• Corpus ID: 239885780

Transformations for Piola-mapped elements

  title={Transformations for Piola-mapped elements},
  author={Francis R. A. Aznaran and Robert C. Kirby and Patrick E. Farrell},
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… 

Robust Approximation of Generalized Biot-Brinkman Problems

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

. 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,

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

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.



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

  • Jun Hu
  • Mathematics
    SIAM 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

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

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.

Finite element appoximation and augmented Lagrangian preconditioning for anisothermal implicitly-constituted non-Newtonian flow

2-field and 4-field finite element approximations of a system describing the steady state of an incompressible heat-conducting fluid with implicit non-Newtonian rheology with robust convergence behaviour when applied to the Navier-Stokes and power-law systems are devised.

Code Generation for Generally Mapped Finite Elements

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

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

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

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

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