Inequalities: for each inequality give its statement, proof (at least in some simplified case), and examples where it is used
trace inequality
Sobolev's inequality
Poincaré's inequality
Friedrichs' inequality
Methods: for each method discuss its strengths and weaknesses, what kinds of results can be proved with it, give examples where it is used
maximum principle
Fourier transform
spectral method (eigenfunction expansion, Fourier series)
energy method
method of characteristics
make sure you have good skills in using the energy method and the
spectral method
Weak formulation: give several examples of the weak formulation of elliptic boundary value problems and apply the Lax-Milgram lemma
Existence, uniqueness and regularity:
give the statements and proofs of existence, uniqueness and regularity of solutions for several examples of elliptic, parabolic and hyperbolic problems
describe the difference between elliptic, parabolic and hyperbolic problems concerning regularity of solutions
explain the role of regularity in numerical analysis
Stability:
give examples of stability estimates for differential equations and
numerical methods
give examples of stability conditions for numerical methods
explain the role of stability in numerical analysis
Error estimates:
give examples, including proofs
Older exams:
901101
solutions,
921104
solutions,
931104
solutions,
940108
These exams were written for another course which only covered
time-dependent problems.