# Differential Equations Seminar

## Francesca R. McFadden, Kevin Williamson, UMBC

Monday, December 5, 2016

11:00 AM - 12:00 PM

11:00 AM - 12:00 PM

Mathematics/Psychology : 401

**Speaker: Francesca R. McFadden**

**Title:**Computational complexity of exact stochastic simulation algorithms for spatially-distributed reaction-diffusion systems

**Abstract:**In this talk, we discuss exact stochastic simulation algorithms for modeling progression of reaction-diffusion systems. We describe algorithms meant to improve the computational complexity for domains showing spatial variation, especially when the spatial domain is subdivided into a large number of numerical cells. Theoretical comparisons of different strategies to get better efficiency are discussed and computational comparisons are shown for a simple example in the quasi-1D domain. Application to 2D and 3D domains is discussed. Different algorithms used are from Hu, Kang, and Othmer (2014).

**Speaker: Kevin Williamson**

**Title:**Numerical Solution of the Stokes-Brinkman Equation Using Mixed Finite Elements

**Abstract:**Flow in porous media has numerous applications, to include reservoir simulation, nuclear waste disposal, and carbon sequestration. Simulations are complicated due to a heterogeneous domain (varying porosity) and the presence of fractures and vugs. The Stokes-Brinkman equation combines Darcy's law (which models the porous region) with Stokes (which models the free-flowing region). This talk will begin with an overview of Stokes-Brinkman and then describe the discretization of the system using mixed finite elements. The resulting linear system is ill-conditioned and requires a preconditioner to speed up convergence to a solution. Two simple preconditioners will be presented and compared.