7.3. The Galerkin Method for ODE Boundary Value Problems#
Version of December 8, 2025
References:
[Johnson, 1994] Numerical solution of partial differential equations by the finite element method.
[Sauer, 2022] Section 7.3, Collocation and the Finite Element Method.
7.3.1. Introduction#
This is for now just a stub; the section will introduce Galerkin methods for solving ODE BVPs in the form
\[
-u'' + q(x)u = f(x),\ a \leq x \leq b
\]
with boundary conditions such as the Homogeneous Dirichlet Boundary conditions
\[
u(a) = u(b) = 0
\]
More generally, the same strategy will be applied to
\[
-(a(x)u')' + q(x)u = f(x),\ a(x) > 0
\]
and then to the related eigenvalue problem
\[
-(a(x)u')' + q(x)u = \lambda u
\]