Structure-preserving discretizations using smooth splines

Finite element exterior calculus (FEEC) is a framework for designing stable and accurate finite element discretizations for a wide variety of systems of PDEs. The involved finite element spaces are constructed using piecewise polynomial differential forms, and stability of the discrete problems is established by preserving at the discrete level the geometric, topological, algebraic and analytic structures that ensure well-posedness of the continuous problem. The framework achieves this using methods from differential geometry, algebraic topology, homological algebra and functional analysis. In this talk I will discuss the use of smooth splines within FEEC, motivated by the fact that smooth splines are the de facto standard for representing geometries of interest in engineering and because they offer superior accuracy in numerical simulations (per degree of freedom) compared to classical finite elements. In particular, I will present new results for smooth splines defined on unstructured meshes (i.e., non-Cartesian and/or locally-refined meshes).