References for the nonlinear reaction-diffusion equation using Finite Element Methods

Question

I want to study how to solve the following PDE

\begin{cases} -\nabla \cdot(\ k(x,y) \ \nabla u \ ) + \beta(x,y)\ u^2 = f(x,y), \ (x,y) \in \Omega \subset \mathbb{R^2} \\ \hspace{0.5cm} u = g_{D} \ \ on \ \ \partial{\Omega}_{D} \\ \hspace{0.2cm} \frac{\partial{u}}{ \partial n } = g_{N} \ \ on \ \ \partial{\Omega}_{N} \end{cases}

I know how to solve the linear version of this PDE with the finite element method.

This is my first time with a nonlinear PDE, so I have to start with theory (from the most basic to the advanced), and then read about implementation details.

I want to read about:

1. Theory: strategy to solve the PDE, models, Newton Method, etc.

2. Implementation.

Can anyone help me with good references (books or papers) where I can learn how to solve this nonlinear PDE?

Answer 1

You are asking about a big area, not a very specific problem. So the best I can offer is for you to look through the lectures on nonlinear problems here: https://www.math.colostate.edu/~bangerth/videos.html It shows you different ways of solving nonlinear problems, and how to implement them.