[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] B Weak Formulation I

• What is a functional?
• What is a vector space? Is (*) a vector space?
• What is an inner product? Is (*) an inner product?
• What is the Cauchy-Schwarz inequality?
• What is a norm? Is (*) a norm?
• What is complete vector space?
• Which vector spaces are complete?
• Is C^(k) complete?
• Is H^(k) complete?
• What is a Hilbert Space? Give an example of a Hilbert Space.
• What is L₂?
• What is H^(k)?
• Compute the L₂-norm and inner-product of (*).
• Compute the H^(k)-norm and inner-product of (*).
• Apply the Poincare-Friedrichs inequality.
• What is a quadratic functional?
• What is a stationary point?
• Compute the directional derivative of (*) functional
• What are the conditions for Lax-Milgram?
• State the Riesz representation theorem.
• Is (*) a bounded linear functional?
• Is (*) a bounded and coercive bilinear form?
• Why is Lax-Milgram so important?

[2] B Weak Formulation II

• 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"?
• Show that (*) satisfies the conditions for Lax-Milgram.
• Is (*) domain Lipschitz? Does (*) satisify the cone condition?
• What is the trace theorem?

[3] 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).
• Show how to handle non-homogeneous boundary conditions.
• 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?
• Apply a simple quadrature rule.

[4] B Error Estimates

• Is (*) a conforming mesh?
• Is (*) a uniform mesh?
• What is the diameter of (*)?
• Is (*) shape regular?
• Are C^(k-1) finite elements in H^(k)?
• What does Céa's Lemma tell us?
• Show the finite element approximation is bounded
• How accurate are H^(k) finite elements?

[5] B Iterative Methods

• Write down the non-linear weak form for (*).
• Write down the linearized bilinear form for (*).
• What would you solve to take one step of Newton's method?
• When does the iteration M x^(k+1) = N^(k) + b converge?
• Take one step of Jacobi/Gauss-Seidel for a 2 x 2 problem.
• Is (*) diagonally dominant?
• How many steps does it take for CG to converge?
• Why is Jacobi and Gauss-Seidel so slow for many large problems?
• How does Multigrid smooth error on all frequencies?