Trace Finite Element Methods for Surface Vector-Laplace Equations

In this paper we analyze a class of trace finite element methods (TraceFEM) for the discretization of vector-Laplace equations. A key issue in the finite element discretization of such problems is the treatment of the constraint that the unknown vector field must be tangential to the surface (``tangent condition''). We study three different natural techniques for treating the tangent condition, namely a consistent penalty method, a simpler inconsistent penalty method and a Lagrange multiplier method. A main goal of the paper is to present an analysis that reveals important properties of these three different techniques for treating the tangent constraint. A detailed error analysis is presented that takes the approximation of both the geometry of the surface and the solution of the partial differential equation into account. Error bounds in the energy norm are derived that show how the discretization error depends on relevant parameters such as the degree of the polynomials used for the approximation of the solution, the degree of the polynomials used for the approximation of the level set function that characterizes the surface, the penalty parameter and the degree of the polynomials used for the approximation of Lagrange multiplier

A Trace Finite Element Method for Vector-Laplacians on Surfaces

We consider a vector-Laplace problem posed on a 2D surface embedded in a 3D domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and analysis of a finite element method for the discretization of this surface partial differential equation. We apply the trace finite element technique, in which finite element spaces on a background shape-regular tetrahedral mesh that is surface-independent are used for discretization. In order to satisfy the constraint that the solution vector field is tangential to the surface we introduce a Lagrange multiplier. We show well-posedness of the resulting saddle point formulation. A discrete variant of this formulation is introduced which contains suitable stabilization terms and is based on trace finite element spaces. For this method we derive optimal discretization error bounds. Furthermore algebraic properties of the resulting discrete saddle point problem are studied. In particular an optimal Schur complement preconditioner is proposed. Results of a numerical experiment are included

Iron-induced relaxation mechanisms in the human substantia nigra: Towards quantifying iron load in dopaminergic neurons

Pathological iron accumulation in the human brain is a biomarker for neurodegeneration. Several diagnostically promising MR- based methods for in vivo iron quantification were proposed, based on the empirical relationship between R 2 * and iron concentration. However, these do not account for different chemical forms and cellular distribution of iron. We combined post mortem MRI, advanced quantitative histology and biophysical modeling to develop a generative theory linking obtained iron concentrations to quantitative MR parameters. The impact of nanoscale molecular interaction of water with iron and of iron-rich dopaminergic neurons was quantified in substantia nigra

Finite element discretization methods for velocity-pressure and stream function formulations of surface Stokes equations

In this paper we study parametric TraceFEM and parametric SurfaceFEM (SFEM) discretizations of a surface Stokes problem. These methods are applied both to the Stokes problem in velocity-pressure formulation and in stream function formulation. A class of higher order methods is presented in a unified framework. Numerical efficiency aspects of the two formulations are discussed and a systematic comparison of TraceFEM and SFEM is given. A benchmark problem is introduced in which a scalar reference quantity is defined and numerically determined.Comment: 26 page

The Heart Game:A New Tool for Digital Patient Education for Patients With Heart Failure

Background The COVID-19 pandemic has shown the need for new ways to perform remote patient education. Patients with heart failure are associated with a high readmission rate. Rehabilitation can prevent hospital readmissions, but there is poor participation in rehabilitation of patients with heart failure. Based on user-driven innovation in the Future Patient research project, we have developed the prototype of the digital Heart Game. The Heart Game is a new approach to digital patient education, where patients can access a digital board game via an app; through the board game, they can be presented with quizzes, activities, and reflection questions, all related to heart failure. Objective The objective of our study was to assess the usability of the Heart Game prototype app. Methods A total of 6 patients with heart failure were recruited. Think-aloud test, where participants were observed during play and sound and iPad screens were recorded, was conducted, followed by a questionnaire immediately after playing. Finally, an interview (n=6) was carried out. Qualitative data were analyzed in NVivo software (version 12.0). Results In total, 6 patients with heart failure (83% male; mean age 66 years) participated in the think-aloud tests. One participant did not complete the game. The game duration was from 9 minutes and 14 seconds to 16 minutes and 13 seconds. Findings from the think-aloud tests were shown in themes: digital games are a new world; practice makes perfect; the Heart Game is illogical; tasks are not generalizable, in understandable formulations, or entertaining; and the Heart Game is relevant, fun, and entertaining to play. Conclusions The usability of the Heart Game shows opportunities for digital patient education. However, there are some challenges in the prototype with illogical game structures in the board game. Further test and development of the prototype needs to be performed. </jats:sec

Location, Location… Mailing Location? The Impact of Address as a Signal

Recently, service providers have begun offering an innovative option for consumers seeking to rent an address. While the traditional option was for a post office box (e.g., PO Box 203, Chicago, IL 60654), the new option features a real street address with a unique suite number (e.g., 301 West Grand Avenue, Suite 203, Chicago, IL 60654). Service providers claim that addresses affect consumer perceptions of credibility and professionalism; however, these arguments have never been substantiated. Therefore, this research aims to examine the relationship between addresses and consumer evaluations of small businesses. Across a series of five experiments, we find evidence in support of the service providers’ claims and apply signaling theory to show why consumers evaluate a small business with a street and suite address more favorably