Introduction to Computational Linear Algebra

202 Introduction to Computational Linear Algebra

7.5.1 The P

1

Finite-Element Spaces

1. Define first the “elements” on the domain Ω of the solution. In (7.42)

Ω = (0, 1), we let Ω :

S

i

[x

i

, x

i+1

] where {x

i

|i = 1, ..., n} is a set of

nodes so that:

(a) x

i

6= x

j

for i 6= j

(b) x

1

= 0 and x

n

= 1

If E

i

is [x

i

, x

i+1

], then

S

i

E

i

= [0, 1] and:

E

i

∩ E

j

=











φ

1 Node

E

i

itself

Remark 7.1 There is no uniformity on the elements, i.e., there exists

i, j such that x

i+1

− x

i

6= x

j+1

− x

j

.

2. Define the P

1

finite-element spaces:

S

1

(Π) = {ϕ ∈ C([0, 1]) | ϕ|

E

i

is a linear polynomial ∀i}.

We can now verify the following results:

Theorem 7.6 Every ϕ ∈ S

1

(Π) is uniquely determined by its values at ...