Numerical Analysis of PDE

Partial differential equations (PDEs) describe how quantities vary in space and time: heat conduction, waves, fluid flow, and much more. But turning a PDE into something a computer can work with is delicate: you must choose a discretization that is accurate, stable, and efficient, and you need theory to know when the computed results can be trusted. This course brings together the analysis of PDEs (existence, uniqueness, and regularity for linear PDEs) with the numerical methods used in practice. You study and compare finite difference and finite element methods for elliptic, parabolic, and hyperbolic equations, and learn how error estimates are derived. The course also covers finite element methods for elliptic eigenvalue problems and introduces the basic theory of numerical methods for semilinear equations. An important component is implementing finite element methods in modern software and analyzing the results in light of the underlying theory.