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{\large Delft University of Technology}\\
{\bf Faculty of Information Technology and Systems}\\
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{\bf 
Numerical Analysis CII (wi4014)\\
Numerical Methods PDE (wi3001)\\
2003/2004 Take Home Exams, Second Series}
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\begin {enumerate}
\item
On the bounded domain $\Omega \subset \IR^2$ we consider the minimization
problem (the so called minimal surface problem) 
\beq
\min_{u} J[u] = \int_\Omega \sqrt {1 + u_x^2 + u_y^2} \ d\Omega,
\eeq

On a part of the boundary, $\Gamma_1$, $u$ is prescribed: $u=g.$

Show by the same method as in Theorem 4.3, that the Euler-Lagrange equation for
this problem is 

\beq
\divg \left( [1 + u_x^2 + u_y^2]^{-\frac12} \grad u \right) = 0
\eeq

Derive also from the minimization problem a boundary condition for $\Gamma_2$,
the part of the boundary where $u$ has not been prescribed.
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\item
Consider the minimization problem:
\beq
 \min_{u \in \Sigma} \int_\Omega \frac12 \left(a_{11}u_x^2 + 2a_{12}u_xu_y +
a_{22} u_y^2 \right) - uf\ d\Omega 
\eeq
with
\beq
\Sigma : \{u \in C^1(\Omega)|u=g,\quad \forall (x,y) \in \Gamma_1.\}
\eeq
De coeffici{e}nts $a_{ij}$ depend on (x,y) and the
symmetric matrix $A$ with co{e}ffici{e}nts $a_{ij}$ is positive
definite for each $(x,y) \in \Omega.$\\
We solve this problem by Finite Element Method and we shall use piecewise quadratic
elements. 
\begin {enumerate}
\item Derive the Euler-Lagrange equations with corresponding boundary
conditions.
\item Show that the Newton-C\^{o}tes formula for the quadratic element is given
by
\beq
\int_e Int(x)\ dx = \frac {|\Delta|}{6} \left[ Int (x_{23}) + Int (x_{13}) + 
Int (x_{12})\right]
\eeq
\item Construct a table of the values of $\phi_{i_x}, \phi_{i_y}, \phi_{ij_x},
\phi_{ij_y}$ in the integration points of an element. Show how to calculate
the integrals $\int_e a_{11} \phi_{i_x} \phi_{j_x}\ de$ and
$\int_e a_{11} \phi_{i_x} \phi_{ij_x}\ de$ using this table.
(Use Newton-C\^{o}tes formulae.)
\item Compute the components $f^e_1$ and $f^e_{12}$ of the element vector.
\end {enumerate}
\end {enumerate}
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{\bf wi3001 to be submitted before Oct .., 2003\\
wi4014 to be submitted before Feb 1, 2004}
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