This is a one-semester course on numerical methods for partial
differential equations. The course will focus on basic concepts of the
finite element method for elliptic boundary value problems. Topics
include: the weak (variational) formulation of protoypical problems,
coercivity and continuity arguments, inf-sup conditions, approximation
theory of finite elements, error analysis, stability, and a discussion
of variational "crimes".
Further, some basic techniques of finite differences and finite
volumes will be introduced and discussed. Maximum principle, energy
type estimates, and Fourier mode analysis will be used for studying
the stability and accuracy of these methods applied to elliptic,
parabolic and hyperbolic problems. The intention is to develop a
fairly strong mathematical knowledge and expertise so that after
completing the class the students can read and understand quite
advances papers on finite element methods for PDEs (and partially
for finite differences, and finite volumes). The programming
assignments will emphasize applications of the numerical techniques
to engineering problems.
Though this course covers most parts of the numerical analysis part
of the qualifying exam in ApplMath/NA it is intended for
engineering students. I recommend the Spring term for math students
who will try to pass the ApplMath/NA qualifier.