Consider the following differential equation \begin{align} \frac{\partial f(u)}{\partial x} &= g(x), \ \ x\in [x_{L},x_{R}] \label{Eq2.2} \\ u(x_{L}) &= g_{1} \end{align} where $f(u)$ is a nonlinear function. Let $\{x_{i}\}_{i=1}^{N+1}$ be a partition of $\Omega$ with $N$ elements such that \begin{equation} x_{L} = x_{1}<x_{2}<\cdots<x_{N}<x_{N+1} = x_{R} \end{equation} where $D_{k} = [x_{k},x_{k+1}]$. The weak formulation of the problem is \begin{equation} \int_{D_{k}} \Bigg[ -f(u)\frac{d \phi_{i}^{k}}{d x}\Bigg] = \int_{D_{k}} g(x)\phi_{i}^{k}-\Big[(f(u))^{*}\phi_{i}^{k} \Big]_{x_{k}}^{x_{k+1}} \label{WeakFormulationNL} \end{equation} where the numerical flux $(f(u))^{*}$ is given by \begin{equation} (f(u))^{*} = \{f(u)\}+\frac{C}{2}[\hspace{-0.6mm}[ u ]\hspace{-0.6mm}] \end{equation} where $\{\cdot\}$ is the average, $[\hspace{-0.6mm}[ \cdot ]\hspace{-0.66mm}]$ is the jump and $C$ is given by $C = \max{ |f(u)| }$. After the space discretization we get a system of the form \begin{equation} -r(u_{h}) = b-w(u_{h}) \end{equation} which can be rewritten as follows \begin{equation} r(u_{h}) + b-w(u_{h}) = 0 \end{equation} The entries of the operators $r(u_{h})$ and $b_{k}$ in the element $D_{k}$ are given by \begin{align} (r_{k})_{i} &= \int_{D_{k}} f(u)\frac{d \phi_{i}^{k}}{d x} \\ (b_{k})_{i} =& \int_{D_{k}} g(x,t)\phi_{i}^{k} \end{align} where $w(u_{h})$ in the element $D_{k}$ is the term $\Big[(f(u))^{*}\phi_{i}^{k} \Big]_{x_{k}}^{x_{k+1}}$. Let $F(u_{h}):= r(u_{h}) + b-w(u_{h})$, then we have a nonlinear system of the form $F(u_{h}) = 0$. I want to rewrite this in the form $G(u_{h}) = u_{h}$ in order to solve the system using a fixed point iteration.
Here's my attempt: let $\lambda \neq 0$ then
\begin{align*}
F(u_{h}) =& 0 \implies \\
r(u_{h}) + b-w(u_{h}) =& 0 \implies \\
r(u_{h}) + b-w(u_{h})+\lambda u_{h} =& \lambda u_{h} \implies \\
\frac{1}{\lambda}\Big[r(u_{h}) + b-w(u_{h})+\lambda u_{h} \Big]=& u_{h}
\end{align*}
Then we define
\begin{equation}
G(u_{h}) = \frac{1}{\lambda}\Big[r(u_{h}) + b-w(u_{h})+\lambda u_{h} \Big]
\end{equation}
My first question is:
Do you think is possible to use the fixed point iteration to solve the nonlinear system?
if it is possible:
Do you think this approach is correct? or Do you know a better way to solve the system? (Without Newton's method)
