Geodesic
Strong-Norm Error Estimates for the Projective-Difference Method for Parabolic Equations with Modified Crank--Nicolson Scheme
Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 913-923

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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. 
@article{MZM_2003_74_6_a11,
     author = {V. V. Smagin},
     title = {Strong-Norm {Error} {Estimates} for the {Projective-Difference} {Method} for {Parabolic} {Equations} with {Modified} {Crank--Nicolson} {Scheme}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {913--923},
     publisher = {mathdoc},
     volume = {74},
     number = {6},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a11/}
}
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V. V. Smagin. Strong-Norm Error Estimates for the Projective-Difference Method for Parabolic Equations with Modified Crank--Nicolson Scheme. Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 913-923. http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a11/