[1] General Introduction to PDEs

• Classification of first order PDEs
  Linear vs. Quasilinear vs. Nonlinear
• Classification of second order PDEs
  Linear vs. Quasilinear vs. Nonlinear
  Elliptic vs. Parabolic vs. Hyperbolic
  Time dependent vs. independent
  Steady State vs. No Steady State
• Example PDEs:
  Advection Equation, Laplace Equation
  Heat Equation, Wave Equation
• Definitions:
  Well Posed
  Characteristic
  Domain of Dependence
  Range of Influence
• How to Compute:
  Characteristics of a first-order PDE
  Characteristics for the wave equation

[2] M&M Parabolic in 1D

• Heat Equation:
  Exact solution
• Finite Differences:
  Forward vs. Backward vs. Centered
  Accuracy using Taylor Series
• Numerical Methods:
  Explicit Method, Implicit Method
  Theta Method, Crank Nicolson
  Method of Lines
• Error and Stability Analysis:
  Truncation Error
  Fourier Analysis
  Maximum Principle
  Global error vs. Truncation Error
• Boundary Conditions:
  Dirichlet, Neumann, Robin
• More general Parabolic problems:
  Setup explicit/implicit methods
  Cost of each timestep
  Upwind scheme
• How to compute:
  Stability using Fourier
  Convergence using maximum principle
  One step with an explicit method

[3] M&M Parabolic in higher dimensions

• Stability of Methods:
  Explicit vs. Crank Nicolson vs. ADI
• Accuracy of Methods:
  Explicit vs. Crank Nicolson vs. ADI
• Cost of Methods:
  Explicit vs. Crank Nicolson vs. ADI
• More general boundaries:
  Dirichlet on a curved boundary
• How to Compute:
  Stability using Fourier
  One step with an explicit method
  Discrete equations at curved boundary

[4] M&M Hyperbolic Problems

• Stability:
  Characteristics
  Courant-Friedrich-Lewy (CFL)
  Domain of Dependence
• Fourier Analysis:
  Upwind Method
  Centered Difference
• Existence and Uniqueness
  Crossing of Characteristics
  Conservation form
  Weak solution in conservation form
  Shock speed from conservation form
• Numerical method properties:
  Upwind Method, Lax-Wendroff
• Know how to compute:
  One step with an explicit method
  Stability using Fourier
  Crossing time of characteristics
  Conservation form
  Shock speed

[6] M&M Elliptic Problems

• Discretization:
  Centered difference scheme
  Sparse structure of equations
  Local order of accuracy
• Global convergence:
  Comparison function
  Conditions for Theorem 6.1
  Result for curved boundaries
  Result for Neumann boundary conditions

[B1] Weak Formulation I

• Definitions:
  Functional
  Vector space
  Inner Product
  Norm
  Cauchy Sequence
  Complete Vector Space
  Hilbert Space
  L₂ space
  Cauchy-Schwarz inequality
  Weak Derivative
  Sobolev spaces
  Poincare-Friedrichs inequality
  Quadratic Functional
  Directional Derivative
  Stationary Point
• Concept of Completion:
  Is C^(k) complete?
  Is H^(k) complete?
• Riesz Representation Theorem:
  Statement and Conditions
• Lax-Milgram Lemma:
  Statement and Conditions

[B2] Weak Formulation II

• Euler-Lagrange Equations:
  Derive from a quadratic functional
  Strong Form vs. Weak Form
  Derivation in 1d/2d/3d
  Gauss divergence Theorem
  Integration by parts in 2d/3d
  Sobolev Spaces in 2d/3d
  Possibility of singular functions
  Lipschitz domain
  Poincare-Friedrichs in 2d/3d
  Trace Theorem
  Cone Condition
• Boundary Conditions:
  Natural vs. Essential Boundary conditions
  How to impose boundary conditions in minimization form

[B3] Ritz-Galerkin Method

• Problem formulation:
  Derive minimization form from a(u,v) = G(v)
  Derive a(u,v) = G(v) from strong form
  Derive discrete equations from a(u,v) = G(v)
• Triangular Finite Element mesh:
  Piecewise linear basis functions
  Mapping to standard triangle
  Assembly of linear system
  When is numerical quadrature necessary?
  Midpoint quadrature rule
  Non-homogeneous Boundary Conditions

[B4] Error Estimates

• Definitions:
  C^(k) finite elements
  Conforming Mesh
  Element diameter
  Mesh diameter
  Shape regular
  Uniform mesh
• Theorems:
  How smooth are H^(k) functions?
  Finite element approximation is bounded
  Céa's Lemma
• Higher-Order Finite Elements:
  Nodal basis
  Interpolation error for H^(k) in L₂-norm

[B5] Iterative Methods

• Nonlinear Problems:
  Nonlinear weak form
  Linearized bilinear form
  Newton's method
• Solving linear systems:
  Fixed point theorem
  Jacobi Iteration
  Gauss Seidel Iteration
  Diagonally Dominant
  Reducible
  Properties of CG

[B6] Multigrid

• Multigrid V-Cycle:
  Frequency of Error on Grids
  Restriction
  Prolongation
  Definition of Coarse Problem