Giselle Saylor

I am an Assistant Professor in the Department of Mathematics and Statistics at Oakland University, Michigan. Previously, I was a Postdoctoral Researcher in the Department of Mathematics at the University of Houston where I worked with Dr. Loic Cappanera. In 2020, I obtained my Ph.D. in Applied Mathematics at the University of Waterloo in Canada under the supervision of Dr. Sander Rhebergen.

Contact me

Oakland University
Department of Mathematics and Statistics
Mathematics and Science Center
344 MSC
email: gsaylor at oakland dot edu

Research

My research interests lie in the design of finite element discretizations for the solution of partial differential equations with applications to realistic and challenging problems in science and engineering. For my projects, I implement novel discretization methods using open-source parallel codes to produce accurate and realistic simulations. Furthermore, I analyze the theoretical convergence of the methods under consideration, which involves rigorous mathematical proofs.

Multiphase flow in porous media

• Highlights. The modeling of multiphase flows in porous media is a computationally and theoretically challenging problem due to the high nonlinear nature of the problem. I am interested in developing and analyzing discontinuous Galerkin discretizations for the three-phase (aqueous, liquid and vapor) case. Our paper https://doi.org/10.1093/imanum/drac053 was the first one to show existence and uniqueness of a DG method for three-phase flows in porous media, and we provided error estimates. Then, in https://doi.org/10.1016/j.amc.2024.128909 we developed a second-order time stepping method for the two-phase flow problem that preserves the energy exactly. We are currently extending this method to the black oil problem, in which phase changes occur.

• Collaborators. Loic Cappanera (University of Ottawa), Catalin Trenchea (University of Pittsburgh), Maicon Correa (University of Sao Paulo), Mark Simmons (graduated from University of Houston).

Phase-field crystal modeling

• Highlights. The phase-field crystal equation is a sixth-order nonlinear PDE that models microstructure evolution of two- phase systems (crystal and liquid) on atomic length and diffusive time scales. The high order in the equation leads to a large linear system. Moreover, higher order in time is highly desirable to be able to capture the evolution of the phases properly, especially for long time simulations. In https://doi.org/10.1016/j.apnum.2026.03.003 we designed a novel hybridizable discontinuous Galerkin method based on reformulation of the PDE as a first order system. We proved existence and uniqueness of a solution and our numerical results demonstrate that with a significantly smaller number of degrees of freedom, our method can achieve comparable results than the state of the art literature. Currently, our Doctoral student David Hennicken is working on a second-order time stepping method that preserves energy exactly.

• Collaborators. Tamas Horvath (Oakland University), Natasha Sharma (University of Texas at El Paso), David Hennicken (current grad student at Oakland University).

Heat transfer due to radiation

• Highlights. The modeling of welding of plates using lasers involves a heat equation with radiation boundary conditions. I recently published a paper https://authors.elsevier.com/sd/article/S0377-0427(26)00379-1 in which I showed that a previously noted issue in which the convergence of the method slows down in some cases, is not due to loss of regularity of the solution, but rather it is an artifact of the linearization scheme. I am currently working on obtaining optimal error estimates, and also on extending the method to include advection effects and a time-dependent problem.

Publications

Teaching

Talks