>Project Abstract for NSF Proposal Number 0071412 (Schatz and Wahlbin). > >Our work deals with the fundamental behavior of the finite element >method. We propose to study the the general question of a posteriori >error estimation. Here we aim to develop a sharp theory which will >predict with precision the error on a single element. Fundamental >technical tools have to be created for this. The work will first be >done in the context of second order elliptic problems. We will also >consider extensions to time-dependent problems involving a second order >elliptic operator, such as parabolic, hyperbolic and partial >integro-differential equations. > >The mathematical field called Numerical Analysis is at the heart of >high-performance computing (which then applies to almost all scientific >and technological endeavours). Our work has, basically, two >interrelated aims: i) we try to gain a detailed and fundamental >understanding of the behavior of existing high-performance methods. ii) >we try to devise even better (more reliable and/or faster) new methods. >The second step in this program is fundamentally dependent on the >insight gained in the first step. > > >**********
