### How to add single-valued unknown (numbers) to variational problem?

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10 weeks ago by
Dear all,

I am trying to solve following system (in domain $\Omega = (0, 1)$ )

$\int_0^1 F_y (y) \ \hat{y} \ dx + G_y(y(0),a) \ \hat{y}(0) = 0 , \\ F_a(y(0),a) = 0 ,$
where $y(x)$ is spatially dependent function, $\hat{y}(x)$ is also spatially dependent test function and  $a$ is only single unknown (a number). The last equation is only an algebraic constraint (without any derivative). The system is coupled, because the value of $a$ influences the Neumann BC in the first equation.

I would like to use NonlinearVariationalProblem and Newton's method to solve this system but I do not know how to add the unknown $a$ (without any testfunctions or whatever) to the form F.
Do you have any suggestion?
Thank you very much!
Community: FEniCS Project

3
10 weeks ago by
You might take a look at the "Real" type element for global scalar unknowns, as demonstrated here

If I understand the present problem correctly, the extra equation would become the weak subproblem $\int_{\partial\Omega}(1-x)F_a(y,a)b\,ds = 0$, where $b$ is a test "function" associated with $a$, and integrating over $\partial\Omega$ and weighting by $1-x$ is effectively a Dirac measure at $x=0$.  (You could also restrict $ds$ to a subdomain at $x=0$ instead of weighting the integral over the whole boundary.)