Recent Results on Smart Algorithms and Adaptive Methods for Two- and Three-dimensional Problems in Computational Fluid Mechanics

Some recent results obtained using adaptive finite clement methods and so-called smart algorithms in twoand three-dimensional problems in /1uid mechanics arc discussed. These include applications of hand h-p-adaptive methods on unstructured meshes. 3. LOCAL APPROXI~IATION OF THE NAVIER-STOKES EQUATIONS ON UNSTRUCTURED MESHES Having an estimate of the error in the solution at a grid celL what can one do to systematically reduce it below some preset level? In general. one can refine the mesh size h (h-methods of adaptivity). increase the density of grid points by relocating nodes (r-methods of adaptivity). increase the local spectral order p of the approximation (p-methods of adaptivity). or use combinations of these techniques (c.g. h-p-techniques). Each of these choices puts new demands on the overall approach to the computational problem. In particular. adaptive tcchniques must ( I ) generally function on unstructured meshes. (2) require elaborate and complicated data structures. (3) employ explicit or iterative solution techniques. since direct solvers are of limited value on dynamically evolving unstructured meshes. (4) cope with special issues of sIability of numerical schemes that must function with continually changing structures and orders, and (5) attempt to minimize the computational overhead of the error estimation and of implementation of the adaptive process itself. These are the major challenges of adaptive methods in computational fluid dynamics. In recent months we have attempted to meet these challenges through a series of studies on each of the above issues. Some of our results have been implemented in a collection of Navier-Stokes solvers that we refer to as the ADAPTnl code: other results deal with new data structures and adaptive methods and require further work before they represent effective tools for complex flow simulations. Some of our results on special issues of adaptivity and on selected applications are briefly summarized. ..