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 Frechet 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.
Highlights
The finite element method (FEM) represents an indispensable computational tool for the accurate, efficient, and rigorous numerical approximation of continuum models arising within a wide range of scientific and engineering application areas
We present a further layer of abstraction for the automatic computation of discontinuous Galerkin finite element methods (DGFEMs) formulations employing symbolic algebra
In this article we have exploited the use of symbolic algebra for the automatic computation of DGFEMs for the numerical approximation of general systems of nonlinear partial differential equations (PDEs)
Summary
The finite element method (FEM) represents an indispensable computational tool for the accurate, efficient, and rigorous numerical approximation of continuum models arising within a wide range of scientific and engineering application areas. Given F. c, which specifies F c (·), the numerical flux functions HLF(·, ·, ·) and HHLLE(·, ·, ·) can be automatically generated in order to yield the discretisation of the convective term present in the underlying PDE problem; we note that the constructors of both classes LocalLaxFriedrichs and HLLE require the symbolic representation of the eigenvalues of B. The class implementation is shown in Listing 7; here, generate_fem_formulation is overridden to construct the DGFEM formulation of the provided convective flux operator on the interior and exterior faces, as well as on the elements in the mesh. The derived classes must manage the function spaces and boundary conditions amongst the aggregated DGFemFormulation members This framework significantly reduces the code required for subsequent development of DGFEM formulations of PDE operators of increasing complexity.
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