UDK 519.632 Vestnik SibGAU. 2014, No. 3(55), P. 73–77
V. A. Kireev
Siberian Federal University 79, Svobodnyi prosp., Krasnoyarsk, 660041, Russian Federation Е-mail: kireevvit@gmail.com
In this paper the collocation method with Hermite bicubic basis functions is considered, applied to the first bound-ary value problem for an elliptic equation in a domain with a curved boundary. The collocation method has some ad-vantages compared with Galerkin finite element method: no need to compute the integrals for the determination of the coefficients of the stiffness matrix. Hermite bicubic functions belongs to the class C1. The consistent with the boundary mesh is constructed for solving the problem. The grid is consistent with the border so that two nodes in irregular cells were lying on a curvilinear boundary. This approach allows to reduce the total number of basis functions in the do-main. As an internal collocation nodes the points of the set of Gauss are taken. The collocation points are distributed evenly on a curvilinear boundary. Under such arrangement, the total number of collocation points equal the total num-ber of basis functions with the given boundary conditions. The problem is reduced to solution of a linear system Au=f where A is a square matrix. The results of numerical experiments of solving Poisson equation with different right sides show that the algorithm of solution has the convergence of high order.
collocation method, Hermite bicubic basis, second order elliptic equation, curved boundary.

1. Russell R. D., Shampine L. F. A collocation method for boundary value problems. Numer. Math, 1972,
vol. 19, p. 1–28.

2. Leyk Z. A Co-collocation-like method for elliptic equations on rectangular regions. Australian Math. Soc. B. Appl. Math, 1997, vol. 38, p. 368–387.

3. Sleptsov A. G. [Collocation-grid solution of an elliptic boundary value problems]. Modelirovanie v mekhanike. 1987, vol. 5(22), no. 2, p. 101–126. (In Russ.).

4. Isaev V. I., Shapeev V. P., Idimeshev S. V. [Variants of the collocation and least squares method of increased accuracy for the numerical solution of the Poisson equation]. Vychislitel'nye tekhnologii. 2011, vol. 16, no. 1, p. 85–93. (In Russ.).

5. Belyaev V. V., Shapeev V. P. [Collocation and least squares method on adaptive grids in the areas with curvilinear boundary]. Vychislitel'nye tekhnologii. 2000, vol. 5, no. 4, p. 13–21. (In Russ.).

6. Houstis E. N., Mitchell W. F., Rice J. R. Collocation Software for Second-Order Elliptic Partial Differential Equations. ACM Transactions on Mathematical Software, 1985, vol. 11, no. 4, p. 379–412.

7. Gileva L., Shaydurov V., Dobronets B. The triangular Hermite finite element complementing the Bogner-Fox-Schmit rectangle. Applied Mathematics. 2013, vol. 5, no. 12A, p. 50–56.

8. Zav'yalov. Yu. S., Kvasov B. I., Miroshnichenko V. L. Metody splayn-funktsiy [Spline-function methods]. Moscow, Nauka Publ., 1980, 352 p.

9. Strang G., Fix. G. J. An analysis of the finite element method. Englewood Cliffs, N. J., Prentice-Hall, Inc., 1973.

10. Dobronets B. S. Combined bicubic Hermite finite element method. The First China-Russia Conference
on Numerical Algebra with Applications in Radiative Hydrodynamics.Beijing, China, October 16–18, 2012,
p. 19.

11. Mateescu G., Ribbens C. J., Watson L. T. A Domain Decomposition Preconditioner for Hermite Collocation Problems. Computer Science, Virginia Tech, Technical Report TR-02-02, 2002.

12. Prenter P. M., Russell. R. D. Orthogonal collocation for elliptic partial differential equations. SIAM J. Numer. Anal., 1976, vol. 13, no. 6, p. 923–939.

13. Bialecki B., Xiao-Chuan C. H1-norm error bounds for piecewise Hermite bicubic orthogonal spline collocation schemes for elliptic boundary value problems. SIAM J. Numer. Anal., 1994, vol. 31, no. 4, p. 1128–1146.

Kireev Vitaliy Aleksandrovich – postgraduate student, Institute of Mathematics and fundamental informatics, Siberian Federal University. E-mail: kireevvit@gmail.com.