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 W1,∞(L 2 ) is proved. An L∞(H1 )-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 W1,∞(L 2 )- error estimate seems not to be present in the existing literature of the Crank-Nicolson finite element schemes for parabolic equations.
We consider a family of conforming finite element schemes with piecewise polynomial space of degree k in space for solving the wave equation, as a model for second order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is h k + τ 2 in the discrete norms of L∞(0, T ; H1 (Ω)) and W1,∞(0, T ; L 2 (Ω)), where h and τ are the mesh size of the spatial and temporal discretization, respectively. These error estimates are useful since they allow us to get second order time accurate approximations for not only the exact solution of the wave equation but also for its first derivatives (both spatial and temporal). Even though the proof presented in this note is in some sense standard, the stated error estimates seem not to be present in the existing literature on the finite element methods which use the Newmark method for the wave equation (or general second order hyperbolic equations).