Patrick E. Farrell
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View article: An Augmented Lagrangian Preconditioner for the Control of the Navier–Stokes Equations
An Augmented Lagrangian Preconditioner for the Control of the Navier–Stokes Equations Open
We address the solution of the distributed control problem for the steady, incompressible Navier–Stokes equations. We propose an inexact Newton linearization of the optimality conditions. Upon discretization by a finite element scheme, we …
View article: Multiple solutions to the static forward free–boundary Grad–Shafranov problem on MAST-U
Multiple solutions to the static forward free–boundary Grad–Shafranov problem on MAST-U Open
The Grad–Shafranov (GS) equation is a nonlinear elliptic partial differential equation that governs the ideal magnetohydrodynamic equilibrium of a tokamak plasma. Previous studies have demonstrated the existence of multiple solutions to th…
View article: Time-harmonic waves in Korteweg and nematic-Korteweg fluids
Time-harmonic waves in Korteweg and nematic-Korteweg fluids Open
We derive the Helmholtz–Korteweg equation, which models acoustic waves in Korteweg fluids. We further derive a nematic variant of the Helmholtz–Korteweg equation, which incorporates an additional orientational term in the stress tensor. It…
View article: Analysis and numerical analysis of the Helmholtz-Korteweg equation
Analysis and numerical analysis of the Helmholtz-Korteweg equation Open
We analyse the nematic Helmholtz-Korteweg equation, a variant of the classical Helmholtz equation that describes time-harmonic wave propagation in calamitic fluids in the presence of nematic order. A prominent example is given by nematic l…
View article: Helicity-preserving finite element discretization for magnetic relaxation
Helicity-preserving finite element discretization for magnetic relaxation Open
The Parker conjecture, which explores whether magnetic fields in perfectly conducting plasmas can develop tangential discontinuities during magnetic relaxation, remains an open question in astrophysics. Helicity conservation provides a top…
View article: Time-harmonic waves in Korteweg and nematic-Korteweg fluids
Time-harmonic waves in Korteweg and nematic-Korteweg fluids Open
We derive the Helmholtz--Korteweg equation, which models acoustic waves in Korteweg fluids. We further derive a nematic variant of the Helmholtz-Korteweg equation, which incorporates an additional orientational term in the stress tensor. I…
View article: High-order finite element methods for three-dimensional multicomponent convection-diffusion
High-order finite element methods for three-dimensional multicomponent convection-diffusion Open
We derive and analyze a broad class of finite element methods for numerically simulating the stationary, low Reynolds number flow of concentrated mixtures of several distinct chemical species in a common thermodynamic phase. The underlying…
View article: An augmented Lagrangian preconditioner for the control of the Navier--Stokes equations
An augmented Lagrangian preconditioner for the control of the Navier--Stokes equations Open
We address the solution of the distributed control problem for the steady, incompressible Navier--Stokes equations. We propose an inexact Newton linearization of the optimality conditions. Upon discretization by a finite element scheme, we…
View article: Enforcing conservation laws and dissipation inequalities numerically via auxiliary variables
Enforcing conservation laws and dissipation inequalities numerically via auxiliary variables Open
We propose a general strategy for enforcing multiple conservation laws and dissipation inequalities in the numerical solution of initial value problems. The key idea is to represent each conservation law or dissipation inequality by means …
View article: Finite element methods for multicomponent convection-diffusion
Finite element methods for multicomponent convection-diffusion Open
We develop finite element methods for coupling the steady-state Onsager–Stefan–Maxwell (OSM) equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical …
View article: On multiple solutions of the Grad–Shafranov equation
On multiple solutions of the Grad–Shafranov equation Open
The Grad–Shafranov equation (GSE) for axisymmetric MHD equilibria is a nonlinear, scalar PDE which in principle can have zero, one or more non-trivial solutions. The conditions for the existence of multiple solutions has been little explor…
View article: Kinetic derivation of a compressible Leslie--Ericksen equation for rarified calamitic gases
Kinetic derivation of a compressible Leslie--Ericksen equation for rarified calamitic gases Open
Nematic ordering describes the phenomenon where anisotropic molecules tend to locally align, like matches in a matchbox. This ordering can arise in solids (as nematic elastomers), liquids (as liquid crystals), and in gases. In the 1940s, O…
View article: Discrete breathers in Klein–Gordon lattices: A deflation-based approach
Discrete breathers in Klein–Gordon lattices: A deflation-based approach Open
Deflation is an efficient numerical technique for identifying new branches of steady state solutions to nonlinear partial differential equations. Here, we demonstrate how to extend deflation to discover new periodic orbits in nonlinear dyn…
View article: Colloidal smectics in button-like confinements: Experiment and theory
Colloidal smectics in button-like confinements: Experiment and theory Open
Liquid crystals can self-organize into a layered smectic phase. While the smectic layers are typically straight, forming a lamellar pattern in bulk, external confinement may drastically distort the layers due to the boundary conditions imp…
View article: A full approximation scheme multilevel method for nonlinear variational inequalities
A full approximation scheme multilevel method for nonlinear variational inequalities Open
We present the full approximation scheme constraint decomposition (FASCD) multilevel method for solving variational inequalities (VIs). FASCD is a common extension of both the full approximation scheme (FAS) multigrid technique for nonline…
View article: Transformations for Piola-mapped elements
Transformations for Piola-mapped elements Open
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 util…
View article: Optimization of Hopf Bifurcation Points
Optimization of Hopf Bifurcation Points Open
We introduce a numerical technique for controlling the location and stability properties of Hopf bifurcations in dynamical systems. The algorithm consists of solving an optimization problem constrained by an extended system of nonlinear pa…
View article: Colloidal smectics in button-like confinements: experiment and theory
Colloidal smectics in button-like confinements: experiment and theory Open
Liquid crystals can self-organize into a layered smectic phase. While the smectic layers are typically straight forming a lamellar pattern in bulk, external confinement may drastically distort the layers due to the boundary conditions impo…
View article: Two-component three-dimensional atomic Bose-Einstein condensates supporting complex stable patterns
Two-component three-dimensional atomic Bose-Einstein condensates supporting complex stable patterns Open
We report the computational discovery of complex, topologically charged, and spectrally stable states in three-dimensional multicomponent nonlinear wave systems of nonlinear Schrödinger type. While our computations relate to two-component …
View article: Structural electroneutrality in Onsager–Stefan–Maxwell transport with charged species
Structural electroneutrality in Onsager–Stefan–Maxwell transport with charged species Open
We present a method to embed local electroneutrality within Onsager–Stefan–Maxwell electrolytic-transport models, circumventing their formulation as differential systems with an algebraic constraint. Flux-explicit transport laws are develo…
View article: Multigrid solvers for the de Rham complex with optimal complexity in polynomial degree
Multigrid solvers for the de Rham complex with optimal complexity in polynomial degree Open
The Riesz maps of the $L^2$ de Rham complex frequently arise as subproblems in the construction of fast preconditioners for more complicated problems. In this work we present multigrid solvers for high-order finite element discretizations …
View article: Structural electroneutrality in Onsager-Stefan-Maxwell transport with charged species
Structural electroneutrality in Onsager-Stefan-Maxwell transport with charged species Open
We present a method to embed local electroneutrality within Onsager-Stefan-Maxwell electrolytic-transport models, circumventing their formulation as differential systems with an algebraic constraint. Flux-explicit transport laws are formul…
View article: A new mixed finite-element method for H2 elliptic problems
A new mixed finite-element method for H2 elliptic problems Open
Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective t…
View article: Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes
Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes Open
In recent years a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are p…
View article: Variational and numerical analysis of a<b>Q</b>-tensor model for smectic-A liquid crystals
Variational and numerical analysis of a<b>Q</b>-tensor model for smectic-A liquid crystals Open
We analyse an energy minimisation problem recently proposed for modelling smectic-A liquid crystals. The optimality conditions give a coupled nonlinear system of partial differential equations, with a second-order equation for the tensor-v…