The Nfears Project. the Density Method for the Recovery and Representation of Finite Element Meshes

The goals for the NFEARS project | Providing computationally cheap nite element solutions, within a given tolerance from the exact solution to nonlinear, parameterized boundary value problems | and the means to reach the goals, are described in detail. Also it is explained how one particular subgoal of the project has been realized. This subgoal was to construct a theory for nding near optimal nite element meshes for the given class of problems, and representing such meshes in a fashion suitable for the realization of the general NFEARS goals. 1. The NFEARS project The NFEARS project was instigated in the mid 1980's. It has two major goals: 1. To construct a theory that allows computationally cheap nite element solutions of nonlinear, parameterized boundary value problems; solutions that lie within a user given tolerance from the exact solution, when the error is measured in any of a number of predetermined norms. 2. To construct a software package based on the theory, for the solution of the mentioned class of problems. The package must be adaptive in the sense that it automatically from non optimal user input constructs near optimal nite element meshes, giving nite element solutions as close to the exact solution as required, in the norm required, and as computationally cheap as possible. Here and below an Optimal Finite Element Mesh for the problem considered, in some class of meshes, with respect to some norm, and for a given tolerance, is a mesh with the minimal number of elements in the subclass of the given class of meshes which gives a nite element solution with an error measured in the required norm which is less than (or equal to) the tolerance. The nonlinear, parameterized boundary value problems make a highly relevant problem class: The reaction of physical systems, and mathematical and engineering models of such, to various external quantities like forces or radiation, is a basic problem in applied mathematics. The mathematical models of such problems are often nonlinear boundary value problems with parameters describing the external quantities. Normally interest is centered around stable systems. It must be certiied that a given system remains stable, i.e. that the qualitative behavior does not change, within a pre speciied range of the parameters. In some cases however the bifurcation points, where the stability properties of the system change, is the main interest. As an example consider the \snap-through" problem where a system …