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Galerkin's Method and Finite Element Basis Functions.

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Galerkin's Method and Finite Element Basis Functions.
Homework 2
Galerkin’s Method and Finite Element Basis Functions
1
Consider the again boundary value problem
−
d2u
= 20x3
dx2
subject to
u(0) = u(1) = 0
a) Use the method of Galerkin without integration by parts on the left hand side and the
following two functions
φ1 = sin(πx)
φ2 = sin(2πx)
to obtain an approximate solution to the problem. Evaluate the accuracy of the approximation.
b) Use the method of Galerkin with integration by parts on the left hand side and the
following two functions
φ1 = sin(πx)
φ2 = sin(2πx)
to obtain an approximate solution to the problem. Evaluate the accuracy of the approximation.
1
2
Consider the again boundary value problem
−
d2u
= 20x3
dx2
subject to
u(0) = u(1) = 0
a) Consider the following function
φ2 (x) =
(
2x,
0 ≤ x ≤ 21
2(1 − x), 21 ≤ x ≤ 1
and use the method of Galerkin with integration by parts on the left hand side to obtain an
approximate solution to the problem. Evaluate the accuracy of the approximation. What
happens if you do not use integration by parts on the left hand side?
b) Consider the following functions
3x,
0 ≤ x ≤ 31
φ2 (x) = 3( 23 − x), 13 ≤ x ≤ 32


2
0,
≤x≤1
3



and
0,
0 ≤ x ≤ 13
1
φ3 (x) = 3(x − 3 ), 31 ≤ x ≤ 32


3(1 − x), 32 ≤ x ≤ 1



and use the method of Galerkin with integration by parts on the left hand side to obtain an
approximate solution to the problem. Evaluate the accuracy of the approximation. What
happens if you do not use integration by parts on the left hand side?
c) Find a new approximation using the Galerkin method using the three functions
φ2(x), φ3 (x), φ4(x) that would follow in the pattern above.
2
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