Problem with variational formulation

General Discussion
1

Hi,
i have to program the following system
du1/dt =D1d^2u1/dx^2 +a1u1*(1-u1-u2)
du2/dt=D2d^2u2/dx^2 +a2(u2+epsu1)*(1-u1-u2)
in the space and time intervals: 0<x<L, 0<t<T
with bounadry conditions
x=0,L: du1/dx=du2/dx=0
initial conditions:
u1(x,0)=1, 0<x<x_0, u_1(x,0)=0: x_0<x<L; u2(x,0)=0:0<x<L
So i write the variational formulation like this, but the cod mark an
error in the variational formulation (line 45).
I don’t udnderstand where is the problem in my variational formulation.

Thank you in advance for the help

// Parameters
real L=20.;
real T=1000.;
real a1=1.;
real a2=1.;
real D1=1.e-3;
real D2=1.e-3;
real x0=0.1;
int Nbx=1e4;
real dt=0.1;
int Nbt=1e2;
real eps=0.1;

real Dx=L/Nbx; // calcul de dx

load “msh3”
meshL Th=segment(Nbx,[x*L,0.]);
fespace Vh(Th,typeFE);
Vh Id = (x<=x0) ? 1. : 0.;

Vh u1,u2;
Vh u1H,u2H;

Vh oldu1=Id;
Vh oldu2=0.0;

Vh nith1=0.0; //l’intération de la non linéarité
Vh nith2=0.0; //l’intération de la non linéarité
Vh g1,g2;

  problem systemU1(u1,u1H,solver=LU)=
    int1d(Th)(u1*u1H/dt + D1*dx(u1)*dx(u1H))
     - int1d(Th)(oldu1*u1H/dt+a1*(u1-(g1-u1*u2))*u1H)
    ;

  problem systemU2(u2,u2H,solver=LU)=
    int1d(Th)(u2*u2H/dt + D2*dx(u2)*dx(u2H))
    - int1d(Th)(oldu2*u2H/dt-a2*[u2-u2*u1-g2+eps*u1-eps*g1-eps*u1*u2]*u2H)
    ;
2

Here you have non-linear terms in coupled equations. So, first linearized it using Newton’s method which is also given here ( Newton Method for the Steady Navier-Stokes equations).