Finite Element Exterior Calculus and Applications
The ambition of this project is to set the scene for a number of new research directions based on FEEC by giving ground-breaking contributions to its foundation. The aim is also to use FEEC as a tool, or a guideline, to extend the foundation of numerical PDE to a variety of problems for which this foundation does not exist.
The finite element method is one of the most successful techniques for designing numerical methods for systems of partial differential equations (PDEs). It is not only a methodology for developing numerical algorithms, but also a mathematical framework in which to explore their behavior. The finite element exterior calculus (FEEC) provides a new structure that produces a deeper understanding of the finite element method and its connections to the partial differential equation being approximated. The goal is to develop discretizations which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the partial differential equation. The phrase FEEC was first used in a paper the PI wrote for Acta Numerica in 2006, together with his coworkers, D.N. Arnold and R.S. Falk. The general philosophy of FEEC has led to the design of new algorithms and software developments, also in areas beyond the direct application of the theory. The present project will be devoted to further development of the foundations of FEEC, and to direct or indirect use of FEEC in specifc applications. The ambition is to set the scene for a number of new research directions based on FEEC by giving ground-breaking contributions to its foundation. The aim is also to use FEEC as a tool, or a guideline, to extend the foundation of numerical PDE to a variety of problems for which this foundation does not exist. The more application oriented parts of the project includes topics like numerical methods for elasticity, its generalizations to more general models in materials science such as viscoelasticity, poroelasticity, and liquid crystals, and the applications of these models to CO2 storage and deformations of the spinal cord.
1.2. 2014 - 31.1.2019