Abstract
A parabolic problem in a separable Hilbert space is solved approximately by the projective-difference method. The problem is discretized with respect to space by the Galerkin method and with respect to time by the modified Cranck--Nicolson scheme. In this paper, we establish efficient (in time and space) strong-norm error estimates for approximate solutions. These estimates allow us to obtain the rate of convergence with respect to time of the error to zero up to the second order. In addition, the error estimates take into account the approximation properties of projective subspaces, which is illustrated for subspaces of finite element type.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
V. V. Smagin, “Strong-norm error estimates for the projective-difference method for approximately solving abstract parabolic equations, ” Mat. Zametki [Math. Notes], 62 (1997), no. 6, 898–909.
V. V. Smagin, “Mean-square error estimates for the projective-difference method for parabolic equations, ” Zh. Vychisl. Mat. i Mat. Fiz. [Comput. Math. and Math. Phys.], 40 (2000), no. 6, 908–919.
V. V. Smagin, “Energy error estimates for the projective-difference method with Crank-Nicolson scheme for parabolic equations, ” Sibirsk. Mat. Zh. [Siberian Math. J.], 42 (2001), no. 3, 670–682.
V. V. Smagin, “Projective-difference methods for approximately solving parabolic equations with non-symmetric operators, ” Differentsial'nye Uravneniya [Differential Equations], 37 (2001), no. 1, 115–123.
I. D. Turetaev, “Sharp error gradient estimates for the projective-difference schemes of parabolic equations in an arbitrary domain, ” Zh. Vychisl. Mat. i Mat. Fiz. [U.S.S.R. Comput. Math. and Math. Phys.], 26 (1986), no. 11, 1748–1751.
J.-L. Lions and E. Magenes, Problemes aux limites non homogenes et applications, Dunod, Paris, 1968; Russian translation: Mir, Moscow, 1971.
V. V. Smagin, “On the solvability of an abstract parabolic equation with operator whose domain of definition depends on time, ” Differentsial'nye Uravneniya [Differential Equations], 32 (1996), no. 5, 711–712.
V. V. Smagin, “On the smooth solvability of variational problems of parabolic type, ” in: Proc. Math. Dept. (New Series) [in Russian], vol. 3, Izd. Voronezh Univ., Voronezh, 1998, pp. 67–72.
V. V. Smagin, “Estimates of the rate of convergence of the projective and projective-difference methods for weakly solvable parabolic equations, ” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 188 (1997), no. 3, 143–160.
G. M. Vainikko and P. E. Oya, “On the convergence and the rate of convergence of the Galerkin method for abstract evolution equations, ” Differentsial'nye Uravneniya [Differential Equations], 11 (1975), no. 7, 1269–1277.
G. I. Marchuk, Methods of Computational Mathematics [in Russian] Nauka, Moscow, 1989.
V. V. Smagin, “Error estimates for semidiscrete Galerkin approximations for parabolic equations with boundary condition of Neumann type, ” Izv. Vyssh. Uchebn. Zaved. Mat. [Russian Math. (Iz. VUZ)], 3(406)(1996), 50–57.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Smagin, V.V. Strong-Norm Error Estimates for the Projective-Difference Method for Parabolic Equations with Modified Crank--Nicolson Scheme. Mathematical Notes 74, 864–873 (2003). https://doi.org/10.1023/B:MATN.0000009023.01722.00
Issue Date:
DOI: https://doi.org/10.1023/B:MATN.0000009023.01722.00