Linear Algebra and Its Applications · Problem 6.5.3

Computations with Matrices — Gilbert Strang

Mathematics · finite-element-method-natural-boundary

Problem

Suppose

-u'' = 2

, with the boundary condition

u(1) = 0

changed to

u'(1) = 0

. This ``natural'' condition on

u'

need not be imposed on the trial functions

V

. With

h = \frac{1}{3}

, there is an extra half-hat

V_3

, which goes from 0 to 1 between

x = \frac{2}{3}

and

x = 1

. Compute

A_{33} = \int (V_3')^2 dx

and

f_3 = \int 2V_3 dx

. Solve

Ay = f

for the finite element solution

y_1 V_1 + y_2 V_2 + y_3 V_3