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Syllabus for Mathematics 642:575:

Numerical Solution of Partial Differential Equations

I. Finite Difference methods for Laplace's equation.

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

II. Finite Element Methods for elliptic equations

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

III. Finite Difference Methods for Linear Parabolic and Hyperbolic Problems

- Basic schemes for the heat equation
- Basic schemes for the wave and transport equations
- Consistency, stability, local trunction error, error estimates

IV. Finite Element Methods for Linear Parabolic and Hyperbolic Problems

V. Finite Volume Methods

VI. Finite Difference Methods for Nonlinear Hyperbolic Equations