Patrick E. Farrell’s research while affiliated with University of Oxford and other places

... where N i is the closure of the union of the elements adjacent to x i ; compare the multilevel version of R ⊕ in [8]. Let ψ h be the unique P 1 function with nodal values (R ⊕ ψ)(x i ). ...

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Surface elevation errors in finite element Stokes models for glacier evolution

... Nonetheless, this explicit matrix assembly becomes a computational and storage disadvantage at high discretization orders, where one would prefer a matrixfree approach. To alleviate this, [10,11] have introduced basis-change strategies that substantially improve complexity for relaxation on the patch systems associated with high-order discretization of spaces within the de Rham complex. However, these approaches have not yet been extended to the triangular patches considered here, and applying these methods to coupled problems remains unexplored, highlighting an interesting area for future research. ...

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Monolithic Multigrid Preconditioners for High-Order Discretizations of Stokes Equations

... 8.4]. More significantly, symmetric stress tensors are often crucial coupling variables between different subproblems in multiphysics, such as the Kirchhoff stress for electrical propagation in hyperelastic models of biomechanics [16], the Cauchy stress for thermomechanical coupling in viscoelasticity [34], or the viscous stress in multicomponent convection-diffusion [10]. ...

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Uniformly $hp$-stable elements for the elasticity complex

... In the context of the topology optimization of Stokes flow [9], the issue of approximating multiple minimizers with the finite element method was recently resolved [25][26][27][28]. It was shown that, for every isolated minimizer of the problem, there exists a sequence of finite element solutions to the first-order optimality conditions that strongly converges to the infinite-dimensional minimizer as the mesh size tends to zero. ...

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Numerical analysis of the SIMP model for the topology optimization of minimizing compliance in linear elasticity

... Similarly, researchers have recently started exploring smectic defects from a microscopic viewpoint. Using simulations and experiments, colloidal smectics have been confined within various geometries, such as rectangles (38,60,61), hexagons (44), circles (62)(63)(64), and annuli (64)(65)(66), establishing a correlation between the confining geometry and the types of defects formed. The defects were found to adhere to global, topological constraints (44,46,64,66). ...

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Curvature-directed anchoring and defect structure of colloidal smectic liquid crystals in confinement

... Recently, methods that aim at preserving multiple structures in the mixed finite element setting become increasingly popular. See, for example, the work of Hu et al. for incompressible MHD that preserves cross-and magnetic-helicity, energy and Gauss law of magnetism [5], the work of Gawlik and Gay-Balmaz for incompressible MHD of a variable fluid density that preserves energy, cross-helicity (when the fluid density is constant) and magnetic-helicity, mass, total squared density, pointwise incompressibility, and Gauss's law for magnetism [15], the work of Laakmann et al. [16] and references therein. These methods preserve several physical quantities of interest, but usually are less computationally efficient because of their large (nonlinear) discrete systems to be solved. ...

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Decoupled structure-preserving discretization of incompressible MHD equations with general boundary conditions

... (K; V) := u ∈ L 2 (K; V) : (u, r) Ω = 0 ∀ r ∈ RM .[1] The Hu-Zhang elements are Piola-inequivalent, so that implementation of their bases in Firedrake required special transformations after mapping from the reference cell[9]. Code, scripts, and exact software versions will be provided for reproducibility upon acceptance. ...

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Uniformly $hp$-stable elements for the elasticity complex

... In the aerospace industry, many types of dynamical system solutions have been considered and applied in design optimization. This includes equilibrium point [3,4], bifurcation [5,6,7,8,9], limit cycle oscillation [10,11,12,13,14,15], and chaotic systems [16,17], but these address a broad range of problems encountered in aircraft design. Much of the previous research focused on the optimization of a dynamical system operating at an equilibrium point. ...

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Adjoint-Based Design Optimization of Stability Constrained Systems

... In contrast, there has been significant work on the development of block-structured preconditioners [8,9,10] and monolithic multigrid algorithms for numerical solution of low-order discretizations of the Stokes, Navier-Stokes, and related equations. Monolithic multigrid methods have been considered with several families of relaxation scheme, including Vanka [11,12,13,14], Braess-Sarazin [15,16], Uzawa [17,18,19,20] and distributive [21,22,23,24] relaxation schemes. Many of these works have focused on extremely low-order discretizations, such as the marker-and-cell discretization or lowest-order Taylor-Hood discretizations. ...

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Achieving $h$- and $p$-robust monolithic multigrid solvers for the Stokes equations

... Regarding the well-posedness of the equations, there are different types of literature results: nonlinear friction boundary conditions [2], an implicitly given viscosity [3] with a differentiable shear-thinning term, or a differentiable shear-thinning term with Dirichlet boundary conditions, [4]. One recent publication with more general boundary conditions than we consider in two dimensions uses Newton's method in a finite-dimensional setting, see [5]. However, we need a combination of a differentiable, explicitly given shear-thinning viscosity, and nonlinear friction boundary conditions. ...

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Global convergence of Newton’s method for the regularized p-Stokes equations