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 and
derivation of error estimates.
- An Alternating Direction Method
II. Finite Element Methods for elliptic problems
- 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
(L2 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.
- Nonconforming and mixed Finite Element Methods
- Finite Element Methods for the Stationary Stokes Equations
III Finite Difference Methods for parabolic problems
- Basic schemes for the Heat Equation
- Consistency, stability, local trunction error, error estimates
IV Boundary Integral Methods for Laplace's Equation.
- Single Layer and Double Layer Potentials
- The Boundary Integral Formulation
- Nystr\"om's Method