To solve a classical second-order differential problem

with FEM, we first need to derive its weak formulation. This is achieved by multiplying the equation by a test function and then integrating by parts to get rid of second order derivatives:

(1)

A typical FEM problem then reads like:

What is the difference between imposing Dirichlet boundary conditions (ex. ) and Neumann ones () from a math perspective? **Dirichlet conditions go into the definition of the space , while Neumann conditions do not. Neumann conditions only affect the variational problem formulation straight away.**