In this talk, we first review the de Rham complex and the finite element exterior calculus, a cohomological framework for structure-preserving discretisation of PDEs. From de Rham complexes, we derive other complexes with applications in elasticity, geometry and general relativity. The derivation, inspired by the Bernstein-Gelfand-Gelfand (BGG) construction, also provides a general machinery to establish results for tensor-valued problems (e.g., elasticity) from de Rham complexes (e.g., electromagnetism and fluid mechanics). We discuss some applications and progress in this direction, including mechanics models and the construction of bounded homotopy operators (Poincaré integrals) and finite elements.

Polynomial splines over simplicial meshes in R^n (triangulations in 2D, tetrahedral meshes in 3D, and so on) sometimes have extra orders of smoothness at a vertex. This property is known as supersmoothness, and plays a role both in the construction of macroelements and in the finite element method.
Supersmoothness depends both on the number of simplices that meet at the vertex and their geometric configuration.

In this talk we review what is known about supersmoothness of polynomial splines and then discuss the more general setting of splines whose individual pieces are any infinitely smooth functions.

This is joint work with Kaibo Hu.

Correlation operators are used in data assimilation algorithms
to weight the contribution of prior and observation information.
Efficient implementation of these operators is therefore crucial for
operational implementations. Diffusion-based correlation operators are popular in ocean data assimilation, but can require a large number of serial matrix-vector products. An all-at-once formulation removes this requirement, and offers the opportunity to exploit modern computer architectures. High quality preconditioners for the all-at-once approach are well-known, but impossible to apply in practice for the
high-dimensional problems that occur in oceanography. In this talk we
consider a nested preconditioning approach which retains many of the
beneficial properties of the ideal analytic preconditioner while
remaining affordable in terms of memory and computational resource.