Mathematically correct representation of Newton's method in weak formulation

mathematics
1

Hello everyone,

I would like to know what is actually going on when we use Newton’s method to solve (for example non-linear) variational problem.
Suppose we have a functional F[u, w] = \int_{\Omega} f(u, w) dx representing the weak form of an interest, where u(x) is an unknown function and w(x) is the test function defined on the same space as u . According to the definition of functional derivative of F w.r.t. u in some direction v(x) (a function defined on the same space as u(x)) one should write

\delta F[u; v] = \int_{\Omega} \frac{\delta F}{\delta u} v dx.

where \delta F is the variation on F or functional differential and \frac{\delta F}{\delta u} is the functional derivative. On the other hand we have a Gateaux derivative definition for F w.r.t. u

D_v F[u] = \frac{d}{d \tau} F[u+\tau v] \Bigg\rvert_{\tau = 0}.

As far as I understand it D_v F[u] is equal to \delta F and is just another way of writing functional differential.

According to the UFL documentation Python function ufl.derivative() computes the the functional differential \delta F. Since the form F[u,w] is the weak form of the interest, the ufl.derivative(F, u, v) gives us another form that is called (at least in the tutorials on solving non-linear problems) Jacobian of this weak form F[u, w]. That is used to solve the initial non-linear problem using Newton’s method. According to this method one has to solve subsequent linear problems of the form Ax=b where A is the obtained Jacobian of the weak form and b is euqal to - F[u, w] so that we have

J[u^i, w] \delta u^{i+1} \equiv \frac{\delta F}{\delta u} \Bigg\rvert_{u^i} \delta u^{i+1} = - F[u^i, w].

where \delta u is the same as v.

The last formula confuses me a lot because according to the Newton’s method the Jacobian has to be a functional derivative, but in FEniCS we supply a functional differential as a Jacobian. And to my understanding if we do the latter and want to follow the Newton’s approach we should solve the next formulation

\int_{\Omega} \frac{\delta F}{\delta u} \Bigg\rvert_{u^i} \delta u^{i+1} dx = - \int_{\Omega} F[u^i, w] dx

but not the

\int_{\Omega} \frac{\delta F}{\delta u} \Bigg\rvert_{u^i} \delta u^{i+1} dx = - F[u^i, w].

What have I missed?

2

I would suggest reading Pages 38-46 of: https://launchpadlibrarian.net/83776282/fenics-book-2011-10-27-final.pdf

1 Like
3

Thank you @dokken for providing a right direction. The mystery is solved for me :slightly_smiling_face:

A short answer that I was looking for is that I have confused discrete and continuous approaches.

In the discrete (or algebraic level) approach we are dealing with the Jacobi matrix and the eq.3 (J_{ij} \delta u_j = - F_i) holds. However, in the continuous limit of the discrete approach matrix-vector multiplication of J_ij \delta u_j turns into an inner product \int J \delta u dx. Since both J_{ij} \delta u_j and \int J \delta u dx are the functional differentials (\delta F) in the direction of \delta u, there is no inconsistency. Moreover, according to the definition of the functional differential \delta F = D_{\delta u} F[u] it is a Gateaux derivative of F w.r.t u in the direction \delta u. That is precisely what the function ufl.derivative() does.