Review: Formulating efficient finite‐element codes for flows in regular domains

Many fluid flow problems of current interest occur in domains that are mappable to a rectangle or a box; conformal mappings are particularly useful in this regard. We are concerned here with the efficient solution of such problems using finite elements. The central issue is the element choice, and this issue is addressed in terms of operation counts, computer memory and I/0 requirements, and the extent to which code vectorization is possible. It is concluded that rectangular (box) elements generally lead to more efficient algorithms that triangular (tetrahedral) elements. A synthesis of algorithms, based on bilinear (trilinear) elements, is presented. The algorithms have the attributes of simplicity, accuracy, stability and straightforward incorporation of boundary conditions. For bilinear and trilinear elements, it is found that product and first‐derivative terms are well‐handled by the Galerkin FE method, but that it is advantageous to go outside of the Galerkin framework when treating second‐derivative terms. It is particularly important to consider the form of the governing equations, vis‐à‐vis the choice of staggered, non‐staggered and/or mixed‐order elements, and to choose an appropriate time scheme. The described techniques have been successfully applied to a variety of problems in regular domains, including the solution of the three‐dimensional time‐dependent hydrostatic primitive equations; these are stiff and include first and second derivative terms, non‐linearities and variable coefficients due to a conformal mapping.