9706353 Keith Miller Miller plans to work on local plus global adaptivity for moving node finite element methods. He plans first (in collaboration with Neil Carlson of Purdue) to combine the local adaptivity of 2-D gradient-weighted Moving Finite Elements (through the continuous movement of its nodes) with an improved global adaptivity (through addition, deletion and reconnection of nodes, based upon enriched geometric criteria and upon local error estimates). Here they are building upon their longstanding and fruitful collaboration in developing the GWMFE method. This method is especially suited to problems which develop sharp moving fronts, especially problems where one needs to resolve the fine-scale structure of the fronts. GWMFE does an extremely fine job of moving its nodes around locally to resolve the sharp features of the solution, but clearly global adaptivity is needed if we are to have truly robust and flexible codes. Here they would be making fundamental changes and additions to the partial global adaptivity introduced by Miller's student Kuprat in his thesis of '92, which added so greatly to the robustness of the method. Miller plans second to develop a revised GWMFE with streamline diffusion; this should correct some problems which the present method has on certain steady-state convection problems in which the GWMFE grid drifts downstream with the flow rather than coming to a steady-state. This revised GWMFE works extremely well in 1-D; success in extending it to multidimensions would be a significant advance for slow-transient and near steady-state fluid computations. Finite element (FE) methods with a triangular grid in 2-D typically compute a piecewise linear approximation to the solution of a partial differential equation (PDE) or system of PDEs; that is, the graph of the approximate solution is an evolving surface with planar triangular faces. For standard FE methods the grid is specified and fixed; however, for the GWMFE method the grid is allowed to deform and the nodes of the grid decide for themselves how to move. On problems with sharp moving fronts the nodes can thus automatically concentrate in the front and move with it. In this way one attains high resolution in the critical regions of the solution while using far fewer nodes and far larger time steps than with standard methods. Examples are the highly nonlinear diffusion of doped arsenic ions in the manufacture of silicon chips, the drift-diffusion equations for the nanosecond evolution of holes, electrons and voltages as a semiconductor device switches states, and the "black oil" equations for flooding of oil reservoirs. The GWMFE method has been strikingly successful on these and many other problems even within the constraints of a logically-fixed grid; that is, the grid deforms automatically, but the number and interconnections of its nodes remain fixed. However, a logically-fixed grid is inadequate for many important problems and it is apparent that the efficiency and robustness of the method can be greatly enhanced by adding global adaptivity (the insertion and deletion of nodes as needed). This requires a good deal of code development, but the combination of local plus global adaptivity in GWMFE should yield a combined method which is accurate and robust, which uses far fewer nodes, and which thereby renders efficiently computable important classes of problems with moving sharp features which are presently inaccessible to numerical computation.
