Evolving surface finite element method for the Cahn-Hilliard equation

We use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport for- mulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subse- quence, of the finite element scheme. We conclude the paper by deriving error estimates and present various numerical examples

The Blackmail of Democracy: A genealogy of British/Pakistani democracy promotion

This thesis begins from the premise that identity is only possible as a function of difference. If someone is British, That is because they are not French or Pakistani. What matters, however, is not the fact of these divisions but how they operate and with what consequences. For contemporary practices of thought, the identification of others by means of temporal distinctions has become extremely important. To explore this, I work genealogically to draw on empirical material from colonia and post-­‐colonial Britain and Pakistan, including legislation, political discourse, government projects and broader cultural representations. I make two main arguments. First, I show the importance of these modes of “temporal othering”. I empirically examine the temporal distinctions that constitute a British, democratic, national identity by dint of positing an “other” that is barbaric, alien, despotic, violent and – most importantly – backward. It is in encountering and constantly re-­‐narrating these threats to democracy that the British come to have a sense of an imagined, democratic community that has emerged -­‐ through a seamless, progressive history -­‐ by virtue of what it is opposed to. Relatedly, democracy is understood as the endpoint of history, with consequences for overseas Democracy Promotion. Second, I argue that it is possible to narrate alternative versions of history. In examining the emergence of such teleological versions of history, I show that teleology isn’t the driving force of history, but rather emerges from the messiness of historical events. Furthermore, the practices that it legitimates are deeply involved in promoting the violence and social marginalisation for which democracy is thought to be the remedy. However, I show that the version of history that currently pervades practices of thought about British identity and democracy promotion is contestable and that therefore it might be possible to think, act and live differently

POD for optimal control of the Cahn-Hilliard system using spatially adapted snapshots

The present work considers the optimal control of a convective Cahn-Hilliard system, where the control enters through the velocity in the transport term. We prove the existence of a solution to the considered optimal control problem. For an efficient numerical solution, the expensive high-dimensional PDE systems are replaced by reduced-order models utilizing proper orthogonal decomposition (POD-ROM). The POD modes are computed from snapshots which are solutions of the governing equations which are discretized utilizing adaptive finite elements. The numerical tests show that the use of POD-ROM combined with spatially adapted snapshots leads to large speedup factors compared with a high-fidelity finite element optimization

Evolving surface finite element method for the Cahn-Hilliard equation

We use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport for- mulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes accord- ing to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subse- quence, of the finite element scheme. We conclude the paper by deriving error estimates and present various numerical examples

A computational approach to an optimal partition problem on surfaces

We explore an optimal partition problem on surfaces using a computational approach. The problem is to minimise the sum of the first Dirichlet Laplace--Beltrami operator eigenvalues over a given number of partitions of a surface. We consider a method based on eigenfunction segregation and perform calculations using modern high performance computing techniques. We first test the accuracy of the method in the case of three partitions on the sphere then explore the problem for higher numbers of partitions and on other surfaces

Mapping the autistic advantage from the accounts of adults diagnosed with autism: A qualitative study

This is the final version. Available on open access from Mary Ann Liebert via the DOI in this recordBackground: Autism has been associated with specific cognitive strengths. Strengths and weaknesses have traditionally been conceptualized as dichotomous. Methods: We conducted 28 semi-structured interviews with autistic adults. Maximum variation sampling was used to ensure diversity in relation to support needs. We asked which personal traits adults attributed to their autism, and how these have helped in the workplace, in relationships, and beyond. Data were collected in two stages. Responses were analyzed using content and thematic techniques. Results: The ability to hyperfocus, attention to detail, good memory, and creativity were the most frequently described traits. Participants also described specific qualities relating to social interaction, such as honesty, loyalty, and empathy for animals or for other autistic people. In thematic analysis we found that traits associated with autism could be experienced either as advantageous or disadvantageous dependent on moderating influences. Moderating influences included the social context in which behaviors occurred, the ability to control behaviors, and the extent to which traits were expressed. Conclusions: Separating autistic strengths from weaknesses may be a false dichotomy if traits cannot be isolated as separate constructs of strengths or deficits. If attempts to isolate problematic traits from advantageous traits are ill conceived, there may be implications for interventions that have reduction in autistic traits as a primary outcome measure.Wellcome Trus

Unfitted finite element methods using bulk meshes for surface partial differential equations

In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the n-dimensional hypersurface, Γ⊂Rn+1, is embedded in a polyhedral domain in Rn+1 consisting of a union, Th, of (n+1)-simplices. The finite element approximating space is based on continuous piece-wise linear finite element functions on Th. Our first method is a sharp interface method, \emph{SIF}, which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, Γh, of Γ. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes \emph{SIF} from the method of [42]. The second method, \emph{NBM}, is a narrow band method in which the region of integration is a narrow band of width O(h). \emph{NBM} is similar to the method of [13]. but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface L2 and H1 norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection diffusion conservation law. Numerical results are given which illustrate the rates of convergence

Evolving surface finite element methods for random advection-diffusion equations

In this paper, we introduce and analyse a surface finite element discretization of advection-diffusion equations with uncertain coefficients on evolving hypersurfaces. After stating unique solvability of the resulting semi-discrete problem, we prove optimal error bounds for the semi-discrete solution and Monte Carlo samplings of its expectation in appropriate Bochner spaces. Our theoretical findings are illustrated by numerical experiments in two and three space dimensions

Stability and error analysis for a diffuse interface approach to an advection-diffusion equation on a moving surface

In this paper we analyze a fully discrete numerical scheme for solving a parabolic PDE on a moving surface. The method is based on a diffuse interface approach that involves a level set description of the moving surface. Under suitable conditions on the spatial grid size, the time step and the interface width we obtain stability and error bounds with respect to natural norms. Furthermore, we present test calculations that confirm our analysis