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 and 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: statement of general results; proof for piecewise linear elements using multipoint Taylor formulas.
  6. Solution of the Discrete Equations by Multigrid
  7. Nonconforming and mixed finite element methods
  8. Finite element methods for the Stationary Stokes equations

III Finite Difference Methods for 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