9706768 Neil N. Carlson ABSTRACT (Moving Node Finite Element Methods) The gradient-weighted moving finite element method (GWMFE) is an adaptive method well-suited to problems which develop sharp moving fronts, especially those problems where one must resolve the fine-scale structure of the front. Three works related to the method are planned. First, development of GWMFE in 3-D will continue. The recently completed scalar code will be extended to systems of equations and the direct linear solver will be replaced by iterative techniques based on the work of Miller and Xaba. The code will be applied to a variety of difficult PDEs and PDE systems: Stefan-type problems, drift-diffusion equations from semiconductor device modeling, the time dependent Ginzburg-Landau equation, are some possible examples. A parallel version of the code will also be developed. Second, a new moving mixed finite element method will be implemented in 2-D and 3-D. Here, in the spirit of the mixed finite element method, the diffusive flux is approximated in an independent finite element space (BDM or RT) and a coupled system of ODEs is then obtained via the same variational argument used in the moving finite element method. This new method gives a more rigorous treatment of diffusion terms, and in 1-D gives superior grid adaptivity. Third, in collaboration with Miller (Berkeley), global adaptivity will be added to GWMFE in 2-D. This global adaptivity will be achieved through the periodic addition, deletion, and reconnection of nodes based upon geometric criteria, which are easily identifiable through simple eigenvalue analysis, and upon local projection error estimates. This complements the very fine local adaptivity GWMFE achieves through the continuous motion of its nodes, and is essential for a robust method. Many physical processes are modeled by partial differential equations. A solution of such an equation might give, for example, the concentration of of arsenic in a semicon ductor substrate at each instant of time and each point in one, two, or three dimensional space. Because it is generally impossible to find the exact solution of such an equation, approximate solutions must be sought using some computer-based technique. For a large class of difficult problems standard techniques are extremely wasteful and inefficient, making computations in two, and especially three, dimensions very difficult if not altogether impossible. More sophisticated adaptive techniques seek to overcome this difficulty. The gradient-weighted moving finite element method (GWMFE) is one such technique, and it offers some significant advantages over more conventional adaptive methods. This project seeks to advance GWMFE and related methods in several directions. First, a full implementation of GWMFE in the computationally challenging three dimensional case will be completed, and applied to a variety of challenging problems. Second, a variant of the method which gives a more accurate treatment of some problems will be investigated. And last, a complementary global adaptivity will be added to GWMFE in two dimensions which will substantially enhance the robustness of the method. GWMFE has been applied to a variety of difficult problems with great success, including problems from semiconductor device manufacture and device simulation, groundwater transport and oil reservoir engineering, and time dependent models of superconductors. Being able to computationally model problems like these is becoming ever more important to scientists and engineers. GWMFE shows great promise in making some of these difficult problems efficiently computable in two and three dimensions, some of which are currently inaccessible with existing techniques.
