[ Introduction to Numerical Methods ]

I. Rayleigh-Ritz
Shape of hanging string.
Why are droplets spherical?
Droplets on substrates

II. Calculus Review
How to find minima of functions
Mutlivariable functions
Constrained minimization
Lagrange multipliers

III. Linear Algebra Review
Quadratic forms
Positive definite quadratic forms
Quadratic form minimization

IV. Finite Elements in 1D
Introduction. Laplace's equation.
The finite element basis
The stiffness matrix

V. Finite elements in 2D and 3D
Triangular meshes
The FEM basis
Tetrahedral meshes

VI. Calculus of variations
Variational formulation
Derivation of Laplace's equation
Neumann or naturatl BC

VII. Quadratic finite elements
Quadratic elements
The FEM basis
Convergence implications

VIII. Problems from Physics and Engineering
The Poisson equation
Equations of elasticity
The heat equation

IX. Numerical Linear Algebra
Conjugate gradient method
Power iteration
Overview of other methods

X. Additional Topics
Singular finite elements