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Code generation based software platforms, such as Firedrake, have becomepopular tools for developing complicated finite element discretisations ofpartial differential equations. We extended the code generation infrastructurein Firedrake with optimisations that can exploit the structure inherent to somefinite elements. This includes sum factorisation on cuboid cells forcontinuous, discontinuous, H(div) and H(curl) conforming elements. Ourexperiments confirm optimal algorithmic complexity for high-order finiteelement assembly. This is achieved through several novel contributions: theintroduction of a more powerful interface between the form compiler and thelibrary providing the finite elements; a more abstract, smarter library offinite elements called FInAT that explicitly communicates the structure ofelements; and form compiler algorithms to automatically exploit this exposedstructure.

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