# Is it possible to use a fixed point iteration for solving this nonlinear system?

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 $$\begin{equation} x_{L} = x_{1} where $$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)

• So by simple integration $f(u(x))=f(u(x_L))+\int_{x_L}^xg(s)\,ds$? The second boundary condition is then in general not possible to meet. // Please define all your symbols, like $\{f(u)\}$ or $[[u]]$. Oct 22, 2021 at 10:09
• $\{\cdot\}$ is the average and $[[ \cdot ]]$ is the jump. Oct 22, 2021 at 14:22
• For this one-dimensional advection equation, you can only pose boundary conditions on one side (typically the left side). Oct 22, 2021 at 22:25
• @WolfgangBangerth Thank you professor. I'm not sure if a fixed point iteration is suitable for solving this nonlinear system. According to your experience, is it possible to solve the system using a fixed point iteration? Oct 22, 2021 at 22:46
• This might be of help. wiki.math.ntnu.no/_media/ma2501/2014v/fixedpoint.pdf Oct 27, 2021 at 1:55