Math Help Forum - View Single Post - Poisson Equation, Energy Function (need help)

jeneverboy · #2 November 5th, 2009, 08:21 AM

Hi, i am dealing with the same sort of topic. First let me see if I can make something more clear.

(1) is a PDE (partial differential equation) and (3) is the minimization problem which is derived from (1). Finding a solution for (3) will take some numerical methods. But this solution is also a solution for (1).

Now from (3) to (1):
$E(u) = 1/2 \int_{\Omega} \vert \vert \nabla u \vert \vert - 2G\theta u d \Omega$

Let $\hat{u}(x,y)$ be the function minimizing this
$u=u+\epsilon \phi$

Substitute this in (3) minimizing problem
$E(u)=1/2 \int_{\Omega} \vert \vert \nabla (u+\epsilon \phi) \vert \vert - 2 G\theta (u+\epsilon \phi) d \Omega$

Now differentiate with respect to $\epsilon$ , and then E(u) reaches a minimum for $\epsilon=0$ . You should get the following:
$1/2 \int_{\Omega} \nabla u \cdot \nabla \phi - 2 G\theta \phi d \Omega = 0$ . I can't derive this but you will get the following;

Use the divergence theorem to get rid of $\nabla \phi$
$1/2 \int_{\Omega} (-div \nabla u - 2 G\theta) \phi d \Omega + \int_{\Gamma} \phi \nabla u \cdot n d \Gamma= 0$

And use DuBois' Lemma to derive the PDE:
$-div \nabla u - 2 G \theta =0$
with the boundary condition:
$\frac{\partial u}{\partial n} = 0 \; \in \; \Gamma$ Which is the natural boundary condition. The essential boundary condition was already given, u=0 on Gamma.

I skipped some essential steps cause these are not clear to me either, but I am pretty shure this is the way to go from PDE to minimizing problem.

From minimizing to FEM is the next step. Hope someone can contribute to this!!

Bye Alex

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