Sun Lee: Multiple Solutions of Nonlinear Differential Equations

The study of nonlinear partial differential equations (PDEs) has led to significant advancements in fields such as physics, biology, ecology, and quantum mechanics. However, identifying multiple solutions to nonlinear PDEs remains a challenge, particularly when suitable initial guesses are not readily available. To address this issue, we introduce the Companion-Based Multilevel Finite Element Method (CBMFEM), a novel framework that utilizes polynomial-based companion matrices to efficiently generate multiple initial guesses. Our approach employs conforming finite elements on nested meshes to systematically refine and capture diverse solutions for semilinear elliptic equations with polynomial nonlinearities.

Bio: Sun Lee is a graduate student from Department of Mathematics at Penn State.