9704852 Parter These investigations will focus on three specific projects: (1) First Order Systems Least Square (FOSLS) methods with a special emphasis on problems in elasticity, (2) the general over-lapping grid problem, and (3) preconditioning strategies for Spectral Collocation Methods. At this time computational methods for problems in elasticity are primarily based on mixed methods which lead to indefinite problems which have proven difficult to solve. The FOSLS approach leads to large systems which are generally much easier to solve. However, despite the efforts of several independent groups, a useful FOSLS formulation of the elasticity problems for general boundary conditions is still a major problem. The overlapping grid method has proven itself an effective tool for problems set in regions with complicated geometry. Further, the analysis of these methods has been lacking until a recent breakthrough for difference equations of positive type. Since there are limitations on the order of accuracy of such difference methods there is a need for further analysis. Spectral Collocation methods are extremely accurate and very badly conditioned. Hence preconditioning strategies are essential. There are over twenty-five years of experience and study of preconditioning methods. And, still, there is a need for better approaches. Moreover, the mathematical justification for some of these methods is incomplete. The basic theme of this research can be stated simply "Find effective numerical methods to solve the important boundary-value problems of mechanics and material science, and provide mathematical proofs of their validity." In the problems of elasticity most of the methods now in use are expensive to implement. Hence, a new formulation is sought which will provide accurate approximations which can be computed at a reasonable cost. In the other two cases one is dealing with established methods which are either difficult to impleme nt or whose mathematical basis is incomplete. This research is aimed at a more complete mathematical understanding which will both clarify the existing methods and provide a basis for the development of new, more effective methods.
