Numerical Integration

Numerical Integration

Newton's Method

Solving Non-linear Systems

Newton for a Simple Equation

Newton for a Simple Equation

Chain Rule

Chain Rule

Jacobian Free Newton Krylov

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".