CAREER: Primal-Dual Weak Galerkin Finite Element Methods

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Partial differential equations (PDEs) are an important mathematical tool for modeling many scientific problems and the most common approach for solving these equations usually involves numerical methods. The goal of this project is to extend the theory and develop novel numerical methods of one such method known as the PD-WG algorithm. The resulting computational codes will be made publicly available. In addition, the PI will apply the new methods to several problems in biology and physical sciences. One exciting application is the mathematical modeling of ion channels which are proteins with a hole down their middle, and which serve an important function as gatekeepers for cells. The project also has an educational component that integrates the PI's research activities with training of students at all levels, from K-12, undergraduate, to graduate students, with a focus on serving underrepresented minorities.The goal of this project is to develop novel numerical methods through a coupling of the original/primal governing equations and their dual. The resulting numerical methods are known as "Primal-Dual Weak Galerkin (PD-WG) finite element methods". The main research of this CAREER project will be focused on five primary areas: (1) developing robust numerical schemes by using PD-WG techniques for various PDE problems including the elliptic Cauchy problem, convection dominated convection-diffusion equations, and general linear or nonlinear PDEs; (2) establishing stability and mathematical convergence (including superconvergence) theory for the newly developed PD-WG methods by developing new mathematical tools; (3) numerical simulations for biological problems modeled by the nonlinear Poisson-Nernst-Planck (PNP) equations; (4) devising new concepts in numerical PDEs by integrating key ideas of optimization with WG methods; and (5) developing application-oriented software packages using the PD-WG methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.