User:Eml5526.s11.team5.JA/Mtg12

 Mtg 12: Wed, 26 Jan 11

[[media: Fe1.s11.mtg12.djvu| Page 12-1 ]]

G1DM1.0/D1: p. 9-1

select $$ \phi \ $$ such that $$ b_j'(x= 0) \ne 0 \ $$

Consider $$ \phi = \frac{\pi}{4} \ $$ and $$ \phi = \frac{\pi}{2} \ $$

1)  let $$ \underline{n= 2}  \Rightarrow \underbrace{ndof}_{\color{Blue}nunber \ of \ degrees \ of \ freedom } = \underbrace{n+1}_{ \mathbf{d} = \left \{ d_j, j = 0, ....,n  \right \}_ {\color{Red}(n+1)x1}} = 2 +1 = 3 \ $$

[[media: Fe1.s11.mtg12.djvu| Page 12-2 ]]

2)  Find 2 equations that enforce boundary conditions for $$ u^h(x) = \sum_{j = {\color{Red}0}}^{n} d_jb_j(x) \ $$

3)  find $$ \underbrace{1}_{\color{Blue} dof - 2 = 3-2} \ $$   more equation to solve for $$ \mathbf{d} = \left \{ d_j \right \}_{\color{Red}3x1} (j = 0,1,2) \ $$ by projecting the  residue  $$ P(u^h) \ $$ on a basis function $$ b_k(x) \ $$ with $$ k = 0, 1, 2 \ $$ such that the additional equation is lineally independent from the above 2 eqations in part  2)

Note:

Residue $$ {\color{Blue}P(u^h)} \ $$ : [[media: Fe1.s11.mtg6.djvu|   page 6-4 ]]


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$$ P(\underbrace{u}_{\color{Blue}exact \ solution}) = 0 \ $$, [[media: Fe1.s11.mtg6.djvu| (6) page 6-4 ]]

$$
 *  $$ \displaystyle {\color{Red} (1) }
 * }


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$$ u^h \cong u \ (u^h \ {\color{Red}approximate} \ u ) \ $$

$$
 *  $$ \displaystyle {\color{Red} (2)}
 * }

$$ P(u^h) \ $$ = residue, "error"


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$$ \Rightarrow \underbrace{ P(u^h)}_{\color{Blue}'error', \ residue} \ne 0 \forall x \in \Omega \ $$

$$
 *  $$ \displaystyle {\color{Red}(3)}
 * }


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Projection:  $$ \int_0^1 b_k(x)P(u^h) dx = 0 \ $$ [[media: Fe1.s11.mtg10.djvu| (2)  page 10-4 ]]

$$
 *  $$ \displaystyle {\color{Red}(4)}
 * }

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[[media: Fe1.s11.mtg7.djvu| (2) page 7-2 ]] $$ := <\mathbf{b}_i, \mathbf{P(v)} > \ $$

$$
 *  $$ \displaystyle
 * }




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$$  = \int_0^1 b_k(x)P(u^h) dx \ $$

$$
 *  $$ \displaystyle
 * }

4)  Display 3 equations in matrix form


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$$ \mathbf{K} \mathbf{d} = \mathbf{F} \ $$

$$
 *  $$ \displaystyle
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Observe symmetric properties of $$ \mathbf(K) \ $$ (symmetric?)

5)  Solve for $$ \mathbf{d} \ $$

6)  Construct $$ u^h_{\color{Blue}n}(x) \ $$ and plot $$ u^h_{\color{Blue}n}(x) \ $$ (approx.) versus $$ \underbrace{u(x)}_{\color{Blue}exact} \ $$

7)  Repeat 1) though 6)  for

7.1)  n = 4 7.2)   n = 6


 * }