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Synopsis: This course is appropriate for mathematics students who are interested in applications and numerical analysis, as well as other students with a strong mathematical background. The topic is computational fluid dynamics (CFD), focusing on the case of incompressible, viscous flows approximated using finite element methods. An introduction to some Hilbert spaces and functional-analytic concepts will be provided in conjunction with Galerkin-finite element methods to approximate fluid flows, and techniques to analyze the numerical properties of the ensuing algorithms. The partial differential equations (Navier-Stokes) used to model the fluid behavior are discussed, including both physical and analytical concepts. The Kolmogorov theory of turbulence is presented, along with methods to approximate the large scales of turbulent flows. Upon completion of the course, students should have a basic understanding of fluid mechanics, the Navier-Stokes equations, finite-element algorithms to approximate flows, and methods to analyze convergence and stability properties of algorithms. A computational project will be required, which will provide some hands-on CFD experience.Homework: Homework will be assigned periodically throughout the semester to cover the critical theoretical concepts for the course.