Adaptive computation with splines on triangulations with hanging vertices. (English) Zbl 1385.65019
Fasshauer, Gregory E. (ed.) et al., Approximation theory XV: San Antonio 2016. Selected papers based on the presentations at the international conference, San Antonio, TX, USA, May 22–25, 2016. Cham: Springer (ISBN 978-3-319-59911-3/hbk; 978-3-319-59912-0/ebook). Springer Proceedings in Mathematics & Statistics 201, 197-218 (2017).
Summary: It is shown how computational methods based on Bernstein-Bézier methods for polynomial splines on triangulations can be carried over to compute with splines on triangulations with hanging vertices. Allowing triangulations with hanging vertices provides much more flexibility than using ordinary triangulations and allows for simple adaptive algorithms based on local refinements. The use of these techniques is illustrated for two application areas of splines – namely, function fitting and the solution of boundary value problems.
For the entire collection see [Zbl 1378.65012].
For the entire collection see [Zbl 1378.65012].
MSC:
65D07 | Numerical computation using splines |
35J25 | Boundary value problems for second-order elliptic equations |
65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
65N15 | Error bounds for boundary value problems involving PDEs |
Keywords:
spline functions; hanging vertices; adaptive methods for fitting; adaptive finite elements; Ritz-Galerkin method; error estimates; numerical experiment; Bernstein-Bézier methods; algorithm; boundary value problemsSoftware:
SplinePakReferences:
[1] | M. Ainsworth, J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis (Wiley-Interscience, New York, 2000) · Zbl 1008.65076 · doi:10.1002/9781118032824 |
[2] | M. Ainsworth, R. Rankin, Constant free error bounds for nonuniform order discontinuous Galerkin finite-element approximation on locally refined meshes with hanging nodes. IMA J. Numer. Anal. 31, 254-280 (2011) · Zbl 1208.65156 · doi:10.1093/imanum/drp025 |
[3] | M. Ainsworth, L. Demkowicz, C.-W. Kim, Analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes. Comput. Methods Appl. Mech. Eng. 196, 3493-3507 (2007) · Zbl 1173.76424 · doi:10.1016/j.cma.2006.10.020 |
[4] | I. Babuška, W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736-754 (1978) · Zbl 0398.65069 · doi:10.1137/0715049 |
[5] | R.E. Bank, R.K. Smith, A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30(4), 921-935 (1993) · Zbl 0787.65078 · doi:10.1137/0730048 |
[6] | C. Carstensen, J. Hu, A. Orlando, Framework for the a posteriori error analysis of nonconforming finite elements. SIAM J. Numer. Anal. 45(1), 68-82 (2007) · Zbl 1165.65072 · doi:10.1137/050628854 |
[7] | A. Cohen, N. Dyn, F. Hecht, J.-M. Mirebeau, Adaptive multiresolution analysis based on anisotropic triangulations. Math. Comp. 81, 789-810 (2012) · Zbl 1242.65293 · doi:10.1090/S0025-5718-2011-02495-6 |
[8] | L. Demaret, N. Dyn, M.S. Floater, A. Iske, Adaptive Thinning for Terrain Modelling and Image Compression, Advances in Multiresolution for Geometric Modelling, Math. Vis. (Springer, Berlin, 2005) · Zbl 1066.94504 |
[9] | L. Demaret, N. Dyn, A. Iske, Image compression by linear splines over adaptive triangulations. Signal Process 86, 1604-1616 (2006) · Zbl 1172.94312 · doi:10.1016/j.sigpro.2005.09.003 |
[10] | C. Erath, D. Praetorius, A posteriori error estimate and adaptive mesh refinement for the cell-centered finite volume method for elliptic boundary value problems. SIAM J. Numer. Anal. 47(1), 109-135 (2008) · Zbl 1188.65148 · doi:10.1137/070702126 |
[11] | A. Iske, Multiresolution Methods in Scattered Data Modelling (Springer, Berlin, 2004) · Zbl 1057.65004 · doi:10.1007/978-3-642-18754-4 |
[12] | M.-J. Lai, C. Mersmann, An Adaptive Triangulation Method for Bivariate Spline Solutions of PDEs, in Approximation Theory XV: San Antonio 2016, Springer Proc. in Math. and Stat. vol. 201, ed. by G.E. Fasshauer, L.L. Schumaker (Springer-Verlag, 2017) pp. 155-175 · Zbl 1385.65059 |
[13] | M.-J. Lai, L.L. Schumaker, Spline Functions on Triangulations (Cambridge University Press, Cambridge, 2007) · Zbl 1185.41001 · doi:10.1017/CBO9780511721588 |
[14] | W.F. Mitchell, A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Softw. 15, 326-347 (1989) · Zbl 0900.65306 · doi:10.1145/76909.76912 |
[15] | L.L. Schumaker, Computing bivariate splines in scattered data fitting and the finite element method. Numer. Algorithms 48, 237-260 (2008) · Zbl 1146.65019 · doi:10.1007/s11075-008-9175-x |
[16] | L.L. Schumaker, Splines on spherical triangulations with hanging vertices. Comput. Aided Geom. Design 30, 263-275 (2013) · Zbl 1268.41006 · doi:10.1016/j.cagd.2013.01.001 |
[17] | L.L. Schumaker, Spline Functions: Computational Methods (SIAM, Philadelphia, 2015) · Zbl 1333.65018 · doi:10.1137/1.9781611973907 |
[18] | L.L. Schumaker, L. Wang, Spline spaces on TR-meshes with hanging vertices. Numer. Math. 118, 531-548 (2011) · Zbl 1225.41003 · doi:10.1007/s00211-010-0353-0 |
[19] | L.L. Schumaker, L. Wang, Splines on triangulations with hanging vertices. Constr. Approx. 36, 487-511 (2012) · Zbl 1262.41008 · doi:10.1007/s00365-012-9167-x |
[20] | P. Šolń, J. Červený, I. Doležel, Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM. Math. Comput. Simul. 77(1), 117-132 (2008) · Zbl 1135.65394 · doi:10.1016/j.matcom.2007.02.011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.