods Appl. Mech. Eng.269 (2014) 471-514. · Zbl 1296.65021
[31] Johannessen, K. A., Remonato, F. and Kvamsdal, T., On the similarities and differences between classical hierarchical, truncated hierarchical and LR B-splines, Comput. Methods Appl. Mech. Eng.291 (2015) 64-101. · Zbl 1425.65027
[32] Kuru, G., Verhoosel, C. V., Van der Zee, K. G. and van Brummelen, E. Harald, Goal-adaptive isogeometric analysis with hierarchical splines, Comput. Methods Appl. Mech. Eng.270 (2014) 270-292. · Zbl 1296.65162
[33] Mørgenstern, P., Globally structured three-dimensional analysis-suitable T-splines: Definition, linear independence and \(m\)-graded local refinement, SIAM J. Numer. Anal.54 (2016) 2163-2186. · Zbl 1386.65077
[34] Mørgenstern, P. and Peterseim, D., Analysis-suitable adaptive T-mesh refinement with linear complexity, Comput.-Aided Geom. Design34 (2015) 50-66. · Zbl 1375.65035
[35] Morin, P., Nochetto, R. H. and Siebert, K. G., Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal.38 (2000) 466-488. · Zbl 0970.65113
[36] Nochetto, R. H. and Veeser, A., Primer of adaptive finite element methods, in Multiscale and Adaptivity: Modeling, Numerics and Applications, , Vol. 2040 (Springer2012), pp. 125-225. · Zbl 1252.65192
[37] Rudin, W., Functional Analysis, 2nd edn. (McGraw-Hill, 1991). · Zbl 0867.46001
[38] S. Schimanko, Adaptive Isogeometric Boundary Element Method for the Hyper-Singular Integral Equation, Master’s Thesis (Institute for Analysis and Scientific Computing, TU Wien, 2016).
[39] Schumaker, L., Spline Functions: Basic Theory (Cambridge Univ. Press, 2007). · Zbl 1123.41008
[40] Scott, M. A., Li, X., Sederberg, T. W. and Hughes, T. J. R., Local refinement of analysis-suitable T-splines, Comput. Methods Appl. Mech. Eng.213 (2012) 206-222. · Zbl 1243.65030
[41] Speleers, H. and Manni, C., Effortless quasi-interpolation in hierarchical spaces, Numer. Math.132 (2016) 155-184. · Zbl 1335.65021
[42] Stevenson, R., Optimality of a standard adaptive finite element method, Found. Comput. Math.7 (2007) 245-269. · Zbl 1136.65109
[43] Stevenson, R., The completion of locally refined simplicial partitions created by bisection, Math. Comput.77 (2008) 227-241. · Zbl 1131.65095
[44] Verfürth, R., A posteriori Error Estimation Techniques for Finite Element Methods (Oxford Univ. Press, 2013). · Zbl 1279.65127
[45] Vuong, A.-V., Giannelli, C., Jüttler, B. and Simeon, B., A hierarchical approach to adaptive local refinement in isogeometric analysis, Comput. Methods Appl. Mech. Eng.200 (2011) 3554-3567. · Zbl 1239.65013
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