- 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.