# Code Generation for Generally Mapped Finite Elements

@article{Kirby2019CodeGF, title={Code Generation for Generally Mapped Finite Elements}, author={Robert C. Kirby and Lawrence Mitchell}, journal={ACM Transactions on Mathematical Software (TOMS)}, year={2019}, volume={45}, pages={1 - 23} }

Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite-element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second-order model problem, we find that new elements give smooth solutions at a mild increase in… Expand

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#### References

SHOWING 1-10 OF 74 REFERENCES

A general approach to transforming finite elements

- Mathematics
- 2017

The use of a reference element on which a finite element basis is constructed once and mapped to each cell in a mesh greatly expedites the structure and efficiency of finite element codes. However,… Expand

Nonconforming finite element methods for eigenvalue problems in linear plate theory

- Mathematics
- 1979

The paper deals with nonconforming finite element methods for approximating fourth order eigenvalue problems of typeΔ2w=λΔw. The methods are handled within an abstract Hilbert space framework which… Expand

Exposing and exploiting structure: optimal code generation for high-order finite element methods

- Computer Science
- ArXiv
- 2017

The code generation infrastructure in Firedrake was extended with optimisations that can exploit the structure inherent to some finite elements, including sum factorisation on cuboid cells for continuous, discontinuous, H(div) and H(curl) conforming elements. Expand

Simple and extensible plate and shell finite element models through automatic code generation tools

- Computer Science
- 2018

It is shown that by using a high-level mathematical modelling strategy and automatic code generation tools, a wide range of advanced plate and shell finite element models can be generated easily and efficiently, including the linear and non-linear geometrically exact Naghdi shell models, the Marguerre-von Karman shallow shell model, and the Reissner-Mindlin plate model. Expand

A discontinuous Galerkin formulation for a strain gradient-dependent damage model

- Mathematics
- 2004

Abstract The numerical solution of strain gradient-dependent continuum problems has been dogged by continuity demands on the basis functions. For most commonly accepted models, solutions using the… Expand

Feel++ : A computational framework for Galerkin Methods and Advanced Numerical Methods

- Computer Science
- 2012

An overview of a unified framework for finite element and spectral element methods in 1D, 2D and 3D in \cpp called \Feel is presented, including examples using the mortar, Schwartz (non)overlapping, three fields and two ficticious domain-like methods. Expand

Multigrid methods for the biharmonic problem discretized by conforming C 1 finite elements on nonnested meshes

- Mathematics
- 1995

We consider multigrid algorithms for the biharmonic problem discretized by conforming C 1 finite elements. Most finite elements for the biharmonic equation are nonnested in the sense that the coarse… Expand

Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements

- Mathematics
- 2010

A key challenge while employing non-interpolatory basis functions in finite-element methods is the robust imposition of Dirichlet boundary conditions. The current work studies the weak enforcement of… Expand

A domain-specific language for the hybridization and static condensation of finite element methods

- Computer Science
- ArXiv
- 2018

A domain-specific language (DSL) for concisely expressing localized linear algebra on finite element tensors, and its integration within a code-generation framework is introduced. Expand

Solver Composition Across the PDE/Linear Algebra Barrier

- Mathematics, Computer Science
- SIAM J. Sci. Comput.
- 2018

Improvements in the Firedrake finite element library are presented that allow for straightforward development of the building blocks of extensible, composable preconditioners that decouple the solver from the model formulation. Expand