Optimal-order isogeometric collocation at Galerkin superconvergent points. (English) Zbl 1439.65187
Summary: In this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least-squares collocation method presented in [C. Anitescu et al., ibid. 284, 1073–1097 (2015; Zbl 1425.65193)] and the variational collocation method presented in [H. Gomez and L. De Lorenzis, “The variational collocation method”, ibid. 309, 152–181 (2016; doi:10.1016/j.cma.2016.06.003)]. The focus is on smoothest B-splines/NURBS approximations, i.e, having global \(C^{p - 1}\) continuity for polynomial degree \(p\). Within the framework of Gomez and De Lorenzis [loc. cit.], we select as collocation points a subset of those considered in [Anitescu, loc. cit.], which are related to the Galerkin superconvergence theory. With our choice, that features local symmetry of the collocation stencil, we improve the convergence behavior with respect to Gomez and De Lorenzis [loc. cit.], achieving optimal \(L^2\)-convergence for odd degree B-splines/NURBS approximations. The same optimal order of convergence is seen in [Anitescu, loc. cit.], where, however a least-squares formulation is adopted. Further careful study is needed, since the robustness of the method and its mathematical foundation are still unclear.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65D07 | Numerical computation using splines |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
Citations:
Zbl 1425.65193References:
| [1] | Bickley, W., Piecewise cubic interpolation and two-point boundary problems, Comput. J., 11, 2, 206-208 (1968) · Zbl 0155.48004 |
| [2] | Fyfe, D., The use of cubic splines in the solution of two-point boundary value problems, Comput. J., 12, 2, 188-192 (1969) · Zbl 0185.41404 |
| [3] | Houstis, E. N.; Vavalis, E.; Rice, J. R., Convergence of \(O(h^4)\) cubic spline collocation methods for elliptic partial differential equations, SIAM J. Numer. Anal., 25, 1, 54-74 (1988) · Zbl 0637.65113 |
| [4] | Russell, R. D.; Shampine, L. F., A collocation method for boundary value problems, Numer. Math., 19, 1, 1-28 (1972) · Zbl 0221.65129 |
| [5] | De Boor, C.; Swartz, B., Collocation at gaussian points, SIAM J. Numer. Anal., 10, 4, 582-606 (1973) · Zbl 0232.65065 |
| [6] | Arnold, D. N.; Wendland, W. L., The convergence of spline collocation for strongly elliptic equations on curves, Numer. Math., 47, 3, 317-341 (1985) · Zbl 0592.65077 |
| [7] | Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194, 39, 4135-4195 (2005) · Zbl 1151.74419 |
| [8] | Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (2009), John Wiley & Sons · Zbl 1378.65009 |
| [9] | Auricchio, F.; Beirão da Veiga, L.; Hughes, T.; Reali, A.; Sangalli, G., Isogeometric collocation methods, Math. Models Methods Appl. Sci., 20, 11, 2075-2107 (2010) · Zbl 1226.65091 |
| [10] | Auricchio, F.; Beirão da Veiga, L.; Kiendl, J.; Lovadina, C.; Reali, A., Locking-free isogeometric collocation methods for spatial Timoshenko rods, Comput. Methods Appl. Mech. Engrg., 263, 113-126 (2013) · Zbl 1286.74057 |
| [11] | Auricchio, F.; Beirão da Veiga, L.; Hughes, T. J.R.; Reali, A.; Sangalli, G., Isogeometric collocation for elastostatics and explicit dynamics, Comput. Methods Appl. Mech. Engrg., 249-252, 2-14 (2012) · Zbl 1348.74305 |
| [12] | Beirao da Veiga, L.; Lovadina, C.; Reali, A., Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods, Comput. Methods Appl. Mech. Engrg., 241, 38-51 (2012) · Zbl 1353.74045 |
| [13] | Kiendl, J.; Auricchio, F.; Beirao da Veiga, L.; Lovadina, C.; Reali, A., Isogeometric collocation methods for the reissner-mindlin plate problem, Comput. Methods Appl. Mech. Engrg., 284, 489-507 (2015) · Zbl 1425.65199 |
| [14] | Reali, A.; Gomez, H., An isogeometric collocation approach for Bernoulli-Euler beams and Kirchhoff plates, Comput. Methods Appl. Mech. Engrg., 284, 623-636 (2015) · Zbl 1423.74553 |
| [15] | De Lorenzis, L.; Evans, J.; Hughes, T.; Reali, A., Isogeometric collocation: Neumann boundary conditions and contact, Comput. Methods Appl. Mech. Engrg., 284, 21-54 (2015) · Zbl 1423.74947 |
| [16] | Reali, A.; Hughes, T. J., An introduction to isogeometric collocation methods, (Isogeometric Methods for Numerical Simulation (2015), Springer), 173-204 · Zbl 1327.74144 |
| [17] | Casquero, H.; Liu, L.; Zhang, Y.; Reali, A.; Gomez, H., Isogeometric collocation using analysis-suitable T-splines of arbitrary degree, Comput. Methods Appl. Mech. Engrg., 301, 164-186 (2016) · Zbl 1425.65195 |
| [18] | Matzen, M.; Cichosz, T.; Bischoff, M., A point to segment contact formulation for isogeometric, NURBS based finite elements, Comput. Methods Appl. Mech. Engrg., 255, 27-39 (2013) · Zbl 1297.74084 |
| [19] | Gomez, H.; Reali, A.; Sangalli, G., Accurate, efficient, and (iso) geometrically flexible collocation methods for phase-field models, J. Comput. Phys., 262, 153-171 (2014) · Zbl 1349.82084 |
| [20] | Manni, C.; Reali, A.; Speleers, H., Isogeometric collocation methods with generalized B-splines, Comput. Math. Appl., 70, 7, 1659-1675 (2015) · Zbl 1443.65107 |
| [21] | Gomez, H.; De Lorenzis, L., The variational collocation method, Comput. Methods Appl. Mech. Engrg., 309, 152-181 (2016) · Zbl 1439.74489 |
| [22] | Schillinger, D.; Evans, J. A.; Reali, A.; Scott, M.; Hughes, T., Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations, Comput. Methods Appl. Mech. Engrg., 267, 170-232 (2013) · Zbl 1286.65174 |
| [24] | de Boor, C., A practical guide to splines, (Applied Mathematical Sciences, vol. 27 (2001), Springer-Verlag: Springer-Verlag New York) · Zbl 0987.65015 |
| [25] | Anitescu, C.; Jia, Y.; Zhang, Y. J.; Rabczuk, T., An isogeometric collocation method using superconvergent points, Comput. Methods Appl. Mech. Engrg., 284, 1073-1097 (2015) · Zbl 1425.65193 |
| [26] | Piegl, L.; Tiller, W., The NURBS Book (2012), Springer |
| [27] | Wahlbin, L. B., Superconvergence in Galerkin Finite Element Methods (1995), Springer · Zbl 0826.65092 |
| [28] | Montardini, M., A new isogeometric collocation method based on Galerkin superconvergent points (2016), Università di Pavia, (Master’s Thesis) |
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