The Optimal Relaxation Parameter for the SOR Method Applied to the Poisson Equation in Any Space Dimensions

Abstract—The finite difference discretization of the Poisson equation with Dirichlet boundary conditions leads to a large, sparse system of linear equations for the solution values at the interior mesh points. This problem is a popular and useful model problem for performance comparisons of iterative methods for the solution of linear systems. To use the successive overrelaxation (SOR) method in these comparisons, a formula for the optimal value of its relaxation parameter is needed. In standard texts, this value is only available for the case of two space dimensions, even though the model problem is also instructive in higher dimensions. This note extends the derivation of the optimal relaxation parameter to any space dimension and confirms its validity by test calculations in three dimensions

Examining the Electrical Excitation, Calcium Signaling, and Mechanical Contraction Cycle in a Heart Cell

As the leading cause of death in the United States, heart disease has become a principal concern in modern society. Cardiac arrhythmias can be caused by a dysregulation of calcium dynamics in cardiomyocytes. Calcium dysregulation, however, is not yet fully understood and is not easily predicted; this provides motivation for the subsequent research. Excitation-contraction coupling (ECC) is the process through which cardiomyocytes undergo contraction from an action potential. Calcium induced calcium release (CICR) is the mechanism through which electrical excitation is coupled with mechanical contraction through calcium signaling. The study of the interplay between electrical excitation, calcium signaling, and mechanical contraction has the potential to improve our understanding of the regular functioning of the cardiomyocytes and help us understand how any dysregulation can lead to potential cardiac arrhythmias. ECC, of which CICR is an important part, can be modeled using a system of partial differential equations that link the electrical excitation, calcium signaling, and mechanical contraction components of a cardiomyocyte. We extend a previous model [Angeloff et al., Spora, 2016] to implement a seven-variable model that includes for the first time the mechanical component of the ECC. We study how the interaction of electrical and calcium systems can impact the cardiomyocyte\u27s levels of contraction

Examining the Effect of Introducing a Link from Electrical Excitation to Calcium Dynamics in a Cardiomyocyte

Calcium dysregulation is a significant cause of fatal cardiac arrythmias, but it is an incompletely understood phenomenon and difficult to predict. Cardiac calcium levels can be modeled as a system of partial differential equations linking the electrical excitation, calcium signaling, and mechanical contraction dynamics of the heart. A complete calcium induced calcium release model uses reaction-diffusion equations to fully link these three systems. Simulations examine the effect of introducing the link from calcium signaling to electrical excitation. In particular, we perform a parameter study on the strength of the feedback connection with both links between calcium signaling and electrical excitation enabled. Simulations indicate that the feedback and feedforward between electrical excitation and calcium signaling can influence the voltage in a physiologically realistic way

Uniform resolvent convergence for strip with fast oscillating boundary

In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin boundary condition. In all cases we describe the homogenized operator, establish the uniform resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. These results are obtained as the order of the amplitude of the oscillations is less, equal or greater than that of the period. It is shown that under the homogenization the type of the boundary condition can change

Estrategias pedagógicas de acompañamiento pastoral en el caso de familias con violencia intrafamiliar

La violencia intrafamiliar es un fenómeno que aqueja a un gran número de familias en Colombia, los problemas sociales, económicos y familiares han sido factores importantes para que la familia se desintegre y caiga en una pérdida de valores que no permite la sana convivencia en el hogar. Es así, como se encuentra que la Institución Educativa Yermo y Parres de Bogotá, no es un caso aislado a dicho fenómeno, por el contrario, se observa que existen varias familias que viven a diario la violencia intrafamiliar en este contexto educativo. En ese orden de ideas, el siguiente proyecto propende trazar un mapa que evidencie de la manera en que las estrategias pedagógicas, de acompañamiento pastoral, que ayudan a la resolución de conflictos y violencia intrafamiliar de algunas familias de los estudiantes de grado 8° de la institución Educativa Yermo y Parres en la zona de Engativá de Bogotá. De igual manera busca diseñar una estrategia de acompañamiento pastoral, basada en una cartografía social, en donde se pueden analizar aspectos importantes como los actores, espacios, factores entre otros, que generan la violencia intrafamiliar en el grupo de familias intervenido y de la misma forma, a través de la capacitación , para la resolución de conflictos, promoviendo la solidaridad con las familias; Esta cartografía social se manifiesta como contribución a la institución educativa, para implementarla no solo desde la escuela de padres, sino como generador para futuros proyectos.Recursos propiosComunidad Educativa Colegio Yermo y Parres grado octavoDomestic violence is a phenomenon that afflicts a large number of families in Colombia, social, economic and family problems have been important factors for the family to disintegrate and fall into a loss of values that does not allow the healthy coexistence in the home. This is how it is found that the Yermo and Parres Educational Institution of Bogotá, is not an isolated case to this phenomenon, on the contrary, it is observed that there are several families that live daily intrafamily violence in this educational context. In that order of ideas, the following project tends to draw a map that evidences the way in which the pedagogical strategies, of pastoral accompaniment, affect in a positive way the resolution of conflicts and intrafamily violence of some families of the 8th grade students. the Educational Institution Yermo y Parres in the Engativá area of Bogotá. Likewise, it seeks to design a strategy of pastoral accompaniment, based on a social cartography, where important aspects such as actors, spaces, factors, among others, can be analyzed that generate intrafamily violence in the group of families intervened and in the same way , through training, to resolve conflicts, promoting solidarity with families; This social cartography is manifested as a contribution to the educational institution, to implement it not only from the school of parents, but as a generator for future projects.Licenciado (a) en Ciencias ReligiosasPregrad

A Homogenization Technique for the Development of Mesoscopic Scale Models for Chemical Vapor Deposition

This dissertation presents a problem arising from the simulation of gas flow over microstructured surfaces. For the industrial application under consideration, the problem is appropriately given as a time-dependent nonlinear reaction-diffusion equation on a domain, which includes a flux condition on a boundary surface consisting of a microscopic fine structure. An equivalent problem for the bulk solution is derived, which incorporates all physical quantities of interest and which is accessible to efficient numerical simulation at the same time. This is achieved by applying a homogenization technique to the boundary condition, which eliminates the microscopic scale while retaining its effect on the bulk solution. The derivation presented in this dissertation is valid for a three-dimensional domain and a general boundary surface given in parameterized form. The underlying application area in semiconductor manufacturing is the modeling of chemical vapor deposition in single wafer reactors..