We consider the mixed finite element method for locally refined triangulations. A local projection operator\(\hat \Pi _h \) is defined to satisfy the commutation property that is required in the general theory of mixed methods. Our results can be applied to every known space of arbitrary order over rectangles or triangles. Optimal-order error estimates and superconvergence for the flux along the Gauss lines are established.

In the multi‐dimensional numerical simulation of certain multi‐phase fluid flow processes, many phenomena are sufficiently localized and transient that self‐adaptive local grid refinement techniques are necessary to resolve the local physical behaviour. For large‐scale simulation problems, efficiency is the key to the choice of specific adaptive strategies. Several different types of self‐adaptive methods will be discussed and briefly compared. The differences in complexity of the data structure and efficiency of the solution techniques will be explained. Concepts for incorporating these methods in existing large‐scale simulators will also be presented. Man man‐years have been invested in developing large‐scale numerical simulators which do an excellent job of resolving most of the larger length‐scale phenomena in our production processes. Although adaptive local refinement methods are critical for many applications, most techniques which have been developed to date either cannot be incorporated in our codes or are so computationally complex that they destroy the efficiency of existing simulators. Recently, new preconditioning techniques have been developed which allow grid refinement methods to be used efficiently in exisiting codes without disrupting the basic solution process. These techniques are highly amenable to vectorization and parallelization of the algorithms. They are also a first step in allowing complex well models to be incorporated into large field‐scale codes and applications.