ABSTRACT
The chapter considers the coverage to finite element and finite volume methods; these are also more naturally suited to multi-dimensional problems in complicated geometries and, in some sense too, to problems in which both diffusion and convection play a significant role. There have been very important developments in algorithms for approximating unsteady convection-dominated problems, particularly hyperbolic conservation laws. Fortunately, there are at least two texts which cover the finite difference methodology as well as the basic theory of hyperbolic conservation laws, namely E. Gdlewski and A. Raviart and R. J. LeVeque; and results obtained with a wide variety of methods for a set of simple convection-diffusion problems are presented in C. B. Vreugdenhil and B. Koren. By far the most common adaptive recovery techniques for unsteady problems consist of constructing a discontinuous piecewise linear approximation in one dimension from a piecewise constant approximation, as first used by B. van Leer.
