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The purpose of this paper is to describe and analyse a class of finite element fractional step methods for solving the incompressible Navier-Stokes equations. The objective is not to reproduce the extensive contributions on the subject, but to report on long-term experience with and provide a unified overview of a particular approach: the characteristic-based split method. Three procedures, the semi-implicit, quasi-implicit and fully explicit, are studied and compared.

This work provides a thorough assessment of the accuracy and efficiency of these schemes, both for a first and second order pressure split.

In transient problems, the quasi-implicit form significantly outperforms the fully explicit approach. The second order (pressure) fractional step method displays significant convergence and accuracy benefits when the quasi-implicit projection method is employed. The fully explicit method, utilising artificial compressibility and a pseudo time stepping procedure, requires no second order fractional split to achieve second order or higher accuracy. While the fully explicit form is efficient for steady state problems, due to its ability to handle local time stepping, the quasi-implicit is the best choice for transient flow calculations with time independent boundary conditions. The semi-implicit form, with its stability restrictions, is the least favoured of all the three forms for incompressible flow calculations.

A comprehensive comparison between three versions of the CBS method is provided for the first time.

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To determine where, when, how, and wherefore European social theory hit upon the formula of “the True, the Good, and the Beautiful,” and how its structural position as a skeleton for the theory of action has changed.

Genealogy, library research, and unusually good fortune were used to trace back the origin of what was to become a ubiquitous phrase, and to reconstruct the debates that made deploying the term seem important to writers.

The triad, although sometimes used accidentally in the renaissance, assumed a key structural place with a rise of Neo-Platonism in the eighteenth century associated with a new interest in providing a serious analysis of taste. It was a focus on taste that allowed the Beautiful to assume a position that was structurally homologous to those of the True and the Good, long understood as potential parallels. Although the first efforts were ones that attempted to emphasize the unification of the human spirit, the triad, once formulated, was attractive to faculties theorists more interested in decomposing the soul. They seized upon the triad as corresponding to an emerging sense of a tripartition of the soul. Finally, the members of the triad became re-understood as values, now as orthogonal dimensions.

This seems to be the first time the story of the development of the triad – one of the most ubiquitous architectonics in social thought – has been told.