[1] General Introduction to PDEs

• Is (*) PDE linear? quasilinear? nonlinear?
• Is (*) PDE elliptic? parabolic? hyperbolic?
• When is a PDE well posed?
• What are the characteristics of this PDE?
• Are the boundary conditions specified correctly?
• How do the characteristics help determine if a PDE is well posed?
• What is the domain of dependence?
• What is the range of influence?
• Is (*) an advection equation?
• Is (*) the Laplace equation? a heat equation? a wave equation?

[2] M&M Parabolic in 1D

• Derive the analytical solution for the 1D Heat equation
• Write down the forward/backward/centered difference approximation
• Derive the local accuracy of a finite difference method
• Write down the theta method.
• For which theta is this method explicit?
• For which theta is this method unconditionally stable?
• For which theta is this method second-order accurate?
• Derive the local truncation error
• Is (*) finite difference method stable?
• Show (*) is stable using Fourier analysis
• Does (*) method have a maximum principle?
• Show (*) converges globally using a maximum principle.
• Are (*) boundary conditions Dirichlet? Neumann? Robin?
• Compute one step with method (*).

[3] M&M Parabolic in higher dimensions

• What is the ADI method?
• How stable/accurate/expensive is the ADI method?
• How does the ADI method compare to Crank Nicolson?
• How does the Explicit method compare to ADI and Crank Nicolson?
• Write down a finite difference approximation at a curved boundary.
• Compute the stability condition for method (*) using Fourier
• How sparse is the linear system solve for ADI and Crank Nicolson?

[4] M&M Hyperbolic Problems

• What is the upwind method for a quasilinear advection equation?
• How about the Lax-Wendroff method?
• What are the characteristics?
• What is the Courant-Friedrich-Lewy (CFL) condition?
• What is the domain of dependence?
• Compute the stability condition of (*) using Fourier
• What happens when characteristics cross?
• Write (*) PDE in conservation form.
• What equation does the weak solution satisfy?
• What is the shock speed for the weak solution?

[6] M&M Elliptic Problems

• Write down a finite difference method for Laplace/Poisson
• What is the local order of accuracy of (*) method?
• What is the sparse structure of the system of equations?
• Prove global convergence for centered difference
  for Poisson on a square domain with Dirichlet boundary conditions.
• For a general elliptic problem, what four conditions must be satisfied to prove convergence using Theorem 6.1?
• What is the global accuracy with curved boundaries (without proof)?
• What about Neumann boundary conditions (without proof)?

[1] A&B Quadratic Functionals

• What is a functional?
• What is a quadratic functional?
• What is a stationary point?
• What is a (strong) local/global minimizer?
• Compute the directional derivative of (*) functional

[2] A&B Variational Formulation I

• Derive the weak form of (*) in strong form
• Derive the Euler-Lagrange equations for (*) functional
• What is the Gauss divergence theorem?
• Integrate the 2d/3d integral (*) by parts to get the weak form.
• How do boundary conditions in the minimization form translate to boundary conditions in the strong form?
• Which boundary conditions are "natural" in minimization form?
• Which boundary conditions are "essential"?

[3] A&B Variational Formulation II

• Is (*) a vector space?
• Is (*) an inner product?
• Is (*) a norm?
• What is a Cauchy sequence?
• What is a Hilbert Space?
• When is a space "complete"?
• What is L₂
• What is the Cauchy-Schwarz inequality?
• Is (*) a bounded linear functional?
• State the Riesz representation theorem.
• When are two norms equivalent?
• Is C^(k) complete?
• Is H^(k) complete?
• What is H^(k)?
• When is a bilinear form "coercive"?
• State the Lax-Milgram Lemma
• Does (*) PDE satisfy Lax-Milgram? Prove or disprove.
• Why is Lax-Milgram so important?

[4] A&B Ritz-Galerkin Method

• What is a(u,v) = G(v) in minimization form?
• What is a(u,v) and G(v) for (*) in strong form?
• Given a finite subspace with basis functions, derive the discrete equations for a(u,v) = G(v).

[5] A&B Galerkin Finite Element Method

• What are the piecewise linear basis functions on a triangle?
• Construct the mapping to a standard triangle.
• Outline how you would assemble the global system of equations.
• When is numerical integration necessary?