For course description and justification, please see Syllabus and Comments.
Familiarity with basic undergraduate numerical analysis and partial differential equations are assumed. Also, basic concepts from real analysis (Inner product space, normed spaces, Banach and Hilbert spaces) are also needed.
Review of relevant topics from PDEs and Analysis. Motivation and showcasing of applications
from various branches of mathematics.
Divided differences. Finite difference methods for parabolic and hyperbolic PDEs.
Weak derivatives and function spaces. Variational formulation of PDEs.
Galerkin projection and the construction of finite element methods for elliptic problems.
Error analysis for finite element methods.
Finite element methods for parabolic problems.
Discontinuous Galerkin methods and basic finite volume concepts for hyperbolic PDEs.
Numerical methods for nonlinear PDEs.
A brief excursion to linear and nonlinear solvers.
Applications.