Project Grant 2208289

This three-year National Science Foundation project grant of $188,904 will support the development of new finite element methods for numerically challenging computations at Oklahoma State University from September 2022 through August 2025. Under the Mathematical and Physical Sciences program (CFDA 47.049), the university will advance computational mathematics and provide more efficient tools for problems involving interfaces between multiple physical systems, materials with changing properties, and other difficulties faced by current methods. Specifically, the project aims to create a finite element method that directly approximates dual variables without approximating primary variables, resulting in smaller problem sizes and symmetric positive-definite matrices. The researchers will derive error estimates including adaptive posteriori techniques and apply the new dual-based approach to problems in discontinuous coefficients, sign-changing coefficients, linear elasticity, and Stokes equations. If successful, the method could have wide-ranging applications in fields utilizing finite element modeling, from biomedical to structural engineering.