Algorithms and Numerical Analysis for Partial Differential Equations

Cornell University, Ithaca NY

Investigators

Abstract

The proposed study is concerned with several aspects of the analysis of the finite element method. To begin with the problem of a posteriori error estimation that can accurately predict the error on a single element using a class of averaging operators will be considered. These methods do not seem to work well for problems on polygonal domains for which the pollution error dominates. Accurate local estimators and indicators will be sought to correct this not only for quasi-uniform meshes but also for refined grids. In order to carry out the above programs, it is necessary to develop fundamental technical tools such a sharp a priori estimates, both local and global, both for smooth and nonsmooth problems where both quasi-uniform and highly refined grids are present. These problems are an integral part of this proposal. The socalled superapproximation property of many finite elements has proved to be a powerful tool in the analysis of the finite element method. Applications will be sought for a new form of this property that does away with mesh restrictions for some problems. There are a very large number of important physical problems that arise in science and industry. For many of these problems it is cost effective to make a mathematical model and to solve the problem on a computer. For example consider the problem of designing an aircraft (or an automobile) so that is as fuel efficient and fast as possible. Suppose that in order to achieve this we wish to minimize the drag and still give the aircraft enough lift to do its job. Instead of making a physical model that may be difficult and costly to change in order to arrive at an optimal design, many aircraft companies now almost completely design airplanes inside and out on computers. The mathematical equations that must be solved are complicated but at the heart of many solution methods there is required the solution of a socalled elliptic differential equation. This type of equation arises in a very large number of applications. This project is concerned with obtaining more cost effective and efficient solutions of elliptic problems in order to obtain more accurate and efficient numerical schemes. In order to do so, we have to confront some fundamental mathematical problems which will be done in this proposal.