Solving boundary value problems for partial differential equations in triangular domains by the least squares collocation method

Authors

• V.P. Shapeev
• V.A. Belyaev

Keywords:

least squares collocation method
boundary value problem
triangular domain
higher order approximation
Poisson’s equation
biharmonic equation

Abstract

A high-accuracy new version of the least squares collocation method (LSC) is proposed and implemented for the numerical solution of boundary value problems for PDEs in triangular domains. The implementation of this approach and numerical experiments are performed using the examples of the biharmonic and Poisson equations. The solution of the biharmonic equation with high accuracy is used to simulate the stress-strain state of an isotropic triangular plate under the action of a transverse load. The differential problems are projected onto the space of fourth-degree polynomials by the LSC method. The boundary conditions for the approximate solution are given exactly on the boundary of the computational domain, which allows us theoretically and indefinitely to increase the order of accuracy of the LSC. The new version of the LSC utilizes a regular grid with rectangular cells inside the domain of the solution. It is relatively easy to use a «single» layer of irregular cells that are cut off by the boundary from the rectangular cells of the initial regular grid. Triangular irregular boundary cells are joint to the adjacent quadrangular or pentagonal cells. Thus, a separate piece of the analytical solution is constructed in combined cells. The collocation and matching points situated outside the domain are used to approximate the differential equations in the boundary cells crossed by the boundary. These two methods allows us to reduce significantly the condition number of the system of linear algebraic equations in the approximate compared to the case when the triangular cells are used as independent ones for constructing an approximate solution of the problem and when the extraboundary part of the boundary cells is not used. The advantage of the proposed approach is shown in comparison with the approach using the mapping of the triangular domain onto the rectangular one. It is also shown that the approximate solution converges with a high order and is coincident with the analytical solution of the test problems with a high accuracy.

2018-03-15

Section

Section 1. Numerical methods and applications

References

1. A. G. Sleptsov, “Collocation Grid Solution of Elliptic Boundary Value Problems,” Modelir. Mekhan. 5 (2), 101-126 (1991).
2. 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).
3. 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).
4. 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).
5. 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
6. 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).
7. 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).
8. 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).
9. 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).
10. 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).
11. 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).
12. 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).
13. 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)].
14. 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)].
15. V. Shapeev, “Collocation and Least Residuals Method and Its Applications,” EPJ Web of Conferences 108 (2016).
doi 10.1051/epjconf/201610801009
16. 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)].
17. Y. Saad, Numerical Methods for Large Eigenvalue Problems (Manchester Univ. Press, Manchester, 1991).
18. A. G. Sleptsov, “On Convergence Acceleration of Linear Iterations, II,” Modelir. Mekhan. 3 (5), 118-125 (1989).
19. 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).
20. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959; Fizmatgiz, Moscow, 1963).