The Pythagorean Theorem — Lesson 4

In this lesson, we mathematically derive the equivalent "Pythagorean" theorem for the finite element method. This theorem states that for problems consisting of only homogeneous Dirichlet boundary conditions, the energy norm of the exact solution is the sum of the energy norm of the finite element solution and the energy norm of the error. This theorem also implies that the finite element solution underestimates the energy norm of the problem being solved.

What does it mean for the Dirichlet boundary conditions to be “homogeneous”? Why does this imply that $S^{h} = V^{h}?$