SC doctoral students James Cheung and Evan Cresswell are winners of the 2017 SIAM-SEAS awards for best graduate oral presentation and graduate poster presentation. Based on his paper entitled Boundary Condition Approximation for Lagrange Finite Elements by Polynomial Extension, Cheung’s award for best oral presentation recognizes effective presentations delivered by impactful, confident and engaging speakers. Cresswell’s award for best graduate poster presentation was for his research, Computational Model for Local Calcium Dynamics in Astrocytes.

Abstracts of Cheung’s and Cresswell’s award winning research presentations are below.

James Cheung

Boundary Condition Approximation for Lagrange Finite Elements by Polynomial Extension

Curved domains pose a great problem to high order finite element methods because the accuracy of a numerical solution ultimately depends on how well the discrete domain approximates the continuous domain. As such, the majority of the higher order finite element codes utilize some sort of coordinate transformation to fit a polygonal mesh to the curved domain up to an acceptable order of accuracy. While these methods can recover optimal theoretical accuracy for the underlying approximation space, they are often difficult to implement. In this talk, we present a new approach to recovering optimal accuracy for higher order finite elements with Lagrange elements for Dirichlet and Neumann problems for scalar elliptic PDE on curved domains. Instead of cumbersome coordinate transforms, we utilize a simple Taylor series to extend the underlying polynomial approximation space to approximate the given boundary condition on the curved boundary. This approximation allows us to retain our polygonal mesh, while at the same time, achieve optimal accuracy.

Evan Cresswell

Computational Model for Local Calcium Dynamics in Astrocytes

Several contemporary studies show that astrocytes, a type of glial cell, are fundamental to several neural functions ranging from metabolic support to higher cognition such as recollection memory. This has resulted in the introduction of astrocytic dynamics into neural modeling. Most cellular function in the astrocyte is triggered by an increase or decrease in calcium concentration within the cytosol. Previous work considered astrocytic dynamics by representing calcium concentration as a point source or a completely spatial model in the cell. We now know, more than ever, that the role of the astrocyte takes many different perspectives. This work, which is inspired by in vivo recordings of astrocytes in the ferret visual cortex, puts forward a novel approach to modeling the different levels of astrocytic calcium activity in the astro-neural system. In the model, we introduce a compartmentalized astrocyte with the purpose of understanding intra-cellular calcium dynamics. Compartmentalizing the astrocyte’s cytosol into the soma and individual branches captures the effect of the local dynamics while still holding on to the analytical power of ODE’s. With this model we investigate the interaction between local and global cellular dynamics within the astrocyte in response to neural activity. This allows us to better understand the effect that astrocytes can have on both individual neurons and populations of neighboring neurons.