Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method (Gottfried Wilhelm von Leibniz, 1646–1719).When, several years ago, I saw for the first time an instrument which, when carried, automatically records the number of steps taken by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only addition and subtraction, but also multiplication and division, could be accomplished by a suitably arranged machine easily, promptly and with sure results…. For it is unworthy of excellent men to lose hours like slaves in the labour of calculations, which could safely be left to anyone else if the machine was used…. And now that we may give final praise to the machine, we may say that it will be desirable to all who are engaged in computations which, as is well known, are the managers of financial affairs, the administrators of others estates, merchants, surveyors, navigators, astronomers, and those connected with any of the crafts that use mathematics (Leibniz).

This paper is the first part in a series of papers on adaptive finite element methods for parabolic problems. In this paper, an adaptive algorithm is presented and analyzed for choosing the space and time discretization in a finite element method for a linear parabolic problem. The finite element method uses a space discretization with meshsize variable in space and time and a third-order accurate time discretization with timesteps variable in time. The algorithm is proven to be (i) reliable in the sense that the L2-error in space is guaranteed to be below a given tolerance for all timesteps and (ii) efficient in the sense that the approximation error is for most timesteps not essentially below the given tolerance. The adaptive algorithm is based on an a posteriori error estimate which proves (i), and sharp a priori error estimates are used to prove (ii). Analogous results are given for the corresponding stationary (elliptic) problem. In the following papers in this series extensions are made, e.g., to timesteps variable also in space and to nonlinear problems.Key words, adaptive finite element procedures, a priori error estimates, a posteriori error estimates, automatic error control, discontinuous Galerkin method, elliptic problems, parabolic problems

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We prove Lp stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain. Using finite element analysis techniques, we obtain L2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials. L estimates for p =* 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree. The latter estimates are proved by combining finite difference and finite element analysis techniques.

The OH stretching region of water molecules in the vicinity of nonionic surfactant monolayers has been investigated using vibrational sum frequency spectroscopy (VSFS) under the polarization combinations ssp, ppp, and sps. The surface sensitivity of the VSFS technique has allowed targeting the few water molecules present at the surface with a net orientation and, in particular, the hydration shell around alcohol, sugar, and poly(ethylene oxide) headgroups. Dramatic differences in the hydration shell of the uncharged headgroups were observed, both in comparison to each another and in comparison to the pure water surface. The water molecules around the rigid glucoside and maltoside sugar rings were found to form strong hydrogen bonds, similar to those observed in tetrahedrally coordinated water in ice. In the case of the poly(ethylene oxide) surfactant monolayer a significant ordering of both strongly and weakly hydrogen bonded water was observed. Moreover, a band common to all the surfactants studied, clearly detected at relatively high frequencies in the polarization combinations ppp and sps, was assigned to water species located in proximity to the surfactant hydrocarbon tail phase, with both hydrogen atoms free from hydrogen bonds. An orientational analysis provided additional information on the water species responsible for this band.