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I have the following energy functional of $p$-Laplacian equation: $$ E(u) = \frac{1}{p} \int_{\Omega} |\nabla u|^p dx $$ for $2.8 \leq p \leq 5$.

My goal is to minimize the energy functional by using nonlinear conjugate gradient method ( https://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient_method ) .

To apply nonlinear conjugate gradient method, I have found that

$$ E'(u) = \int_{\Omega} |\nabla u|^{p-2} \nabla u \cdot \nabla v dx $$

Now I took five nodes $u_1, u_2, u_3, u_4, u_5$ in one dimension. We know $u_1=0=u_5$ and want to find $u_2,u_3,u_4$. By NCG method we will get $$ \begin{bmatrix} u_2^{n+1}\\ u_3^{n+1}\\ u_4^{n+1} \end{bmatrix} = \begin{bmatrix} u_2^{n}\\ u_3^{n}\\ u_4^{n}\end{bmatrix} - \alpha \cdot \begin{bmatrix} E_2'(u_2^{n})\\ E_3'(u_3^{n})\\ E_4'(u_4^{n}) \end{bmatrix} $$

Is my attempt correct? I am confused about how to manage integral sign of derivative of energy functional ?

User124356
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  • It’s always good to see how far you’ve gotten trying to adapt the $p=2$ case to the more general case. Makes it easier for us to guide you more effectively. – Deane Mar 26 '23 at 18:20
  • @Deane for $p>2$ this problem is a non-linear optimization problem and I can proceed as above. – User124356 Mar 26 '23 at 18:24
  • The sign issue should be handled the same way it is for $p=2$. Is that the only snag you’re encountering? Otherwise, your formula for $E’$ looks good. – Deane Mar 26 '23 at 18:40
  • @Deane one question is coming to my mind. In the above 1-d case $E_i'(u_i^n)= |\nabla u_i^n| \nabla u_i^n \cdot \nabla v_i^n$. I thought integral sign replace with summation but in this case we are calculating $E'$ at each node. Could you please clarify this point? – User124356 Mar 26 '23 at 22:07

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