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# Non-linear static problems

Here we propose to solve the following non-linear academic problem of minimization of a functional:

$J(u) = \int_\Omega \frac{1}{2} f(|\nabla u|^2) - u*b$

where $$u$$ is function of $$H^1_0(\Omega)$$ and $$f$$ defined by:

$f(x) = a*x + x-ln(1+x),\ f'(x) = a+\frac{x}{1+x},\ f''(x) = \frac{1}{(1+x)^2}$

## Newton-Raphson algorithm

Now, we solve the Euler problem $$\nabla J (u) = 0$$ with Newton-Raphson algorithm, that is:

$u^{n+1} = u^n - ( \nabla^2 J (u^{n}))^{-1}*\nabla J(u^n)$
 1// Parameters
2real a = 0.001;
3func b = 1.;
4
5// Mesh
6mesh Th = square(10, 10);
7Th = adaptmesh(Th, 0.05, IsMetric=1, splitpbedge=1);
8plot(Th, wait=true);
9
10// Fespace
11fespace Vh(Th, P1);
12Vh u=0;
13Vh v, w;
14
15fespace Ph(Th, P1dc);
16Ph alpha; //to store |nabla u|^2
17Ph dalpha ; //to store 2f''(|nabla u|^2)
18
19// Function
20func real f (real u){
21    return u*a + u - log(1.+u);
22}
23func real df (real u){
24    return a +u/(1.+u);
25}
26func real ddf (real u){
27    return 1. / ((1.+u)*(1.+u));
28}
29
30// Problem
31//the variational form of evaluate dJ = nabla J
32//dJ = f'()*(dx(u)*dx(vh) + dy(u)*dy(vh))
33varf vdJ (uh, vh)
34    = int2d(Th)(
35          alpha*(dx(u)*dx(vh) + dy(u)*dy(vh))
36        - b*vh
37    )
38    + on(1, 2, 3, 4, uh=0)
39    ;
40
41//the variational form of evaluate ddJ = nabla^2 J
42//hJ(uh,vh) = f'()*(dx(uh)*dx(vh) + dy(uh)*dy(vh))
43//  + 2*f''()(dx(u)*dx(uh) + dy(u)*dy(uh)) * (dx(u)*dx(vh) + dy(u)*dy(vh))
44varf vhJ (uh, vh)
45    = int2d(Th)(
46          alpha*(dx(uh)*dx(vh) + dy(uh)*dy(vh))
47        + dalpha*(dx(u)*dx(vh) + dy(u)*dy(vh))*(dx(u)*dx(uh) + dy(u)*dy(uh))
48    )
49    + on(1, 2, 3, 4, uh=0)
50    ;
51
52// Newton algorithm
53for (int i = 0; i < 100; i++){
54    // Compute f' and f''
55    alpha = df(dx(u)*dx(u) + dy(u)*dy(u));
56    dalpha = 2*ddf(dx(u)*dx(u) + dy(u)*dy(u));
57
58    // nabla J
59    v[]= vdJ(0, Vh);
60
61    // Residual
62    real res = v[]'*v[];
63    cout << i << " residu^2 = " << res << endl;
64    if( res < 1e-12) break;
65
66    // HJ
67    matrix H = vhJ(Vh, Vh, factorize=1, solver=LU);
68
69    // Newton
70    w[] = H^-1*v[];
71    u[] -= w[];
72}
73
74// Plot
75plot (u, wait=true, cmm="Solution with Newton-Raphson");

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