Discussion Forum

Hello everyone,

I am searching for numerical solution of the following problem:
u_{xx}+u_{yy}=0 in D=[a,b]x[c,d] (that is, 2D Laplace equation in the rectangle)
(u_x)^2+(u_y)^2 =((cosh u)/aa)^2 on \delta D (that is, a nonlinear combination of the partial derivatives and function
values on the rectangle boundary).
Evidently, the main problem is nonlinear boundary conditions.

Trying to use Comsol 3.5 to solve this problem in D=[-1,1]x[1,1] with parameter aa=4, I specified:
1. In Model Navigator: COMSOL Multiphysics -> PDE Modes -> PDE, General Form -> Stationary analysis
Then in the opened Problem Window:
2. Option -> Axes/Grid settings: x from -1 to 1 and y from -1 to 1
3. Chose the centered square form and draw it in the picture window
4. Physics -> Subdomain Settings -> Coefficients: \Gama = -ux , -uy ; F=0 ; ea=0 ; da=0
Note. Actually ea and da should not appear for the stationary problem
5. Physics -> Subdomain Settings -> Init: u(t0)=0, or u(t0)=y, or u(t0)=log(tan(y./8.+3.141593/4))
(the last one satisfies the boundary conditions)
Note. ut(t0)=0 was set in all experiments, although this value should not appear for this problem
6. Physics -> Boundary Settings -> for all 4 boundaries -> Dirichlet boundary condition:
R= ux.^2.+uy.^2.-(cosh(u)./4).^2
Note. G=0 was set in almost all experiments, although this value should not appear for this problem

Depending on specified initial guess ui defined in Physics -> Subdomain Settings->Init I have received
the following results:
1. for ui=0 - the solution is 0; however it is not real solution because the boundary conditions are not
satisfied for u=0.
2. for ui=y - the error message: Failed to find a solution: Maximum number of Newton iterations reached.
Returned solution has not converged.
3. for u(t0)=log(tan(y./8.+3.141593/4)) - the error message: Failed to find a solution: Maximum number of
Newton iterations reached. Returned solution has not converged.
Note. I have tried with refined meshes and also with heat equation expecting to obtain convergence to
stationary regime. Nothing works.

So now my questions:
1. Are all my above specifications correct for the described stationary problem?
2. Why it appears the option for choice of G, if I only need to specify R in the boundary condition R=0?
3. Is my choice for G=0 correct? I have changed this choice to G=1 in some experiments and have
received different results. So it affects somehow the computed solution.
4. There is a way to solve the above problem using Comsol?

Any help is very appreciated.

Sincerely,
Anton Gorodec