Automating the Finite Element Method

TL;DR

This paper reviews research on fully automating the finite element method, enabling automatic discretization, code generation, and parametrization over variational problems, finite elements, and meshes.

Contribution

It introduces algorithms for automatic computation of discrete systems and demonstrates automatic code generation for flexible finite element implementations.

Findings

Algorithms enable automatic discretization of variational problems.

Automatic code generation allows parametrization over multiple variables.

The approach enhances flexibility and efficiency of finite element software.

Abstract

The finite element method can be viewed as a machine that automates the discretization of differential equations, taking as input a variational problem, a finite element and a mesh, and producing as output a system of discrete equations. However, the generality of the framework provided by the finite element method is seldom reflected in implementations (realizations), which are often specialized and can handle only a small set of variational problems and finite elements (but are typically parametrized over the choice of mesh). This paper reviews ongoing research in the direction of a complete automation of the finite element method. In particular, this work discusses algorithms for the efficient and automatic computation of a system of discrete equations from a given variational problem, finite element and mesh. It is demonstrated that by automatically generating and compiling efficient low-level code, it is possible to parametrize a finite element code over variational problem and finite element in addition to the mesh.