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Article information
2006 , Volume 11, Special issue, p.84-91
Shapeev V.P., Shapeev A.V.
Solutions of the elliptic problems with singularities using finite difference schemes with high order of approximation
Numerical solutions of boundary-value problems with singularities are examined. We consider finite difference schemes with high order of approximation for two specific different elliptic equations, namely Poisson equation and an equation with a small parameter at the highest derivative. For the boundary-value problem for the Poisson equation the singularity arises due to a discontinuity of the second-order derivative at the corner point of the domain boundary. For the equation with a small parameter at the highest derivative an inner thin boundary layer is a manifestation of the singularity. Behavior of the solution and convergence of the numerical solution on a sequence of grids has been analyzed using exact solutions. In both cases the convergence order of the high-order methods was higher compared to the low-order methods. A derivation of the finite difference schemes and analysis of the solutions were carried out with the help of computer algebra software package Mathematica.
[full text] Author(s):
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: 7128-5536Shapeev A.V.
Office: Institute of Theoretical and Applied Mechanics of SB RAS
Address: Russia, Novosibirsk, Novosibirsk, Institutskaya Str., 4/1
Bibliography link:
Shapeev V.P., Shapeev A.V. Solutions of the elliptic problems with singularities using finite difference schemes with high order of approximation // Computational technologies. 2006. V. 11. Special issue, devoted to N.N. Yanenko's 85-th anniversary, part 2. P. 84-91
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