A~cascadic multigrid algorithm in the finite element method for the three-dimensional Dirichlet problem in a~curvilinear boundary domain

L. V. Gilyova ; V. V. Shaidurov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 5 (2002) no. 2, pp. 127-147.

Résumé

The paper deals with the three-dimensional Dirichlet problem for a second order elliptic equation in a domain with a smooth curvilinear boundary. To construct a finite element scheme, the embedded subspaces of basic functions are used without strict embeddedness of a sequence of spatial triangulations. It is proved that the discretization error is of the same order as in the case of the standard piecewise linear elements on a polyhedron. For solving the obtained system of linear algebraic equations on a sequence of grids, the cascadic organization of two iterative processes is applied providing a simple version of a multigrid method without any preconditioning or restriction to coarser grids. The cascadic algorithm starts on the coarsest grid where the grid problem is directly solved. On finer grids, approximate solutions are obtained by an iterative process, where interpolation of an approximate solution from the previous coarser grid is taken as an initial guess. It is proved that the convergence rate of this algorithm does not depend on the number of unknown values as well as on the number of grids.

Bibliographie

[1] Deuflhard P., Cascadic conjugate gradient methods for elliptic partial differential equations. I. Algorithm and numerical results, Technical Report SC 93-23, Konrad-Zuse-Zentrum, Berlin, 1993 | MR

[2] Deuflhard P., “Cascadic conjugate gradient methods for elliptic partial differential equations. Algorithm and numerical results”, Proceedings of 7-th International Conference on Domain Decomposition Methods 1993, Contemporary Math., 180, AMS, Providence, 1994, 29–42 | MR

[3] Shaidurov V. V., Some estimates of the rate of convergence for the cascadic conjugate-gradient method, Preprint Otto-von-Guericke-Universitat no 4, Magdeburg, 1994 | MR

[4] Shaidurov V. V., “Some estimates of the rate of convergence for the cascadic conjugate-gradient method”, Computers Math. Applic., 31:4–5 (1996), 161–171 | DOI | MR | Zbl

[5] Shaidurov V. V., “The convergence of the cascadic conjugate-gradient method under a deficient regularity”, Problems and Methods in Mathematical Physics, Teubner, Stuttgart, 1994, 185–194 | MR | Zbl

[6] Bornemann F. A., On the convergence of cascadic iterations for elliptic problems, Preprint; SC 94-8, Konrad-Zuse-Zentrum, Berlin, 1994

[7] Shaidurov V. V., Mnogosetochnye metody konechnykh elementov, Nauka, M., 1989 | MR

[8] Shaidurov V. V., “Cascadic algorithm with nested subspaces in domains with curvilinear boundary”, Advanced Mathematics: Computations and Applications, eds. A. S. Alekseev and N. S. Bakhvalov, NCC Publisher, Novosibirsk, 1995, 588–595 | MR

[9] Gileva L. V., “Kaskadnyi mnogosetochnyi algoritm v metode konechnykh elementov dlya trekhmernoi zadachi Dirikhle”, Sib. zhurn. vychisl. matematiki / RAN. Sib. otd-nie. — Novosibirsk, 1:3 (1998), 217–226 | MR

[10] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[11] Gileva L. V., Shaidurov V. V., “Obosnovanie asimptoticheskoi ustoichivosti algoritma triangulyatsii trekhmernoi oblasti”, Sib. zhurn. vychisl. matematiki / RAN. Sib. otd-nie. — Novosibirsk, 3:2 (2000), 123–136

[12] Syarle F., Metod konechnykh elementov dlya ellipticheskikh zadach, Mir, M., 1980 | MR

[13] Korn G., Korn T., Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov, Nauka, M., 1984 | MR

[14] Voevodin V. V., Kuznetsov V. V., Matritsy i vychisleniya, Nauka, M., 1984 | MR | Zbl

[15] Bey J., “Tetrahedral grid refinement”, Computing, 55 (1995), 355–378 | DOI | MR | Zbl