A question related with $p$-Laplacian and conjugate gradient method

Question

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 ?

Answer 1

You seem to be misunderstanding the difference between the derivative of $E$ and the gradient of $E$. I am going to assume that $u$ lies in an appropriate Sobolev space $V$, which embeds continuously into $L^2(\Omega)$.

You have successfully computed the derivative of $E$ with respect to $u$ defined by a directional derivative: $$ \delta E(u)v := \int_{\Omega}|\nabla u|^{p-2} \nabla u\cdot \nabla v \ dx.$$

However, notice that the derivative $\delta E(u)$ is a linear functional on $V$, i.e., it is an element of $V^*$, not $V$, so we need to do some more work to get something that we can add or subtract from $u$ for use in optimization algorithms. The gradient what we want.

The gradient of $E$ with respect to the $L^2$ inner product is the unique element $g\in V$ that satisfies $$ \langle g,v\rangle_{L^2} = \delta E(u)v \ \ \ \ \forall v\in V. $$

That is, the gradient is the thing you integrate against $v$ to obtain the action of the derivative. Existence is guaranteed by the Riesz representation theorem. Some integration by parts and use of homogenous boundary conditions will show that that gradient is given by $$ g = -\nabla\cdot(|\nabla u|^{p-2}\nabla u). $$

This is an element of $V$, so you can compute it over your discretization and use it in whatever optimization algorithm you choose.