Introduction to Computational Linear Algebra

Sparse Systems to Solve Poisson Differential Equations 197

7.4.2 Variational Formulation to One-Dimensional Poisson

Problems

In the case of the one-dimensional model and if we let Ω = (0, L), the differ-

ential equation in (7.1) can be rewritten as:

−

d

dx

(a(x)

du

dx

) + b(x)u = f(x), x ∈ Ω u(0) = α, u

0

(L) = β. (7.42)

Although (7.42) can be handled using finite-difference discretizations, we find

out now that it can be dealt with more naturally if it is put in variational

form. We associate with this boundary value problem the two sets:

1. The set of “test functions”:

T = {ϕ ∈ C

1

(Ω) ∩ C(Ω)|ϕ(0) = 0}

2. The set of “admissible functions”:

U

ad

= {ϕ ∈ C

1

(Ω) ∩ C(Ω)|ϕ