Geodesic
Hp-version of the least-squares collocation method with Gaussian points
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 2, pp. 1181-1201 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper addresses new hp-version of the least-squares collocation method (hp-LSCM) with Gaussian points. The optimal order of convergence of the developed method for solving boundary value problems for Poisson's equation, biharmonic equation, and for a system of partial differential equations of the Reissner–Mindlin plate problem is shown numerically. An algorithm for obtaining a system of linear algebraic equations with a invertible quadratic matrix in hp-LSCM is given. The advantages of the developed collocation method in comparison with previous versions of the hp-LSCM and isogeometric collocation method are shown.
Keywords: least-squares collocation method, Gaussian points, biharmonic equation, Kirchhoff – Love theory, Reissner – Mindlin theory, plate bending.
Mots-clés : optimal convergence, Poisson's equation 
@article{SEMR_2024_21_2_a51,
     author = {L. S. Bryndin and V. A. Belyaev},
     title = {Hp-version of the least-squares collocation method with {Gaussian} points},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1181--1201},
     year = {2024},
     volume = {21},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2024_21_2_a51/}
}
TY  - JOUR
AU  - L. S. Bryndin
AU  - V. A. Belyaev
TI  - Hp-version of the least-squares collocation method with Gaussian points
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2024
SP  - 1181
EP  - 1201
VL  - 21
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SEMR_2024_21_2_a51/
LA  - ru
ID  - SEMR_2024_21_2_a51
ER  - 
%0 Journal Article
%A L. S. Bryndin
%A V. A. Belyaev
%T Hp-version of the least-squares collocation method with Gaussian points
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2024
%P 1181-1201
%V 21
%N 2
%U http://geodesic.mathdoc.fr/item/SEMR_2024_21_2_a51/
%G ru
%F SEMR_2024_21_2_a51
L. S. Bryndin; V. A. Belyaev. Hp-version of the least-squares collocation method with Gaussian points. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 2, pp. 1181-1201. http://geodesic.mathdoc.fr/item/SEMR_2024_21_2_a51/

[1] V.A. Belyaev, L.S. Bryndin, S.K. Golushko, B.V. Semisalov, V.P. Shapeev, “h-, p-, and hp-versions of the least-squares collocation method for solving boundary value problems for biharmonic equation in irregular domains and their applications”, Comput. Math. Math. Phys., 62:4 (2022), 517–537 | DOI | Zbl

[2] L.S. Bryndin, V.A. Belyaev, V.P. Shapeev, “Development and verification of a simplified hp-version of the least-squares collocation method for irregular domains”, Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Model. Program., 16:3 (2023), 35–50 | DOI | Zbl

[3] W. Shao, X. Wu, S. Chen, “Chebyshev tau meshless method based on the integration-differentiation for biharmonic-type equations on irregular domain”, Eng. Anal. Bound. Elem., 36:12 (2012), 1787–1798 | DOI | Zbl

[4] D. Schillinger, J.A. Evans, A. Reali, M.A. Scott, T.J.R. Hughes, “Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations”, Comput. Methods Appl. Mech. Eng., 267 (2013), 170–232 | DOI | Zbl

[5] V.A. Kireev, “Hermite bicubic collocation method in domain with curvilinear boundary”, Vestnik SibGAU, 15:3 (2014), 73–77 https://journals.eco-vector.com/2712-8970/article/view/504066

[6] J. Kiendl, F. Auricchio, L. Beirão da Veiga, C. Lovadina, A. Reali, “Isogeometric collocation methods for the Reissner-Mindlin plate problem”, Comput. Methods Appl. Mech. Eng., 284 (2015), 489–507 | DOI | Zbl

[7] C. de Boor, B. Swartz, “Collocation at Gaussian points”, SIAM J. Numer. Anal., 10:4 (1973), 582–606 | DOI | Zbl

[8] G. Fairweather, A. Karageorghis, J. Maack, “Compact optimal quadratic spline collocation methods for the Helmholtz equation”, J. Comput. Phys., 230:8 (2011), 2880–2895 | DOI | Zbl

[9] B. Bialecki, G. Fairweather, A. Karageorghis, J. Maack, “A quadratic spline collocation method for the Dirichlet biharmonic problem”, Numer. Algorithms, 83:1 (2020), 165–199 | DOI | Zbl

[10] X. Yang, H. Zhang, D. Xu, “WSGD-OSC scheme for two-dimensional distributed order fractional reaction-diffusion equation”, J. Sci. Comput., 76:3 (2018), 1502–1520 | DOI | Zbl

[11] R.I. Fernandes, G. Fairweather, “An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems”, J. Comput. Phys., 231:19 (2012), 6248–6267 | DOI | Zbl

[12] B. Bialecki, G. Fairweather, “Orthogonal spline collocation methods for partial differential equations”, J. Comput. Appl. Math., 128:1-2 (2001), 55–82 | DOI | Zbl

[13] S. Timoshenko, S. Woinowsky-Krieger, Theory of plates and shells, McGraw-Hill Book Comp., New York etc., 1959 | Zbl

[14] J.N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis, 2nd edn., CRC Press, Boca Raton, 2004 | DOI | Zbl

[15] S.V. Idimeshev, Modified method of collocations and least residuals and its application in the mechanics of multilayer composite beams and plates, Dissertation, Cand. Sci. (Phys.-Math.), Institute of Computing Technologies, Novosibirsk, 2016

[16] V.A. Belyaev, “Solving the Poisson equation with singularities by the least-squares collocation method”, Numer. Anal. Appl., 13:3 (2020), 207–218 | DOI

[17] G. Chen, Z. Li, P. Lin, “A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow”, Adv. Comput. Math., 29:2 (2008), 113–133 | DOI | Zbl

[18] C.C. Ike, “Mathematical solutions for the flexural analysis of Mindlin's first order shear deformable circular plate”, Math. Models Engineering, 4:2 (2018), 50–72 | DOI

[19] O. Garcia, E.A. Fancello, C.S. de Barcellos, C.A. Duarte, “hp-clouds in Mindlin's thick plate model”, Int. J. Numer. Methods Eng., 47:8 (2000), 1381–1400 | 3.0.CO;2-9 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | Zbl

[20] C. Tiago, V.M.A. Leit$\tilde{\rm{a}}$o, “Eliminating shear-locking in meshless methods: a critical overview and a new framework for structural theories”, Advances in meshfree techniques, Computational Methods in Applied Sciences, 5, eds. Leit$\tilde{\rm{a}}$o V.M.A. et al., Springer, Dordrecht, 2007, 123–147 | DOI | Zbl

[21] A.G. Sleptsov, Yu.I. Shokin, “An adaptive grid-projection method for elliptic problems”, Comput. Math. Math. Phys., 37:5 (1997), 558–571 | Zbl

[22] S.K. Golushko, S.V. Idimeshev, V.P. Shapeev, “Application of collocations and least residuals method to problems of the isotropic plates theory”, Computational Technologies, 18:6 (2013), 31–43

[23] E.V. Vorozhtsov, V.P. Shapeev, “On the efficiency of combining different methods for acceleration of iterations at the solution of PDEs by the method of collocations and least residuals”, Appl. Math. Comput., 363 (2019), 124644 | DOI | Zbl

[24] T.A. Davis, “Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization”, ACM Trans. Math. Softw., 38:1 (2011), 8 | DOI | Zbl

[25] J.M. Ortega, Introduction to parallel and vector solution of linear systems, Plenum Press, New York etc., 1988 | Zbl

[26] V.P. Shapeev, L.S. Bryndin, V.A. Belyaev, “The hp-version of the least-squares collocation method with integral collocation for solving a biharmonic equation”, Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki, 26:3 (2022), 556–572 | DOI | Zbl

[27] P.M. Prenter, R.D. Russel, “Orthogonal collocation for elliptic partial differential equations”, SIAM J. Numer. Anal., 13:6 (1976), 923–939 | DOI | Zbl

[28] L. Yunhua, On Shear Locking in Finite Elements, Royal Institute of Technology, Stockholm, 1997 https://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-739