Ari Stern, associate professor of mathematics and statistics, won a $237,648 grant from the National Science Foundation for a project titled “Structure-Preserving Hybrid Finite Element Methods.” Stern’s work aims to produce new computational methods and improved understanding of current methods for a wide variety of high-value scientific applications, with important ramifications for computational physics and engineering.
Finite element methods are used to solve computing problems in areas ranging from materials and manufacturing to biomedical engineering and biotechnology. By approximating solutions to the equations that describe scientific problems, researchers can then simulate the behavior of complex physical systems. However, classical methods currently used by scientists and engineers aren’t always able to capture fundamental features of the systems they’re meant to simulate.
To address this shortcoming, Stern is expanding his investigations into hybrid methods, which he has already shown can be used to solve complex problems by breaking them up into smaller pieces, then using additional variables to “glue” the pieces back together. Stern explains that these methods are “hybrid” because they involve two kinds of variables – those in the original problem and those introduced as glue. This combined approach promises advantages in both performance and accuracy over classical methods.
“Over the last few decades, there hasn't been much overlap between the research community working on high-performance finite element methods and the one working on ‘structure-preserving’ methods for conservative systems,” Stern said. “One of the biggest impacts of my research program will be in bringing these two lines of research together and using that to develop methods that are both high-performance and structure-preserving.”
The specific advantages of Stern’s hybrid methods include increased computational efficiency, additional insights about the system gleaned from the “glue” variables, and greater accuracy of the simulation with little additional computational effort.
“I'm especially excited about applications in computational electromagnetics, where my collaborators and I have just introduced some new hybrid methods that I hope to see more widely adopted,” Stern said. “We already have some exciting results about their structure-preserving properties, but there's still a lot more to explore.”
