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

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..