Geodesic
Solving elliptic equations in polygonal domains by the least squares collocation method
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 12 (2019) no. 3, pp. 140-152

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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.
Keywords: least squares collocation method, polygonal multiply connected domain, nonhomogeneous biharmonic equation, stress-strain state.
Mots-clés : Poisson's equation 
@article{VYURU_2019_12_3_a11,
     author = {V. P. Shapeev and L. S. Bryndin and V. A. Belyaev},
     title = {Solving elliptic equations in polygonal domains by the least squares collocation method},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {140--152},
     publisher = {mathdoc},
     volume = {12},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2019_12_3_a11/}
}
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V. P. Shapeev; L. S. Bryndin; V. A. Belyaev. Solving elliptic equations in polygonal domains by the least squares collocation method. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 12 (2019) no. 3, pp. 140-152. http://geodesic.mathdoc.fr/item/VYURU_2019_12_3_a11/