Outline

References:

• Theory:
  • The first five chapters of Partial Differential Equations, Methods and Applications by Robert McOwen (Prentice Hall, 1996).
• Numerics:
  • The Mathematical Theory of Finite Element Methods by Susanne C. Brenner and L. Ridgway Scott (Springer-Verlag, 1994).
  • Numerical Solution of Partial Differential Equations by Morton and Mayers.

1.

Theory:

(a)

First-Order Equations:

i.

The Cauchy problem

ii.

The method of characteristics

iii.

Complete Integral and General solutions

(b)

Higher-Order Equations:

i.

The Cauchy problem: Normal Form, Power series and the Cauchy-Kovalevski theorem

ii.

Second order equations: Classification by characteristics, canonical forms, general solutions, first-order systems

iii.

Linear equations and generalized solutions: Adjoints and weak solutions, distributions, convolutions and fundamental solutions

(c)

The Wave Equation:

i.

1-D wave equation: Initial and boundary value problems, weak solutions, nonhomogeneous equations (Duhamel's principle)

ii.

Higher dimensions: Spherical means and the Cauchy problem, Three- and two-dimensional wave equations, Huygens principle

iii.

Energy Methods

iv.

Dispersion, dissipation, domain of dependence

(d)

The Laplace Equation:

i.

Separation of variables, Green's identity and uniqueness, mean values and maximum principles

ii.

Potential theory and Green's functions: Fundamental solutions, Poisson Kernel, Dirichlet problem on a half-space and a on a ball, harmonic functions

iii.

Subharmonic functions, Perron's method

iv.

Eigenvalues and eigenfunction expansions

v.

Convergence theory for classical Fourier series; Smoothness and rate of decay of the Fourier coefficients.

(e)

The Heat Equation:

i.

The heat equation in a bounded domain: Eigenfunction expansion, maximum principle, uniqueness

ii.

Initial value problems: Fourier transforms, fundamental solutions, nonhomogeneous equation

iii.

Regularity and similarity: Smoothness of solutions, Scale invariance and similarity methods

2.

Numerics:

(a)

Finite difference methods

i.

Common finite difference methods for the heat, wave, advection, and Poisson equations

ii.

Incorporation of Dirichlet and Neumann boundary conditions and initial conditions

iii.

Stability, consistency, and convergence

• Definition of each concept
• Lax equivalence theorem and proof

iv.

Fourier stability analysis for one-step schemes

v.

Advantages of implicit methods for the heat equation

(b)

Finite element methods

i.

Derivation of the weak or variational formulation of a second-order or fourth-order elliptic BVP

• Incorporation of boundary conditions (essential versus natural)
• Existence and uniqueness of solutions to the weak form; Riesz representation theorem; Lax-Milgram Theorem

ii.

Galerkin and related methods for approximation

• Best approximation property of the Galerkin approximation for symmetric problems; Céa's Lemma
• Derivation of formulas for stiffness matrix and load vector

iii.

Type of finite elements

• Lagrange triangles and tetrahedra
• tensor product rectangles and boxes
• isoparametric elements

iv.

Convergence of Galerkin finite element methods

• Role of interpolation theory
• Role of elliptic regularity in obtaining L² error estimates

The student is not expected to prove results from interpolation theory or elliptic regularity theorems, but rather to understand how these results are used to prove the convergence of Galerkin finite element methods.

v.

Method of lines and FEM for parabolic problems

• Weak form of a parabolic IBVP
• Finite element semi-discretization in space