A.A. Sobolev*, M.R. Timerbaev**
Kazan Federal University, Kazan, 420008 Russia
E-mail: *andreyasob@yandex.ru, **marat.timerbayev@sofoil.com
Received April 12, 2017
Abstract
The paper deals with the construction of high-order accuracy finite element schemes for the fourth-order ordinary differential equation with degenerate coefficients on the boundary. The method for solving the problem is based on both multiplicative and additive-multiplicative separation of singularities. For the given class of smoothness of the right-hand sides, the optimal convergence rate has been proved.
Keywords: two-point boundary value problem, finite element schemes, weight function space, multiplicative and additive-multiplicative decomposition of singularity
References
1. Tayupov Sh.I., Timerbaev M.R. Finite element schemes of a high accuracy order for two-pointed heterogenous boundary-value problem with designation. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2006, vol. 148, no. 4, pp. 63–75. (In Russian)
2. Sobolev A.A., Timerbaev M.R. On finite element method of high-order accuracy for two-point degenerated Dirichlet problem of 4th order. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2010, no. 1, pp. 235–244. (In Russian)
3. Timerbaev M.R. Multiplicative extraction of singularities in FEM solvers for degenerate elliptic equations. Differ. Equations, 2000, vol. 36, no. 7, pp. 1086–1093. doi: 10.1007/BF02754511.
4. Kudryavtsev L.D. On equivalent norms in weighted spaces. Tr. MIAN im. Steklova, 1984, vol. 170, pt. 10, pp. 161–190. (In Russian)
5. Nikol'skii S.M. Approximation of Functions of Several Variables and Imbedding Theorems. Moscow, Nauka, 1977. 456 p. (In Russian)
6. Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Elsevier Sci. Publ., 1978. 500 p.
7. Timerbayev M.R. Weighted estimates of solution of the Dirichlet problem with anisotropic degeneration on a part of boundary. Russ. Math., 2003, vol. 47, no. 1, pp. 58–71.
8. Timerbaev M.R. On FEM schemes for two-point boundary-value Dirichlet problems of the fourth order with weak degeneration. Issled. Prikl. Mat. Inf., 2004, no. 25, pp. 78–85. (In Russian)
9. Ciarlet P. The Finite Element Method for Elliptic Problems. North Holland, 1978. 529 p.
For citation: Sobolev A.A., Timerbaev M.R. High-order accuracy approximation forthe two-point boundary value problem of the fourth order with degenerate coefficients. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 4, pp. 493–508. (In Russian)
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