Mathematics Commons^™

Signal Transmission In Epithelial Layers, Filippo Posta

Dissertations

Cell signaling is at the basis of many biological processes such as development, tissue repair, and homeostasis. It can be carried out by different mechanisms. Here we are focusing on ligand mediated cell-to-cell signaling in which a molecule (ligand) is free to move into the extra-cellular medium. On the cell layer surface, it can bind to its molecule-specific receptors located on the cell plasma membrane. This mechanism is the subject of many experimental and theoretical studies on many model biological systems, such as the follicular epithelium of the Drosophila egg, which motivates this work.

Here, we present a general mathematical …

An Optimal-Order Error Estimate For A Family Of Ellam-Mfem Approximations To Porous Medium Flow, Hong Wang

Faculty Publications

Mathematical models used to describe porous medium flow lead to coupled systems of time-dependent nonlinear partial differential equations, which present serious mathematical and numerical difficulties. Standard methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion along with spurious grid-orientation effect. The ELLAM-MFEM time-stepping procedure, in which an Eulerian–Lagrangian localized adjoint method (ELLAM) is used to solve the transport equation and a mixed finite element method (MFEM) is used for the pressure equation, simulates porous medium flow accurately even if large spatial grids and time steps are used. In this paper we prove an optimal-order error estimate for …

Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

The finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither Hermitian nor definite when the coefficient function is complex valued. For such problems, under a uniqueness assumption, we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero. Then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number, providing an efficient solver for computing discrete approximations. Numerical experiments, while confirming the theory, …