Volume 12, no. 3Pages 140 - 152
Solving Elliptic Equations in Polygonal Domains by the Least Squares Collocation MethodV.P. Shapeev, L.S. Bryndin, V.A. Belyaev
The paper considers a new version of the least squares collocation (LSC) method for the numerical solution of boundary value problems for elliptic equations in polygonal domains, in particular, in multiply connected domains. The implementation of this approach and numerical experiments are performed on the examples of the inhomogeneous biharmonic and Poisson equations. As an application, we use the nonhomogeneous biharmonic equation to simulate the stress-strain state of isotropic elastic thin plate of polygonal form under the action of transverse load. The new version of the LSC method is based on the triangulation of the original domain. Therefore, this approach is fundamentally different from the previous more complicated versions of the LSC method proposed to solve the boundary value problems for partial derivative equations in irregular domains. We make the numerical experiments on the convergence of the approximate solution to various problems on a sequence of grids. The experiments show that the solution to the problems converges with high order and, in the case of the known analytical solution, matches with high accuracy with the analytical solution to the test problems. Full text
- least squares collocation method; polygonal multiply connected domain; Poisson's equation; nonhomogeneous biharmonic equation; stress-strain state.
- 1. Guo Chen, Zhilin Li, Ping Lin. A Fast Finite Difference Method for Biharmonic Equations on Irregular Domains and its Application to an Incompressible Stokes Flow. Advances in Computational Mathematics, 2008, vol. 29, no. 2, pp. 113-133. DOI: 10.1007/s10444-007-9043-6
2. Shapeev V.P., Belyaev V.A. [Solving the Biharmonic Equation with High Order Accuracy in Irregular Domains by the Least Squares Collocation Method]. Numerical Methods and Programming, 2018, vol. 19, no. 4, pp. 340-355. (in Russian)
3. Timoshenko S.P., Woinowsky-Krieger S. Theory of Plates and Shells. N.-Y., Toronto, London, McGraw-Hill, 1959.
4. Shapeev V.P., Belyaev V.A. Solving Boundary Value Problems for Partial Differential Equations in Triangular Domains by the Least Squares Collocation Method. Numerical Methods and Programming, 2018, vol. 19, no. 1, pp. 96-111. (in Russian)
5. Sorokin, S.B. Preconditioning in the Numerical Solution to a Dirichlet Problem for the Biharmonic Equation. Numerical Analysis and Applications, 2011, vol. 4, no. 2, pp. 167-174. DOI: 10.1134/S1995423911020078
6. Shapeev V.P., Belyaev V.A. Versions of High Order Accuracy Collocation and Least Residuals Method in the Domain with a Curvilinear Boundary. Computational Technologies, 2016, vol. 21, no. 5, pp. 95-110. (in Russian)
7. Belyaev V.A., Shapeev V.P. The Versions of Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Trapezoidal Domains. Computational Technologies, 2017, vol. 22, no. 4, pp. 22-42. (in Russian)
8. Belyaev V.A., Shapeev V.P. Versions of the Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Convex Quadrangular Domains. Modeling and Analysis of Information Systems, 2017, vol. 24, no. 5, pp. 629-648. DOI: 10.18255/1818-1015-2017-5-629-648 (in Russian)
9. Belyaev V.A., Shapeev V.P. Solving the Dirichlet Problem for the Poisson Equation by the Least Squares Collocation Method with Given Discrete Boundary Domain. Computational Technologies, 2018, vol. 23, no. 3, pp. 15-30. DOI: 10.25743/ICT.2018.3.15956 (in Russian)
10. Fedorenko R.P. The Speed of Convergence of One Iterative Process. Computational Mathematics and Mathematical Physics, 1964, vol. 4, no. 3, pp. 227-235. DOI: 10.1016/0041-5553(64)90253-8
11. Saad, Y. Numerical Methods for Large Eigenvalue Problems. Manchester, Manchester University Press, 1991.
12. Golushko S.K., Idimeshev S.V., Shapeev V.P. Application of Collocations and Least Residuals Method to Problems of the Isotropic Plates Theory. Computational Technologies, 2013, vol. 18, no. 6, pp. 31-43. (in Russian)