9403512 Schatz New algorithms involving finite element methods for the numerical approximation of various problems will be introduced and analyzed. New techniques and methods of analysis of the behavior of existing approximation methods will be studied. Three main problems will be considered: (1) A new approach to superconvergence and extrapolation will be studied resulting from new local error estimates. Extensions to problems with non- smooth solutions, parabolic equations, and integral equations will be investigated, as well as applications to domain decomposition methods and multigrid. (2) Algorithms for obtaining optimal order accurate methods for elliptic problems on 3-dimensional polyhedral domains where there is a confluence of edge singularities and vertex singularities in derivatives will be studied. (3) New methods for obtaining local up to boundary estimates for finite element methods using isoparametric elements in R^n will be investigated. The subject of superconvergence and extrapolation for finite element methods has been the object of investigations by many researchers. A new and general theory that unifies both of these subjects and leads to new results has been developed by the principal investigator. These results lead to new highly accurate and efficient methods for solving some important physical problems. The methods are easy to implement, especially on modern supercomputers. They can be totally parallelized and are compatible with recent domain decomposition and multigrid algorithms. Fewer finite elements are needed for these methods to produce the same accuracy as standard finite element methods using the same type of element. The new theory seems to be the most general available at the present time. It is proposed to continue to investigate new facets of the theory and expand its applicability to a variety of new problems such as problems with non smooth solutions and parabolic equations.
