Syllabus for Mathematics 642:575:
Numerical Solution of Partial Differential Equations

I. Finite Difference methods for Laplace's equation.

  1. Derivation of 5-point difference scheme
  2. Discrete Maximum principle
  3. Existence and uniqueness of the approximate solution
  4. Consistency, stability, and convergence
  5. Derivation of error estimates.

II. Finite Element Methods for elliptic equations

  1. Standard variational formulation of second order elliptic boundary value problems; natural and essential boundary conditions, connection with minimization problems.
  2. Construction of finite element spaces in one-dimension: dimension of the spaces, basis functions, degrees of freedom
  3. Construction of Lagrange-type triangular finite element spaces in two dimensions: barycentric coordinates, mapping from the reference to the general triangle.
  4. Error estimates for finite element approximation schemes (L2 function and derivative errors) -- reduction to approximation theory
  5. Approximation theory results for piecewise polynomials
  6. A posteriori error estimates and adaptive methods
  7. Finite element methods for elliptic variational inequalities
  8. Efficient solution of the discrete equations by iterative methods
  9. Solution of the discrete equations by multigrid

III. Finite Difference Methods for Linear Parabolic and Hyperbolic Problems

  1. Basic schemes for the heat equation
  2. Basic schemes for the wave and transport equations
  3. Consistency, stability, local trunction error, error estimates

IV. Finite Element Methods for Linear Parabolic and Hyperbolic Problems and Parabolic Variational Inequalities

V. Finite Volume Methods

VI. Finite Difference Methods for Nonlinear Hyperbolic Equations