Numerical Integration
Newton's Method
Newton for a Simple Equation
Chain Rule
Jacobian Free Newton Krylov
Wrap Up
The Finite Element Method is a way of numerically approximating the solution of PDEs.
Just like polynomial fitting, FEM finds coefficients for basis functions.
The "solution" is the combination of the coefficients and the basis functions, and the solution can be sampled anywhere in the domain.
We compute integrals numerically using quadrature.
Newton's Method provides a mechanism for solving a system of nonlinear equations.
The Jacobian Free Newton Krylov (JFNK) method allows us to avoid explicitly forming the Jacobian matrix while still computing its "action".