DOI: https://doi.org/10.26089/NumMet.v19r431

Solving the biharmonic equation with high order accuracy in irregular domains by the least squares collocation method

Authors

  • V.P. Shapeev
  • V.A. Belyaev

Keywords:

least squares collocation method
nonhomogeneous biharmonic equations
high order approximation
irregular domains
double splines

Abstract

This paper addresses a new version of the least squares collocation (LSC) method of high order accuracy proposed and implemented for the numerical solution of the nonhomogeneous biharmonic equation. The differential problem is projected onto a polynomial space of fourth and eighth degrees by the LSC method. The algorithm implemented is applied in irregular domains. The boundaries of these domains are given by analytical curves, in particular, by splines. The irregular domain is embedded in a rectangle covered by a regular grid with rectangular cells. In this paper we use the irregular cells (i-cells) which are cut off by the domain boundary from the rectangular cells of the initial regular grid. All i-cells are divided into two classes: the independent and dependent ones. The center of a dependent cell is located outside the domain by contrast with the center of an independent cell. The idea of attaching elongated dependent irregular cells to the neighboring ones is used. A separate piece of the analytical solution is constructed in the combined cells. The collocation and matching points located outside the domain are used to approximate the differential equations in the boundary cells. These two approaches allow us to essentially reduce the conditionality of the corresponding system of linear algebraic equations. It is shown that the approximate solutions obtained by the LSC method converge with an increased order and coincide with the analytical solutions of the test problems with high accuracy in the case of the known solution. The numerical results are compared with those found by other authors who used a high order finite difference method. The nonhomogeneous biharmonic equation is used to model the stress-strain state of isotropic thin irregular plates as an application.

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Published

2018-12-24

Issue

Vol. 19 (2018): Issue 4.

Section

Section 1. Numerical methods and applications

Author Biographies

V.P. Shapeev

S.A. Khristianovich Institute of Theoretical and Applied Mechanics of SB RAS
• Chief Researcher

V.A. Belyaev

S.A. Khristianovich Institute of Theoretical and Applied Mechanics of SB RAS
• Senior Assistant

References

  1. L. W. Ehrlich and M. M. Gupta, “Some Difference Schemes for the Biharmonic Equation,” SIAM J. Numer. Anal. 12 (5), 773-790 (1975).
  2. A. Mayo, “The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions,” SIAM J. Numer. Anal. 21 (2), 285-299 (1984).
  3. J. W. Stephenson, “Single Cell Discretizations of Order Two and Four for Biharmonic Problems,” J. Comput. Phys. 55 (1), 65-80 (1984).
  4. J. Shen, “Efficient Spectral-Galerkin Method II. Direct Solvers of Second- and Fourth-Order Equations by Using Chebyshev Polynomials,” SIAM J. Sci. Comput. 16 (1), 74-87 (1995).
  5. I. Altas, J. Dym, M. M. Gupta, and R. P. Manohar, “Multigrid Solution of Automatically Generated High-Order Discretizations for the Biharmonic Equation,” SIAM J. Sci. Comput. 19 (5), 1575-1585 (1998).
  6. M.-C. Lai and H.-C. Liu, “Fast Direct Solver for the Biharmonic Equation on a Disk and Its Application to Incompressible Flows,” Appl. Math. Comput. 164 (3), 679-695 (2005).
  7. G. Chen, Z. Li, and 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), 113-133 (2008).
  8. M. Ben-Artzi, J.-P. Croisille, and D. Fishelov, “A Fast Direct Solver for the Biharmonic Problem in a Rectangular Grid,” SIAM J. Sci. Comput. 31 (1), 303-333 (2008).
  9. M. Ben-Artzi, I. Chorev, J.-P. Croisille, and D. Fishelov, “A Compact Difference Scheme for the Biharmonic Equation in Planar Irregular Domains,” SIAM J. Numer. Anal. 47 (4), 3087-3108 (2009).
  10. S. C. Brenner, “An Optimal-Order Nonconforming Multigrid Method for the Biharmonic Equation,” SIAM J. Numer. Anal. 26 (5), 1124-1138 (1989).
  11. A. Mayo and A. Greenbaum, “Fast Parallel Iterative Solution of Poisson’s and the Biharmonic Equations on Irregular Regions,” SIAM J. Sci. Stat. Comput. 13 (1), 101-118 (1992).
  12. M. R. Hanisch, “Multigrid Preconditioning for the Biharmonic Dirichlet Problem,” SIAM J. Numer. Anal. 30 (1), 184-214 (1993).
  13. C. Davini and I. Pitacco, “An Unconstrained Mixed Method for the Biharmonic Problem,” SIAM J. Numer. Anal. 38 (3), 820-836 (2000).
  14. Y. Jiang, B. Wang, and Y. Xu, “A Fast Fourier-Galerkin Method Solving a Boundary Integral Equation for the Biharmonic Equation,” SIAM J. Numer. Anal. 52 (5), 2530-2554 (2014).
  15. M.-C. Lai and J.-M. Tseng, “A Formally Fourth-Order Accurate Compact Scheme for 3D Poisson Equation in Cylindrical Coordinates,” J. Comput. Appl. Math. 201 (1), 175-181 (2007).
  16. V. P. Shapeev and E. V. Vorozhtsov, “Application of the Method of Collocations and Least Residuals to the Solution of the Poisson Equation in Polar Coordinates,” J. Multidiscip. Eng. Sci. Technol. 2 (9), 2553-2562 (2015).
  17. V. P. Shapeev and V. A. Belyaev, “Solving Boundary Value Problems for Partial Differential Equations in Triangular Domains by the Least Squares Collocation Method,” Vychisl. Metody Programm. 19, 96-111 (2018).
  18. V. A. Belyaev and V. P. Shapeev, “The Collocation and Least Squares Method on Adaptive Grids in a Domain with a Curvilinear Boundary,” Vychisl. Tekhnol. 5 (4), 13-21 (2000).
  19. V. P. Shapeev and V. A. Belyaev, “Versions of High Order Accuracy Collocation and Least Residuals Method in the Domain with a Curvilinear Boundary,” Vychisl. Tekhnol. 21 (5), 95-110 (2016).
  20. V. A. Belyaev and V. P. Shapeev, “The Versions of Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Trapezoidal Domains,” Vychisl. Tekhnol. 22 (4), 22-42 (2017).
  21. V. A. Belyaev and V. P. Shapeev, “Versions of the Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Convex Quadrangular Domains,” Model. Anal. Inform. Sist. 24 (5), 629-648 (2017).
  22. V. A. Belyaev and V. P. Shapeev, “Versions of the Collocation and Least Squares Method for Solving Biharmonic Equations in Non-Canonical Domains,” AIP Conf. Proc. 1893 (2017).
    doi 10.1063/1.5007560
  23. V. Shapeev, V. Belyaev, S. Golushko, and S. Idimeshev, “New Possibilities and Applications of the Least Squares Collocation Method,” EPJ Web of Conf. 173 (2018).
    doi 10.1051/epjconf/201817301012
  24. V. A. Belyaev and V. P. Shapeev, “Solving the Dirichlet Problem by the Least Squares Collocation Method in a Domain with a Discrete Boundary,” Vychisl. Tekhnol. 23 (3), 15-30 (2018).
  25. A. G. Sleptsov, “Collocation Grid Solution of Elliptic Boundary Value Problems,” Modelir. Mekhan. 5 (2), 101-126 (1991).
  26. L. G. Semin, A. G. Sleptsov, and V. P. Shapeev, “Method of Collocations-Least Squares for Stokes Equations,” Vychisl. Tekhnol. 1 (2), 90-98 (1996).
  27. V. I. Isaev, V. P. Shapeev, and S. A. Eremin, “An Investigation of the Collocation and the Least Squares Method for Solution of Boundary Value Problems for the Navier-Stokes and Poisson Equations,” Vychisl. Tekhnol. 12 (3), 53-70 (2007).
  28. V. I. Isaev and V. P. Shapeev, “Development of the Collocations and Least Squares Method,” Tr. Inst. Mat. Mekh. UrO RAN 14 (1), 41-60 (2008) [Proc. Steklov Inst. Math. 261 (Suppl. 1), S87-S106 (2008)].
  29. V. I. Isaev and V. P. Shapeev, “High-Accuracy Versions of the Collocations and Least Squares Method for the Numerical Solution of the Navier-Stokes Equations,” Zh. Vychisl. Mat. Mat. Fiz. 50 (10), 1758-1770 (2010) [Comput. Math. Math. Phys. 50 (10), 1670-1681 (2010)].
  30. S. K. Golushko, S. V. Idimeshev, and V. P. Shapeev, “Application of Collocations and Least Residuals Method to Problems of the Isotropic Plates Theory,” Vychisl. Tekhnol. 18 (6), 31-43 (2013).
  31. S. K. Golushko, S. V. Idimeshev, and V. P. Shapeev, “Development and Application of Collocations and Least Residuals Method to the Solution of Problems in Mechanics of Anisotropic Laminated Plates,” Vychisl. Tekhnol. 19 (5), 24-36 (2014).
  32. V. P. Shapeev and E. V. Vorozhtsov, “CAS Application to the Construction of the Collocations and Least Residuals Method for the Solution of the Bürgers and Korteweg-de Vries-Bürgers Equations,” in Lecture Notes in Computer Science (Springer, Cham, 2014), Vol. 8660, pp. 432-446.
  33. E. V. Vorozhtsov and V. P. Shapeev, “On Combining the Techniques for Convergence Acceleration of Iteration Processes During the Numerical Solution of Navier-Stokes Equations,” Vychisl. Metody Programm. 18, 80-102 (2017).
  34. Y. Saad, Numerical Methods for Large Eigenvalue Problems (Manchester Univ. Press, Manchester, 1991).
  35. R. P. Fedorenko, “The Speed of Convergence of One Iterative Process,” Zh. Vychisl. Mat. Mat. Fiz. 4 (3), 559-564 (1964) [USSR Comput. Math. Math. Phys. 4 (3), 227-235 (1964)].
  36. B.-N. Jiang and L. A. Povinelli, “Least-Squares Finite Element Method for Fluid Dynamics,” Comput. Methods Appl. Mech. Eng. 81 (1), 13-37 (1990).
  37. C. L. Chang and B.-N. Jiang, “An Error Analysis of Least-Squares Finite Element Method of Velocity-Pressure-Vorticity Formulation for Stokes Problem,” Comput. Methods Appl. Mech. Eng. 84 (3), 247-255 (1990).
  38. P. B. Bochev and M. D. Gunzburger, Least-Squares Finite Element Methods (Springer, New York, 2009).
  39. H. Chen, C. Min, and F. Gibou, “A Supra-Convergent Finite Difference Scheme for the Poisson and Heat Equations on Irregular Domains and Non-Graded Adaptive Cartesian Grids,” J. Sci. Comput. 31 (1-2), 19-60 (2007).
  40. X.-D. Liu, R. P. Fedkiw, and M. Kang, “A Boundary Condition Capturing Method for Poisson’s Equation on Irregular Domains,” J. Comput. Phys. 160 (1), 151-178 (2000).
  41. F. Gibou and R. Fedkiw, “A Fourth Order Accurate Discretization for the Laplace and Heat Equations on Arbitrary Domains, with Applications to the Stefan Problem,” J. Comput. Phys. 202 (2), 577-601 (2005).
  42. A. Friedman, Partial Differential Equations of Parabolic Type (Prentice-Hall, Englewood Cliffs, 1964; Mir, Moscow, 1968).
  43. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1977, Dover, New York, 2011).
  44. V. B. Barakhnin and V. P. Shapeev, Introduction to Numerical Analysis (Lan’, Moscow, 2005) [in Russian].
  45. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959; Fizmatgiz, Moscow, 1963).