User:Eml5526.s11.team2.brenner/hwk6

=Problem 6.1 Using Quadratic Lagrange Element Basis Functions (QLEBF) obtain the trial solution from Weak Form=

Problem Statement
Use QLEBF with uniform discretization (equidistant element nodes) for $$ \displaystyle nel = 2,4,6,8 $$

$$ \frac{\partial}{\partial x} \left[ (2+3x) \frac{\partial u}{\partial x} \right] + 5x = 0 \qquad \forall x \in \left[ 0,1 \right] $$,

Essential Boundary Function  $$ u \left(0\right)=4 $$, and

Natural Boundary Function  $$ n(1)a_2(1)\frac{\partial u(x=1)}{\partial x}=12$$

$$\implies (1)(2+3x)\frac{\partial u(x=1)}{\partial x}=12$$

$$\implies \frac{\partial u(x=1)}{\partial x}=\frac{12}{5}

$$

$$ \Gamma_g = \left[0\right] \qquad \Gamma_h = \left[1\right] $$

Find
1) For number of elements $$ \displaystyle nel = 2 $$, compute $$ \displaystyle \tilde{K} = \sum_{e=1}^2 \tilde{K}^e $$ with $$ \displaystyle \tilde{K}^e $$ given by (6.1.1) display $$ \displaystyle \tilde{K}^e, e = 1,2 $$


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$$ \displaystyle \tilde{K_{ij}}^e = \int_{\omega^e}{{b_i}^e}^\prime(x^e)a_2(x^e){{b_j}^e}^\prime(x^e)dx^e $$ (6.1.1)
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2) Compute $$ \displaystyle {k}^e, {L}^e for e = 1,2 $$

3) Compute $$ \displaystyle \tilde{K}^e = {{L}^e}^T {k}^e {L}^e for e =1,2 $$ compare to item 1) above

4) Plot all QLEBF for $$ \displaystyle nel = 2 $$

5) Plot $$ \displaystyle {u^h}_\tilde{n} $$ vs. $$ \displaystyle u $$ and $$ \displaystyle {u^h}_\tilde{n}(0.5) - u(0.5) $$ vs $$ \displaystyle \tilde{n} $$