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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

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Abstract

New h-, p-, and hp-versions of the least-squares collocation method are proposed and implemented. They yield approximate solutions of boundary value problems for an inhomogeneous biharmonic equation in irregular and multiply-connected domains. Formulas for the extension operation in the transition from coarse to finer grids on a multigrid complex are given in the case of applying various spaces of polynomials. It is experimentally shown that numerical solutions of boundary value problems produced by the developed versions of the method have a higher order of convergence to analytical solutions with no singularities. The results are compared with those of other authors produced by applying finite difference, finite element, and other methods based on Chebyshev polynomials. Examples of problems with singularities are considered. The developed versions of the method are used to simulate the bending of an elastic isotropic plate of irregular shape subjected to transverse loading.

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ACKNOWLEDGMENTS

The authors are grateful to S.V. Idimeshev, V.I. Isaev, and V.A. Kolotilov for helpful discussion of the results of this work.

Funding

This work was performed in the framework of the state assignment (state registration nos. 121030500137-5 and AAAA-A19-119051590004-5) and was supported in part by the Russian Foundation for Basic Research (project no. 18-29-18029).

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Authors and Affiliations

  1. Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    V. A. Belyaev, L. S. Bryndin & V. P. Shapeev

  2. Novosibirsk State University, 630090, Novosibirsk, Russia

    V. A. Belyaev, L. S. Bryndin, S. K. Golushko, B. V. Semisalov & V. P. Shapeev

  3. Federal Research Center for Information and Computational Technologies, 630090, Novosibirsk, Russia

    S. K. Golushko

  4. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    B. V. Semisalov

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  1. V. A. Belyaev
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  2. L. S. Bryndin
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  3. S. K. Golushko
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  4. B. V. Semisalov
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  5. V. P. Shapeev
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Correspondence to V. A. Belyaev, L. S. Bryndin, S. K. Golushko, B. V. Semisalov or V. P. Shapeev.

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Belyaev, V.A., Bryndin, L.S., Golushko, S.K. et al. 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. and Math. Phys. 62, 517–537 (2022). https://doi.org/10.1134/S0965542522040029

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