Recovery and Few Parameter Representation of the Optimal Mesh Density Function for near Optimal Finite Element Meshes

A method is presented to recover nearly optimal nite element meshes represented by mesh density functions described by a few parameters. The density representation of nite element meshes is part of a methodology for adaptive solution of linear or non linear parameter dependent problems allowing easy optimization , storage, and comparison of meshes. This gives the possibility of easy prediction of meshes for future parameter values for parametrized problems. Asymptotical results showing the optimality of the recovered meshes are given, and computational examples show the validity of the results also for coarse meshes 1. Introduction It is well-known that for most boundary value problems, in particular problems with singularities, the computational eeort invested in the solution with the nite element method can be signiicantly reduced by using properly graded meshes as compared to uniform meshes. The idea of recovery of optimal nite element meshes is not new. It can be found in the work of Oden et al (for example 15], 16], 18], 23], and 26]), Zienkiewicz et al (for example 29], and 32]), Babuska, Rheinboldt et al (for example 6], 7], and 11]), and many other places. In this article formulas for the recovery of optimal nite element meshes are given for a wide class of problems involving one or more physical parameters (load parameters). As explained in 20] special features prevail in the case of parametrized problems that allow signiicant savings in solution time by taking a novel approach: Normally a sequence of problems are solved (diierent load cases) for which the problems , solutions and optimal meshes vary little in some sense from load case to load