Solving non-linear pde with newton method

Question

I know that to solve a nonlinear pde, you either have to linearize or you have to solve it using Newton's method. I didn't find any clue or example about how to do it with Newton's method. Can any body help me?

P.S. I know the newton's method but I don't get that how am I supposed to use it for pde, specially that I have to use it in matlab. Thanks.

Answer 1

First it is worth noting that Newton's method is a linearisation technique. In this case let us assume that we have a uniformly elliptic nonlinear operator $$F[u]:=\mathcal{F}(D^2u,Du,u,x),$$ and our nonlinear PDE is given in the form $$F[u]=0,$$ subject to some boundary conditions.

Newton's method is given as follows: Given $u_n$ find $u_{n+1}$ such that $$F'[u_n](u_{n+1}-u_n)=-F[u_n].$$

Note that we define $$F'[u](v):=\lim_{\epsilon\to0}\frac{F[u+\epsilon v]-F[v]}{\epsilon}.$$