I am dealing with following problem:

$$ \min_{u,\gamma}\Bigg\{ \frac{1}{1000} \iint_{S_2} {\gamma (x,y)^2 dxdy} + \iint_{S_2} {[u(x,y) - u_0 (x,y)]^2 dxdy} + \iint_{S_2} {[\Delta u(x,y) - \gamma (x,y) u(x,y)]^2 dxdy}\Bigg\} $$

where the unknown functions $u(x, y)$ and $\gamma(x, y)$ are defined on the unit square $S_2$ and the function $u_0(x, y)$ is defined on $S_2$ by $u_0(x, y) = \sin(6\pi x) \sin(2\pi y)$.

This problem should be discretized using 5-point finite differences.

I have no experience discretizing such problems. Could somebody help me out, or point me to right literature to look for some examples?

Note: this problem comes from the paper Recursive trust-region methods for multiscale nonlinear optimization (page 10)

  • $\begingroup$ Could you try to make the equation easier to read using LaTeX? $\endgroup$ – fibonatic Jul 25 '15 at 2:26
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    $\begingroup$ I think there might be a bug in the middle term. As written, that's identically zero. Presumably one of those $u$ terms must be a $u_0$, but I'll let you fix it. $\endgroup$ – Bill Barth Jul 25 '15 at 15:06
  • $\begingroup$ you are right, I was missing subscript in one term $\endgroup$ – SmallElephant Jul 25 '15 at 15:07
  • $\begingroup$ What kind of operation does $\Delta$ perform on $u(x,y)$? $\endgroup$ – fibonatic Jul 27 '15 at 3:36
  • $\begingroup$ @fibonatic do u have any suggestion? I am mostly curious how to deal with last term $\endgroup$ – SmallElephant Jul 27 '15 at 18:01

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