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parabolic equation; finite element method; Crank-Nicolson method; new error estimate
Finite element methods with piecewise polynomial spaces in space for solving the nonstationary heat equation, as a model for parabolic equations are considered. The discretization in time is performed using the Crank-Nicolson method. A new a priori estimate is proved. Thanks to this new a priori estimate, a new error estimate in the discrete norm of $\mathcal {W}^{1,\infty }(\mathcal {L}^2)$ is proved. An $\mathcal {L}^\infty (\mathcal {H}^1)$-error estimate is also shown. These error estimates are useful since they allow us to get second order time accurate approximations for not only the exact solution of the heat equation but also for its first derivatives (both spatial and temporal). Even the proof presented in this note is in some sense standard but the stated $\mathcal {W}^{1,\infty }(\mathcal {L}^2)$-error estimate seems not to be present in the existing literature of the Crank-Nicolson finite element schemes for parabolic equations.
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