Exposing and exploiting structure: optimal code generation for high-order finite element methods

Code generation based software platforms, such as Firedrake, have become popular tools for developing complicated finite element discretisations of partial differential equations. We extended the code generation infrastructure in Firedrake with optimisations that can exploit the structure inherent to some finite elements. This includes sum factorisation on cuboid cells for continuous, discontinuous, H(div) and H(curl) conforming elements. Our experiments confirm optimal algorithmic complexity for high-order finite element assembly. This is achieved through several novel contributions: the introduction of a more powerful interface between the form compiler and the library providing the finite elements; a more abstract, smarter library of finite elements called FInAT that explicitly communicates the structure of elements; and form compiler algorithms to automatically exploit this exposed structure.