This question is from the Finite Element Solution of Partial Differential Equations 1. Let I=[x0?,x1?]. Verify by direct calculation that the basis functions ?0?(x)=x1??x0?x1??x?,?1?(x)=x1??x0?x?x0?? for P1?(I) satisfy ?0?(x)+?1?(x)=1 and x0??0?(x)+x1??1?(x)=x. Give a geometrical interpretation by drawing ?0?(x),?1?(x),?0?(x)+?1?(x),x0??0?(x), solutionspile.com

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This question is from the Finite Element Solution of Partial Differential Equations

$1. Let $ I=\left[x_{0}, x_{1}\right] $. Verify by direct calculation that the basis functions \[ \lambda_{0}(x)=\frac{x_{1}$

1. Let

I = [x_{0}, x_{1}]

. Verify by direct calculation that the basis functions

?_{0} (x) = \frac{x _{1} ? x}{x _{1} ? x _{0}}, ?_{1} (x) = \frac{x ? x _{0}}{x _{1} ? x _{0}}

for

P_{1} (I)

satisfy

?_{0} (x) + ?_{1} (x) = 1

and

x_{0} ?_{0} (x) + x_{1} ?_{1} (x) = x

. Give a geometrical interpretation by drawing

?_{0} (x), ?_{1} (x), ?_{0} (x) + ?_{1} (x), x_{0} ?_{0} (x), x_{1} ?_{1} (x)

and

x_{0} ?_{0} (x) + x_{1} ?_{1} (x)

. 2. Let

0 = x_{0} < x_{1} < x_{2} < x_{3} = 1

, where

x_{1} = 1/6

and

x_{2} = 1/2

, be a partition of the interval

[0, 1]

into three sub-intervals, and let

V_{h}

be the space of contimuons piece-wise linear functions on this partition. (a) Determine analytical expressions for the hat function

?_{1} (x)

and draw it. (b) Draw the function

v (x) = ? ?_{0} (x) + ?_{2} (x) + 2 ?_{3} (x)

and its derivative

v^{?} (x)

. (c) Draw the piecewise constant mesh function

h (x) = h_{i}

on the sub-interval

I_{i}

. (d) What is the dimension of

V_{h}

? (e) Draw the function

f (x) = x ln (x)

and its linear interpolant

? f (x)

on this mesh.

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Expert Answer to - This question is from the Finite Element Solution of Partial Differential Equations      1. Let I=[x

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(Solved): This question is from the Finite Element Solution of Partial Differential Equations 1. Let I=[x ...