Tangential Gradient Boundary Conditions

Question

Hello, this is perhaps a simple question with an obvious answer, but I am attempting to solve what amouts to laplace's equation on a simple mesh. The issue is I have boundary conditions that specify the tangential gradients on the boundary.

For simplicity I have started with a mesh with boundaries parallel to the X and Y axes, and as an example on the top edge (parallel to the x axis) the boundary condition reads

$\frac{du}{dx}=-1$

I couldn't find anything obvious in the documentation for boundary conditions like this, but I gather the way to do this is with lagrange multipliers, is there an example in the documentation of a similar problem that I can follow?

Comments

You may want to revise the formulation of your problem.

For the scalar elliptic PDE

$$ -\Delta u = f $$

the boundary condition $\nabla u \cdot \tau = \sigma_\tau$ will not lead to a well posed boundary value problem.

Umbe

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Apologies, this is true, I am in fact equipped with a mean zero condition for the field which I forgot to mention, although the actual implementation of the boundary conditions is the actual issue that I have

$$\int_{\partial\Omega}{udx}=0$$

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Hi,

Even if you add the zero-mean condition your problem is still ill-posed.

In fact, the weak form of your PDE is
$$
\int_\Omega \nabla u \cdot \nabla v dx - \int_{\partial \Omega} (\nabla u) \cdot {\bf n} v ds = \int_{\Omega} fv dx.$$

Now in order to get riddle of the boundary integral
$$ \int_{\partial \Omega} (\nabla u) \cdot {\bf n} v ds $$

you have either to

• Impose a Dirichlet boundary condition for u, i.e. u = u0 on $\partial \Omega$. Such Dirichlet condition will allow you to take p = 0 on $\partial \Omega$.

• Impose a Neuman boundary condition $(\nabla u) \cdot {\bf n} = \sigma_n$, so that you can move
  $$ \int_{\partial \Omega} (\nabla u) \cdot {\bf n} v ds = \int_{\partial \Omega} \sigma_n v ds $$
  to the r.h.s.

• Impose a Robin boundary condition $(\nabla u) \cdot {\bf n}+ \alpha u = \sigma_n$ ($\alpha > 0$) and write

$$
\int_\Omega \nabla u \cdot \nabla v dx + \int_{\partial \Omega} \alpha u v ds = \int_{\Omega} fv dx. + \int_{\partial \Omega} \sigma_n v ds. $$

Best,

Umbe