Modelling question: example of a physical phenomenon with this jump condition at an interface?

Question

in our finite element class we were talking about interface problems our teacher came up with the following, where $K_i$ are two given functions and $u_i$ is the restriction of the solution $u$ to $\Omega_i$.

$$-\nabla \cdot (K_i \nabla u_i) =f_i \text{ in } \Omega_i$$ $$u_i=0 \text{ on } \partial \Omega \cap \Omega_i$$ $$n_1 \cdot K_1 \nabla u_1 + n_2 \cdot \nabla u_2 = 0 \text{ on } \Gamma:=\partial \bar{\Omega_1} \cap \partial \bar{\Omega_2}$$ $$[u]=0 \text{ on } \Gamma$$

We assumed that $\Omega_2 \subset \Omega_1$, i.e. one domain is included in another.

Question: what is the physical meaning of the condition $$n_1 \cdot K_1 \nabla u_1 + n_2 \cdot K_2 \nabla u_2 = 0 \text{ on } \Gamma$$ ? Can you provide a simple example where this condition means something from the physical standpoint?

For sure it is a continuity condition, but I cannot imagine a physical situation where this occurs. Examples from elasticity/solid mechanics are of course welcome.

Answer 1

$\vec\Phi = K_i \nabla u_i$ is the flux across the interface. For example, if $u$ is the thermal energy density and $K$ the thermal conductivity, then $\vec\Phi$ is the thermal energy flux. Energy conservation then dictates that whatever flows into the interface on one side ($\vec n_1 \cdot K_1 \nabla \vec\Phi_1$) better be equal to what flows out on the other side ($-\vec n_2 \cdot K_2 \nabla \vec\Phi_2$). This then results in the condition you show.

Similarly, if $u$ is the concentration of a substance diffusing in a medium, where the diffusion constant is $K$, you end up with the same condition by considering the conservation of mass of the substance.

Answer 2

In addition to Wolfgang Bangerth's explanation of temperature and concentration, let me give an other application where such interface conditions arise: (linear) elasticity, which has a similar structure to the elliptic equation in your example. Consider the situation with no interface first. Then the equilibrium, constitutive and kinematic equations of linear elasticity read \begin{gather} \sigma\cdot\nabla = \mathbf{0}, \quad \mathbf{x}\in\Omega \\ \sigma = \mathcal{C}:\varepsilon, \quad \mathbf{x}\in\Omega \\ \varepsilon = \nabla^s\mathbf{u}, \quad \mathbf{x}\in\Omega \end{gather} with properly defined boundary conditions, where $\Omega = \Omega_1 \cup \Omega_2$, $\mathcal{C}$ is the Hooke tensor, $\sigma$ is the Cauchy stress tensor, $\varepsilon$ is the linearized strain tensor and $\mathbf{u}$ is the displacement vector.

A physically equivalent formulation is when these equations are written for each subdomain and the subdomains are tied together: \begin{gather} \sigma_i\cdot\nabla = \mathbf{0}, \quad \mathbf{x}\in\Omega_i \\ \sigma_i = \mathcal{C}_i:\varepsilon_i, \quad \mathbf{x}\in\Omega_i \\ \varepsilon_i = \nabla^s\mathbf{u}_i, \quad \mathbf{x}\in\Omega_i \end{gather} with the continuity of the displacement field (no interface separation) and the traction field (Newton's third axiom): \begin{gather} \mathbf{u}_1| = \mathbf{u}_2, \quad \mathbf{x} \in \Gamma \\ \mathbf{t}_i = -\mathbf{t}_j, \quad \mathbf{x} \in \Gamma. \end{gather} By Cauchy's theorem, the traction vector can be expressed with the stress tensor as \begin{equation} \sigma_i|_\Gamma\cdot\mathbf{n}_i = -\sigma_j|_\Gamma\cdot\mathbf{n}_j. \end{equation} If you substitute the constitutive and kinematics equations into this equation, you get the structurally same condition as what your teacher wrote. Here, the fourth-order tensor $\mathcal{C}_i$ corresponds to the scalar $K_i$ and $\mathbf{u}_i$ corresponds to $u_i$.

Remark 1: The above formulae hold even if $\Omega_2 \not\subset \Omega_1$ (i.e. $\Gamma$ is not closed), but $\Omega_1$ and $\Omega_2$ have a common boundary (i.e. $\Gamma \neq \emptyset$).

Remark 2: The same considerations hold for multiple subdomains, in which two neighbouring subdomains $\Omega_i$ and $\Omega_j$ are separated by the interface $\Gamma_{ij}$.

Remark 3: Note that writing the PDEs for each subdomain independently and then tying them together resembles a domain decomposition method. Indeed, if the continuity conditions are imposed by a Lagrange multiplier (which turns out to be equivalent to the interface traction), the resulting mixed formulation is the continuous analogue of the FETI domain decomposition method.