Automatic symbolic computation for discontinuous Galerkin finite element methods

TL;DR

This paper introduces a symbolic algebra-based framework within FEniCS for automating the implementation and assembly of discontinuous Galerkin finite element methods, simplifying complex PDE discretizations and derivative computations.

Contribution

It develops a class structure for automatic DGFEM formulation computation, enhancing abstraction and ease of implementation in high-level languages.

Findings

Simplifies implementation of DGFEMs for various PDEs

Facilitates automatic assembly and differentiation of nonlinear systems

Demonstrates effectiveness through numerical examples

Abstract

The implementation of discontinuous Galerkin finite element methods (DGFEMs) represents a very challenging computational task, particularly for systems of coupled nonlinear PDEs, including multiphysics problems, whose parameters may consist of power series or functionals of the solution variables. Thereby, the exploitation of symbolic algebra to express a given DGFEM approximation of a PDE problem within a high level language, whose syntax closely resembles the mathematical definition, is an invaluable tool. Indeed, this then facilitates the automatic assembly of the resulting system of (nonlinear) equations, as well as the computation of Fr\'echet derivative(s) of the DGFEM scheme, needed, for example, within a Newton-type solver. However, even exploiting symbolic algebra, the discretisation of coupled systems of PDEs can still be extremely verbose and hard to debug. Thereby, in this article we develop a further layer of abstraction by designing a class structure for the automatic computation of DGFEM formulations. This work has been implemented within the FEniCS package, based on exploiting the Unified Form Language. Numerical examples are presented which highlight the simplicity of implementation of DGFEMs for the numerical approximation of a range of PDE problems.