Automated generation and symbolic manipulation of tensor product finite elements

Abstract

We describe and implement a symbolic algebra for scalar and vector-valued finite elements, enabling the computer generation of elements with tensor product structure on quadrilateral, hexahedral and triangular prismatic cells. The algebra is implemented as an extension to the domain-specific language UFL, the Unified Form Language. This allows users to construct many finite element spaces beyond those supported by existing software packages. We have made corresponding extensions to FIAT, the FInite element Automatic Tabulator, to enable numerical tabulation of such spaces. This tabulation is consequently used during the automatic generation of low-level code that carries out local assembly operations, within the wider context of solving finite element problems posed over such function spaces. We have done this work within the code-generation pipeline of the software package Firedrake; we make use of the full Firedrake package to present numerical examples.

Submitted to SISC special issue on CSE Software. Updated version, following reviewer comments

References in corpus (2)

• Optimisations for quadrature representations of finite element tensors through automated code generation
• COFFEE: an Optimizing Compiler for Finite Element Local Assembly

Cited by in corpus (22)

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• A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in Firedrake
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• A Compatible Finite Element Discretisation for the Moist Compressible Euler Equations
• Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond
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• Data-driven stochastic Lie transport modelling of the 2D Euler equations
• Maximum-principle-satisfying discontinuous Galerkin methods for incompressible two-phase immiscible flow
• The scaling and skewness of optimally transported meshes on the sphere
• The Recovered Space Advection Scheme for Lowest-Order Compatible Finite Element Methods
• A scalable DG solver for the electroneutral Nernst-Planck equations
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• Extending FEniCS to Work in Higher Dimensions Using Tensor Product Finite Elements
• asQ: parallel-in-time finite element simulations using ParaDiag for geoscientific models and beyond
• Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements
• Structure Matters: A Scale-Resolved Numerical Operando Approach for Lithium-Sulfur Batteries
• FIAT: improving performance and accuracy for high-order finite elements