Unlike FIAT, “FInAT Is not A Tabulator.” Instead, it provides symbolic expressions for the evaluation of finite elements. Thus FInAT can express the structure that is intrinsic to some finite elements, which a form compiler can exploit to automatically:

• sum factorise continuous, discontinuous, H(div) and H(curl) conforming elements on cuboid cells;
• optimise evaluation at collocated quadrature points when under-integration is requested; and
• optimise evaluation on vector and tensor elements.

Furthermore, FInAT is equipped to provide symbolic expressions for basis transformations when the caller implements an interface capable of providing (symbolic expressions for) the required geometric quantities (such as Jacobians, normals, or tangents), which facilitates the implementation of finite elements such as Hermite, Morley, Bell, and Argyris within an automatic code generation framework.

The goal of FInAT is to be a “single source of truth” for finite elements. This includes not only basis function evaluation, like FIAT, but also structural and algorithmic considerations as well as reference element transformations. Symbolic expression languages and form compilers can then exploit this information in an element-independent manner.

FInAT is integrated with TSFC and is a component of Firedrake, “an automated system for the portable solution of partial differential equations using the finite element method.” To facilitate the exchange of symbolic expressions between the element library on the one hand, and the form compiler on the other, they need to agree on a common language. Where FIAT communicated with the form compiler through numerical arrays, FInAT communicates with the form compiler by exchanging GEM expressions. GEM is the intermediate language used in both TSFC and FInAT to describe tensor algebra.

1. Miklós Homolya, Robert C. Kirby, and David A. Ham (2017). “Exposing and exploiting structure: optimal code generation for high-order finite element methods.” arXiv preprint arXiv:1711.02473.
2. Robert C. Kirby, and Lawrence Mitchell (2019). “Code generation for generally mapped finite elements.” ACM Transactions on Mathematical Software (TOMS), 45(4), pp. 1-23.

All files in this repository are available under the MIT license, see the LICENSE file for details.