Imposing only normal or tangential direction Dirichlet boundary conditions in the weak form of a Poisson equation

Question

Consider the vector Poisson equation

\begin{align} -\Delta u &= f \text{ in } \Omega \subset \mathbb{R}^d \tag{1}\\ u\cdot v &= 0 \text{ on } \partial \Omega \tag{2} \end{align}

where $v = n$ or $v = \tau$ where $n$ is normal to $\partial \Omega$ and $\tau$ is tangent to the boundary and $d = 2,3$.

Before imposing any boundary conditions, the weak form of this equation is: We seek $u\in (H^1(\Omega))^d$ such that for all $v \in (H^1_0(\Omega))^d$

$$(\nabla u, \nabla v) - \int_{\partial \Omega} n^T \cdot \nabla u \cdot v \,ds= (f,v)$$

Now, normally, if we have (2) for both $n$ and $\tau$, then we can impose the homogeneous condition by noticing the boundary term is exactly zero. However, if we only have one of them, then I'm at a loss. Here is what I've tried: We rewrite $u = u_n + u_\tau = (u\cdot n)n + (u\cdot \tau) \tau$ and then whichever component is zero (say we want $u_n = 0$) we substitute that into our boundary integral to get

$$ (\nabla u, \nabla v) - \int_{\partial \Omega} n^T \cdot \nabla u_\tau \cdot v \,ds= (f,v) $$

However, it's not clear to me that $\nabla u_\tau$ makes any sense. For example, if the boundary in question happens to be the interval $x\in[0,1]$ $\nabla u_\tau$ would have to make sense of $$\frac{\partial u_\tau}{\partial y}$$

which does not seem to have any intuitive interpretation when the tangent vector has no dependence on $y$. Does this in fact, have a good interpretation? If not how can this boundary condition be applied?

Answer 1

The issue is that the equation $$ \mathbf{u} \cdot \mathbf{n}=0$$ or $$ \mathbf{u} \cdot \mathbf{t}=0$$ is a constraint equation rather than a normal flux, which is what enters naturally in the weak form of the equation.

The weak form derivation goes something like this. Multiply by $\mathbf{v}$: $$-\int_\Omega \nabla \cdot \nabla \mathbf{u} \cdot \mathbf{v} dV=\int_\Omega \mathbf{f} \cdot \mathbf{v} dV$$ Now, perform partial integration: $$-\int_{\partial\Omega} (\mathbf{n} \cdot \nabla \mathbf{u}) \cdot \mathbf{v}dS+\int_\Omega \nabla \mathbf{u} \cdot \nabla \mathbf{v} dV=\int_\Omega \mathbf{f} \cdot \mathbf{v} dV$$ The interesting term here is the boundary integral: $$-\int_{\partial\Omega} (\mathbf{n} \cdot \nabla \mathbf{u}) \cdot \mathbf{v}dS$$ which shows that we control the flux/force $\mathbf{g}$ on the boundary on the following form: $$-\int_{\partial\Omega} \mathbf{g} \cdot \mathbf{v}dS$$ On strong form this becomes: $$\mathbf{n} \cdot \nabla \mathbf{u}=\mathbf{g}$$ So the weak form shows that, on the boundary, we do not get direct control over $\mathbf{u}$ and its normal and tangential projections. Instead we only get "access" to the normal projection of the gradient: $$\mathbf{a}_n=\mathbf{n} \cdot \nabla \mathbf{u}$$ (Note that this vector can be "leaning" and doesn't have to be in the normal direction.) One formal way to get control over constraints is to add a Lagrange multiplier to the boundary flux. Something like: $$\mathbf{n} \cdot \nabla \mathbf{u}=\mathbf{P} \vec{\lambda}$$ where $\mathbf{P}$ is a matrix, and find a vector $\vec{\lambda}$ so that the constraint is fulfilled. This vector represents the flux that you need in order to fulfill the constraint (to every constraint there is a corresponding flux). Now we are getting into theory of Lagrange multipliers.

Let's now partition the boundary up in two (possibly more as needed) parts. One $\partial \Omega_1$ with a Neumann condition and one $\partial \Omega_2$ with a constraint (general Dirichlet) condition.

We can now use the following energy functional (Dirichlet's principle):

$$E=\frac{1}{2}\int_\Omega \nabla \mathbf{u} \cdot \nabla \mathbf{u} dV-\int_{\Omega} \mathbf{f} \cdot \mathbf{u} dV-\int_{\partial \Omega_1} \mathbf{g} \cdot \mathbf{u} dS-\int_{\partial \Omega_2} (P\mathbf{u}-b) \cdot \mathbf{\lambda} dS$$

where $\lambda=(\lambda_1,\lambda_2)$ is a Lagrange multiplier vector.

To simplify matters, let's only consider part of the functional: $$E=\frac{1}{2}\int_\Omega \nabla \mathbf{u} \cdot \nabla \mathbf{u} dV-\int_{\partial \Omega_2} (P\mathbf{u}-b) \cdot \mathbf{\lambda} dS$$

Now, consider the stationarity of the first variation of this functional: $$\delta E=\int_\Omega \nabla \mathbf{u} \cdot \nabla \delta\mathbf{u} dV-\int_{\partial \Omega_2} P\delta \mathbf{u} \cdot \mathbf{\lambda} dS-\int_{\partial \Omega_2} (P\mathbf{u}-b) \cdot \delta \mathbf{\lambda} dS=0$$

If we set $\mathbf{v}=\delta \mathbf{u}=(\delta u_1,\delta u_2)=(v_1,v_2)$ and $\mu=\delta \lambda=(\mu_1,\mu_2)$ this becomes: $$\int_\Omega \nabla \mathbf{u} \cdot \nabla \mathbf{v} dV-\int_{\partial \Omega_2} P\mathbf{v} \cdot \mathbf{\lambda} dS-\int_{\partial \Omega_2} (P\mathbf{u}-b) \cdot \mathbf{\mu} dS=0$$

If we now have constraints in both tangential and normal direction, assuming 2D and $\mathbf{b}=0$, then we have the constraint relationship:

$$P\mathbf{u}=\begin{bmatrix} n_x & n_{y}\\ t_{x} & t_{y}\\ \end{bmatrix} \begin{bmatrix} u_1\\ u_2\\ \end{bmatrix}= \begin{bmatrix} 0\\ 0\\ \end{bmatrix}$$

Let's say we now only have a normal constraint, then we get: $$P\mathbf{u}=\begin{bmatrix} n_x & n_{y}\\ 0 & 0\\ \end{bmatrix} \begin{bmatrix} u_1\\ u_2\\ \end{bmatrix}= \begin{bmatrix} n_x u_1+n_y u_2\\ 0\\ \end{bmatrix}=\begin{bmatrix} 0\\ 0\\ \end{bmatrix} $$ Here, the constraint matrix is:

$$A=\begin{bmatrix} n_x & n_{y}\\ 0 & 0\\ \end{bmatrix}$$

The two Lagrange multiplier terms are now:

$$\int_{\partial \Omega_2} P\mathbf{v} \cdot \mathbf{\lambda} dS=\int_{\partial \Omega_2} (n_x v_1+n_y v_2) \lambda_1 dS$$

and

$$\int_{\partial \Omega_2} P\mathbf{u} \cdot \mathbf{\mu} dS=\int_{\partial \Omega_2} (n_x u_1+n_y u_2) \mu_1 dS$$

Note that whether you put the constraint in the first or second row of the constraint matrix is somewhat arbitrary.

For the equation that is not acted on by the Lagrange multiplier you can apply a Neumann condition.

I didn't find any quite satisfactory references. These are some that might be helpful: https://users.oden.utexas.edu/~oden/Dr._Oden_Reprints/1982-008.finite_element.pdf

Also the appendix of this book: https://www.amazon.com/Incompressible-Finite-Element-Isothermal-Laminar/dp/0471492507

These references are far from perfect. I will put in more here if I find any.