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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 and discrete Green's functions
- Existence and uniqueness of the approximate solution and
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: statement
of general results; proof for piecewise linear elements using
multipoint Taylor formulas.
- Solution of the Discrete Equations by Multigrid
- Nonconforming and mixed finite element methods
- Finite element methods for the Stationary Stokes equations

III. Finite Difference Methods for 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 Parabolic and Hyperbolic Problems

V. Finite Volume Methods