Abstract
The boundary value problem for the ordinary differential equation of reaction-diffusion on the interval [−1, 1] is examined. The highest derivative in this equation appears with a small parameter ɛ2 (ɛ ∈ (0, 1]). As the small parameter approaches zero, boundary layers arise in the neighborhood of the interval endpoints. An algorithm for the construction of a posteriori adaptive piecewise uniform grids is proposed. In the adaptation process, the edges of the boundary layers are located more accurately and the grid on the boundary layers is repeatedly refined. To find an approximate solution, the finite element method is used. The sequence of grids constructed by the algorithm is shown to converge “conditionally ɛ-uniformly” to some limit partition for which the error estimate O(N −2ln3 N) is proved. The main results are obtained under the assumption that ɛ ≪ N −1, where N is number of grid nodes; thus, conditional ɛ-uniform convergence is dealt with. The proofs use the Galerkin projector and its property to be quasi-optimal.
Similar content being viewed by others
References
V. D. Liseikin, Grid Generation Methods (Springer, Berlin, 1999).
G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations (Ural. Otd. Ross. Akad. Nauk, Yekaterinburg, 1992) [in Russian].
G. I. Shishkin, “Grid Approximation of a Parabolic Convection-Diffusion Equation on a Priori Adapted Grids: ɛ-Uniformly Convergent Schemes,” Zh. Vychisl. Mat. Mat. Fiz. 48, 1014–1033 (2008) [Comput. Math. Math. Phys. 48, 956–974 (2008)].
G. I. Shishkin, “Grid Approximation of Singularly Perturbed Boundary Value Problems on Locally Condensing Grids: Convection-Diffusion Equations,” Zh. Vychisl. Mat. Mat. Fiz. 40, 714–725 (2000) [Comput. Math. Math. Phys. 40, 680–691 (2000)].
G. I. Shishkin, “A posteriori Adaptive Grids (with Respect to the Solution Gradient) in Approximation of Singularly Perturbed Convection-Diffusion Equations,” Vychisl. Tekhnol. 6(1), 72–87 (2001).
G. I. Shishkin, “Difference Approximation of Singularly Perturbed Boundary Value Problems on Locally Refined Grids: Reaction-Diffusion Equations,” Mat. Model. 11(12), 87–104 (1999).
A. B. Vasil’eva and V. F. Butuzov, Asymptotic Expansions of Solutions to Singularly Perturbed Equations (Nauka, Moscow, 1973) [in Russian].
I. A. Blatov and V. V. Strygin, Elements of Spline Theory and the Finite Element Method for Boundary Layer Problems (Voronezh. Gos. Univ., Voronezh, 1997) [in Russian].
S. Demko, “Inverses of Band Matrices and Local Convergence of Spline Projection,” SIAM J. Numer. Anal. 14, 616–619 (1977).
A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations (Nauka, Moscow, 1978; Birkhäuser, Basel, 1989).
I. A. Blatov, “On the Galerkin Finite Element Method for Singularly Perturbed Parabolic Initial Value Problems,” Differ. Uravn. 32, 661–669 (1996).
N. N. Kalitkin, Numerical Methods (Nauka, Moscow, 1977) [in Russian].
G. I. Marchuk, Methods of Numerical Mathematics (Springer-Verlag, New York, 1982; Nauka, Moscow, 1989).
N. S. Bakhvalov, “Optimization of Methods for Boundary Value Problems with Boundary Layers,” Zh. Vychisl. Mat. Mat. Fiz. 9, 841–859 (1969).
V. I. Lebedev, Functional Analysis and Computational Mathematics (Fizmatlit, Moscow, 2000) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © I.A. Blatov, N.V. Dobrobog, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 9, pp. 1550–1568.
Rights and permissions
About this article
Cite this article
Blatov, I.A., Dobrobog, N.V. Conditional ɛ-uniform convergence of adaptive algorithms in the finite element method as applied to singularly perturbed problems. Comput. Math. and Math. Phys. 50, 1476–1493 (2010). https://doi.org/10.1134/S0965542510090022
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542510090022