The focus in this course is to develop a theory and method of computationally solving general differential equations on general geometries. Instead of analytic closed form solutions using (combinations of) known functions, we will develop computational methods which numerically compute the solution using discrete basis functions (usually local polynomials -- "finite elements"). Unlike conventional numerical analysis courses, our main focus is more on formulating and computationally solving real world problems, and a bit less on e.g. error analysis, though this is brought up.
The course will start with simple ODE you may know from examples in calculus, e.g. population dynamics, 1D diffusion, and use these to introduce Galerkins method. Then we will move to solve several important PDE including the heat equation, wave equation, convection-diffusion problems, and elliptic eigenvalue problems. In the process we will cover basic concepts of function spaces as well as special numerical methods for solving the large equation systems resulting from Galerkin's finite element method (FEM). However, on these side topics the course is not as deep as a functional analysis course, nor as general as a typical introduction to numerical methods courses. Instead it is complementary to these courses.
The course examination will include homework problems throughout the term, some simpler programming/use of Matlab, and two exams.
For the majority of the course we will be using: