# Transformations for Piola-mapped elements

@article{Aznaran2021TransformationsFP, title={Transformations for Piola-mapped elements}, author={Francis R. A. Aznaran and Robert C. Kirby and Patrick E. Farrell}, journal={ArXiv}, year={2021}, volume={abs/2110.13224} }

The Arnold–Winther element successfully discretizes the Hellinger–Reissner variational formulation of linear elasticity; its development was one of the key early breakthroughs of the finite element exterior calculus. Despite its great utility, it is not available in standard finite element software, because its degrees of freedom are not preserved under the standard Piola push-forward. In this work we apply the novel transformation theory recently developed by Kirby [SMAI-JCM, 4:197–224, 2018…

## 3 Citations

### Preconditioners for computing multiple solutions in three-dimensional fluid topology optimization

- Computer ScienceArXiv
- 2022

This work develops a nested block preconditioning approach which reduces the linear systems to solving two symmetric positive-deﬁnite matrices and an augmented momentum block, and presents multiple solutions in three-dimensional examples computed using the proposed iterative solver.

### Robust Approximation of Generalized Biot-Brinkman Problems

- EngineeringJournal of Scientific Computing
- 2022

This paper introduces, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized BiotBrinkman equations and demonstrates that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes.

### Finite element methods for multicomponent convection-diffusion

- MathematicsArXiv
- 2022

. We develop ﬁnite element methods for coupling the steady-state Onsager–Stefan– Maxwell equations to compressible Stokes ﬂow. These equations describe multicomponent ﬂow at low Reynolds number,…

## References

SHOWING 1-10 OF 88 REFERENCES

### A New Family of Efficient Conforming Mixed Finite Elements on Both Rectangular and Cuboid Meshes for Linear Elasticity in the Symmetric Formulation

- MathematicsSIAM J. Numer. Anal.
- 2015

A new family of mixed finite elements is proposed for solving the classical Hellinger--Reissner mixed problem of the elasticity equations. For two dimensions, the normal stress of the matrix-valued…

### A Multigrid Preconditioner for the Mixed Formulation of Linear Plane Elasticity

- Computer ScienceSIAM J. Numer. Anal.
- 2006

A multigrid preconditioner for the discrete system of linear equations that results from the mixed formulation of the linear plane elasticity problem using the Arnold-Winther elements is developed.

### Mixed finite element methods for linear elasticity with weakly imposed symmetry

- MathematicsMath. Comput.
- 2007

New finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approxima- tions to both stresses and displacements are constructed.

### Mixed Kirchhoff stress–displacement–pressure formulations for incompressible hyperelasticity

- Mathematics
- 2021

### Code Generation for Generally Mapped Finite Elements

- Computer ScienceACM Trans. Math. Softw.
- 2019

This work describes how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite- element system, and evaluates the new elements, finding that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements.

### L2 best approximation of the elastic stress in the Arnold–Winther FEM

- Mathematics
- 2016

The first part of this paper enfolds a medius analysis for mixed finite element methods (FEMs) and proves a best-approximation result in L 2 for the stress variable independent of the error of the…

### Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow

- MathematicsJ. Num. Math.
- 2022

Two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow are proposed and it is rigorously proved that both methods are stable, result in optimal convergent numerical schemes with respect to appropriate mesh-dependent norms, although the chosen norms do not scale as the usual L2-norm.

### On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows

- Computer ScienceSIAM Rev.
- 2017

Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $H(div)$-conforming finite ...

### A family of conforming mixed finite elements for linear elasticity on triangular grids

- Mathematics
- 2014

This paper presents a family of mixed finite elements on triangular grids for solving the classical Hellinger-Reissner mixed problem of the elasticity equations. In these elements, the matrix-valued…

### An augmented Lagrangian preconditioner for implicitly-constituted non-Newtonian incompressible flow

- Computer ScienceSIAM J. Sci. Comput.
- 2020

An augmented Lagrangian preconditioner for a three-field stress-velocity-pressure discretization of stationary non-Newtonian incompressible flow with an implicit constitutive relation of power-law type is proposed, allowing for the simulation of a wide range of materials.