Final Report Summary - FEEC-A (Finite Element Exterior Calculus and Applications)
The purpose of this project has been to further develop the theoretical foundation for scientific computing. Scientific computing is today an indispensable tool in almost all branches of science and engineering. Previously, the PI had done fundamental work on finite element exterior calculus, where the goal was to develop numerical schemes which are compatible with the geometric, topological, and algebraic properties of the differential equations being approximated. During this project, this foundation has been further developed. In particular, we have constructed a theory where certain properties of the methods can be shown to be almost independent of the degree of the finite element spaces. This has been achieved through a construction, referred to as the bubble transform, which is in fact independent of any choice of finite element spaces. It only depends on the underlying mesh. We have also been involved in constructing new numerical schemes for problems in material science which have already been successfully used for building simulators for certain key processes in the human brain. Some of our results can also be used to explain the advantage of increasing the smoothness of finite element spaces.