Geodesic
Numerical solution of the Poisson equation in polar coordinates by the method of collocations and least residuals
Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 5, pp. 648-664

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A version of the method of collocations and least residuals is proposed for the numerical solution of the Poisson equation in polar coordinates on non-uniform grids. By introducing general curvilinear coordinates the original Poisson equation is reduced to the Beltrami equation. A uniform grid is used in curvilinear coordinates. The grid non-uniformity in the plane of the original polar coordinates is ensured with the aid of functions which control the grid stretching and entering the formulas of the passage from polar coordinates to the curvilinear ones. The method was verified on two test problems having exact analytic solutions. The examples of numerical computations show that if the radial coordinate axis origin lies outside the computational region, the proposed method has the second order of accuracy. If the computational region contains the singularity, the application of a non-uniform grid along the radial coordinate enables an increase in the numerical solution accuracy by factors from 1.7 to 5 in comparison with the uniform grid case at the same number of grid nodes.
Mots-clés : Poisson equation
Keywords: polar coordinates, the method of collocations and least residuals. 
@article{MAIS_2015_22_5_a3,
     author = {E. V. Vorozhtsov and V. P. Shapeev},
     title = {Numerical solution of the {Poisson} equation in polar coordinates by the method of collocations and least residuals},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {648--664},
     publisher = {mathdoc},
     volume = {22},
     number = {5},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2015_22_5_a3/}
}
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E. V. Vorozhtsov; V. P. Shapeev. Numerical solution of the Poisson equation in polar coordinates by the method of collocations and least residuals. Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 5, pp. 648-664. http://geodesic.mathdoc.fr/item/MAIS_2015_22_5_a3/