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.
Simone de Beauvoir’s Th e Second Sex was translated into Czech in 1966, the fi rst translation of the book to be published in a socialist state. It was, like many other translations during this period, a compilation of selections and was edited by the phenomenologist Jan Patočka who, in his postscript, presented the work primarily within its philosophical context. Th e book, which was published in three editions within two years and reached a combined print run of almost one hundred thousand copies, reaped substantial acclaim both among the lay and the academic public. Th e main debate about the book unfolded in the magazines Literární noviny and Vlasta, in which the contributors aired their views on the book from various positions – as advocates of phenomenology, Marxism, and the women’s press.
In this paper we establish some new nonlinear difference inequalities. We also present an application of one inequality to certain nonlinear sum-difference equation.
First, some classic properties of a weighted Frobenius-Perron operator P u ϕ on L 1 (Σ) as a predual of weighted Koopman operator W = uUϕ on L∞(Σ) will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of P u ϕ under certain conditions