Abstract
We solve an abstract parabolic problem in a separable Hilbert space, using the projection-difference method. The spatial discretization is carried out by the Galerkin method and the time discretization, by the Crank–Nicolson scheme. On assuming weak solvability of the exact problem, we establish effective energy estimates for the error of approximate solutions. These estimates enable us to obtain the rate of convergence of approximate solutions to the exact solution in time up to the second order. Moreover, these estimates involve the approximation properties of the projection subspaces, which is illustrated by subspaces of the finite element type.
Similar content being viewed by others
References
Lions J.-L. and Magenes E., Inhomogeneous Boundary Value Problems and Their Applications [Russian translation], Mir, Moscow (1971).
Smagin V. V., “On solvability of an abstract parabolic equation with an operator having time dependent domain,” Differentsial'nye Uravneniya, 32, No. 5, 711–712 (1996).
Smagin V. V., “Coercive error estimates for the projection-difference method for an abstract parabolic equation with an operator having time dependent domain,” Sibirsk. Mat. Zh., 37, No. 2, 406–418 (1996).
Zlotnik A. A. and Turetaev I. D., “On sharp error estimates and optimality of two-level economical methods for solving the heat equation,” Dokl. Akad. Nauk SSSR, 272, No. 6, 1306–1311 (1983).
Zlotnik A. A. and Turetaev I. D., “Sharp error estimates of some two-level methods for solving the three-dimensional heat equation,” Mat. Sb., 128, No. 4, 530–544 (1985).
Turetaev I. D., “Exact gradient error estimates of projection-difference schemes for parabolic equations in an arbitrary domain,” Zh. Vychisl. Matematiki i Mat. Fiziki, 26, No. 11, 1748–1751 (1986).
Smagin V. V., “Mean square error estimates of the projection-difference method for parabolic equations,” Zh. Vychisl. Matematiki i Mat. Fiziki, 40, No. 6, 908–919 (2000).
Smagin V. V., “Estimates in strong error norms for the projection-difference method for an approximate solution to an abstract parabolic equation,” Mat. Zametki, 62, No. 6, 898–909 (1997).
Smagin V. V., “Estimates for the convergence rate of the projection and projection-difference methods for weakly solvable parabolic equations,” Mat. Sb., 188, No. 3, 143–160 (1997).
Va\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{i} \)nikko G. M. and Oya P. È., “On convergence and the rate of convergence of the Galerkin method for abstract evolution equations,” Differentsial'nye Uravneniya, 11, No. 7, 1269–1277 (1975).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Smagin, V.V. Energy Error Estimates for the Projection-Difference Method with the Crank–Nicolson Scheme for Parabolic Equations. Siberian Mathematical Journal 42, 568–578 (2001). https://doi.org/10.1023/A:1010483428596
Issue Date:
DOI: https://doi.org/10.1023/A:1010483428596