Automatic symbolic computation for discontinuous Galerkin finite element methods
For researchers and engineers implementing DGFEMs, this work reduces the complexity and debugging effort of coding these methods, but it is an incremental improvement over existing symbolic algebra tools.
The paper develops a new class structure for automatic computation of discontinuous Galerkin finite element methods (DGFEMs), implemented in FEniCS, which simplifies implementation for coupled nonlinear PDEs. Numerical examples demonstrate ease of use for various PDE problems.
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 Fréchet 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.