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Article information
2017 , Volume 22, ¹ 4, p.22-42
Belyaev V.A., Shapeev V.P.
The versions of collocation and least residuals method for solving problems of mathematical physics in the trapezoidal domains
The paper addresses new versions of collocations and least residuals (CLR) method are proposed and implemented for the numerical solution of boundary value problems for PDE in trapezoidal domains. Their implementation and numerical experiments are performed on the examples of the biharmonic and Poisson equations. The solution of the biharmonic equation is used for simulation of the stress-strain state of an isotropic plate under the action of the transverse load. Three different variants of the construction of the computational grid in the trapezoidal cells are implemented in the present study. First of all, the original domain is covered by a regular grid with trapezoidal cells. In the second variant of the method we apply the idea of using parts of the cells of a regular grid outside the domain, which are cut off by the boundary for the constructing the CLR methods. It is assumed that the solution has no singularities at the boundary and in a certain small neighborhood of it. The differential equation for the problem is true not only in the computational domain, but also in a small neighborhood of its boundary. Then we use the idea of joining the “small” irregular cells to the adjacent cells in order to reduce the condition number for the global system of linear algebraic equations. It is shown that the approximate solutions, obtained by CLR, converge with high order of accuracy, thus accurately match the analytical solutions of test problems.
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Keywords: collocations and least residuals method, boundary value problem, trapezoidal domain, high order approximation, Poissons equation, biharmonic equation
Author(s):
Belyaev Vasily Alexeyevich
Position: assistant
Office: Institute of Theoretical and Applied Mechanics, Novosibirsk State University
Address: 630090, Russia, Novosibirsk, 2 Pirogova Str.
E-mail: belyaevasily@mail.ru
Shapeev Vasily Pavlovich
Dr. , Professor
Position: General Scientist
Office: Institute of Theoretical and Applied Mechanics of SB RAS, Novosibirsk State University
Address: 630090, Russia, Novosibirsk, Institutskaya Str., 4/1
Phone Office: (383) 330 27 13
E-mail: vshapeev@ngs.ru
SPIN-code: 7128-5536 References:
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Bibliography link:
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. V. 22. ¹ 4. P. 22-42
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