fem Weak form and shape functions

[Ehiman1]

Good evening, I began to study the finite element method and I wanted to understand why it reduces the strong formulation of a degree to get to the weak.
Then I wanted to know what degree should have the shape functions? depending on what I choose it? And the system of algebraic equations and 'solvable even if non-linear? thanks to all

 

[csk62]

These are fundamental questions.

The weak form allows for the underlying differential equation(s) to be solved numerically using Gaussian quadrature. Search Galerkin Method.
Think of shape functions as interpolation functions. They must satisfy certain continuity constraints, but they can be up to whatever order. Search p-method
I think your "algebraic equations" may refer to the standard way of visualizing the segmentation of stiffness matrix, displacement and force vectors into a conceptually tractable way, based on BCs.

FEM requires many mathematical preliminaries in order to understand how it works.
If it is helpful, think of the strong form as the exact differential equation, and the weak form as an integral equation.
For an exercise, try transforming the 1D heat equation with prescribed BCs to the weak form.