We consider the essential ideas for numerical methods of linear and nonlinear elliptic differential equations, extendible to systems of Order 2 or 2m. Surprisingly, the general discretization theory shows an appropriate common structure for stability and consistency implying convergence for all up-to-date methods. For fully nonlinear problems a totally new approach is necessary We present
1. Classical, Strong and Weak Solutions, Example: The Laplacian and its Generalizations
2. Relation to Coercive and Elliptic Bilinear Forms
3. Different Discretization Methods, FEMs, Difference, FEMs with Crimes, (DCGMs, Spectral, Wavelet, Radial(?) ) Methods
4. General Discretization Theory, Linearization with Stability and Fredholm Alternative
5. Solving the Highly Nonlinear Systems
6. A FEM for Fully Nonlin. Ell. Equations
7. Example: Navier-Stokes Operator