# Globalized Newton method for minimizing a specific functional: Convergence?

I'm currently working on a generalized p-Laplace equation:

\begin{align} \label{DP} \begin{cases} \text{div} (\sigma |\nabla u|^{p-2} \nabla u) = f &\quad \text{in } \Omega\\ u = g &\quad \text{in } \partial\Omega \end{cases} \end{align}

We can get the solution by minimizing the following functional over all $$v \in W^{1,p}_g(\Omega)$$:

\begin{align} J:~W^{1,p}(\Omega) \rightarrow \mathbb{R}, \quad J(v) = \frac{1}{p}\int_{\Omega} \sigma |\nabla v|^p~d x - \int_{\Omega} fv~ d x \end{align}

First I proved that there exists a unique (basicly because $$x \mapsto |x|^p$$ is strict convex for $$p>1$$) solution $$u$$ which solves the dirichlet problem and minimizes the functional $$J$$. Then I showed that we can get as near as possible with a finite element solution $$u_h$$, meaning $$u_h \rightarrow u$$ for $$h \rightarrow 0$$.

To compute an numerical approx solution I used a globalized Newton method:

We start with the simple Gradient descent method until a specific condition is met, which tells us that from now on the classical Newton method will converge, so we use the newton method from this point.

From my optimization lecture script I know that this method will always converge at least superlinear when the underlying function is twice continuously differentiable.

In my case the function I want to use the globalized Newton method on is the discrete version of $$J$$ (associated with the mesh used in the finite element method):

\begin{align} J_h: \mathbb{R}^n \rightarrow \mathbb{R},~ J(u_1, ..., u_n) = \int \left |\sum_{k=1}^{k=n} u_k \nabla \phi_k \right| ^p - \int f \left ( \sum_{k=1}^{k=n} u_k \phi_k \right ) \end{align}

where $$\phi_k$$ are the classical hat-functions and therefore are lineare on the triangles associated with the triangulation needed for the finite element method. thus i can use a linear combo of the $$\phi_k$$ to approx functions ( $$g \approx \sum_{k=1}^{k=n} c_k \phi_k$$).

As far as I see it my functional $$J_h$$ should twice continuously differentiable withe the gradient and the jacobi matrix being

\begin{align} (\nabla J_h)_i = \frac{J_h(u_1,..., u_n)}{\partial u_i} = \int \left |\sum_{k=1}^{k=n} u_k \nabla \phi_k \right| ^{p-2} \left (\sum_{k=1}^{k=n} u_k \nabla \phi_k \right) \nabla \phi_i - \int f \phi_i \end{align}

and

\begin{align} (\nabla^2 J_h)_{i,j} = \frac{J_h(u_1,..., u_n)}{\partial u_i \partial u_j} = \int (p-1) \left |\sum_{k=1}^{k=n} u_k \nabla \phi_k \right| ^{p-2} \nabla \phi_i \nabla \phi_j \end{align}

Therefore I can "easily" prove that the globalized newton method for my functional $$J_h$$ will converge to the minimizer by stating that $$J_h$$ is strict convex, has a unique minimizer, $$\nabla^2 J_h$$ is continously and then simply refering to the script where it is stated that this algorithm will always converge to a stationary point if the underlying function is twice continuously differentiable.

QUESTION: Are my thoughts correct, or do i miss some ciritcal points?

EDIT: Ok, so now I am quite sure, that my thoughts should be right. The Jacobi Matrix $$\nabla ^2 J_h$$ is continously since it is just a composition of continiously functions... BUT: I forget about the case $$1 < p < 2$$. What if $$\sum u_k \nabla \phi_k = 0$$? I think this can only happen if $$u_1 = u_2 = ... = u_n$$ because in this case $$\sum u_k \phi_k$$ whould be a constant function with gradient being zero. So my start-vector for this algorithm to converge can't be something like $$(1, 1, ..., 1) \in \mathbb{R}^n$$ AND i need somehow to garantuee that in the iterative process my next-iterations points are NEVER $$\alpha (1, 1, ...., 1)$$ with $$\alpha \in \mathbb{R}$$. But how can i prove such thing... and can i then speak of truly "global" convergence if not every point in $$\mathbb{R}^n$$ is allowed as start-input?

I hope it is detailed enough and not to chaotic... Thanks in advance for everyone who takes a look at this! :)