583 research outputs found
Automatic symbolic computation for discontinuous Galerkin finite element methods
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
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Optimisations for quadrature representations of finite element tensors through automated code generation
We examine aspects of the computation of finite element matrices and vectors
which are made possible by automated code generation. Given a variational form
in a syntax which resembles standard mathematical notation, the low-level
computer code for building finite element tensors, typically matrices, vectors
and scalars, can be generated automatically via a form compiler. In particular,
the generation of code for computing finite element matrices using a quadrature
approach is addressed. For quadrature representations, a number of optimisation
strategies which are made possible by automated code generation are presented.
The relative performance of two different automatically generated
representations of finite element matrices is examined, with a particular
emphasis on complicated variational forms. It is shown that approaches which
perform best for simple forms are not tractable for more complicated problems
in terms of run time performance, the time required to generate the code or the
size of the generated code. The approach and optimisations elaborated here are
effective for a range of variational forms
The deal.II Library, Version 9.1
This paper provides an overview of the new features of the finite element library deal.II, version 9.1
Self-Evaluation Applied Mathematics 2003-2008 University of Twente
This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008
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