Divergence Theorem

The divergence theorem states that the volume integral of the divergence of a vector field over a volume $$$V$$$ bounded by a surface $$$S$$$ is equal to the surface integral of the vector field projected on the outward facing normal of the surface $$$S$$$.

$$$$\int_\Omega \nabla F dV = \int_{\partial\Omega} F\cdot\mathbf{n}dS$$$$

Product Rule

Product rule for the product of a scalar $$$a$$$ and a vector $$$\mathbf{b}$$$ is useful to reduce the derivative order on an expression in conjunction with the divergence theorem.

$$$$\nabla (a\mathbf{b}) = \nabla a \cdot \mathbf{b} + a \nabla\cdot\mathbf{b}$$$$

Shuffle the terms (and note that this is valid for a vector $$$\mathbf{a}$$$ and a scalar $$$b$$$ as well)

$$$$-\nabla a \cdot \mathbf{b} = a \nabla\cdot\mathbf{b} - \nabla (a\mathbf{b})$$$$

$$$$-\nabla \cdot \mathbf{a} b = \mathbf{a}\cdot \nabla b - \nabla (\mathbf{a}b)$$$$

The right most term ($$$\nabla(\dots)$$$) can be transformed using the divergence theorem. This can be used to effectively shift a derivative over to the test function when building a residual.

Fundamental Lemma of calculus of variations

The functional derivative in the Cahn-Hilliard equation can be calculated using this rule

$$$$\frac{\delta F}{\delta c} = \frac{\partial f}{\partial c} - \nabla \cdot \frac{\partial f}{\partial\nabla c}$$$$

Weak form of the ACInterface Kernel

The term $$$L\nabla(\kappa\nabla\eta)$$$ is multiplied with the test function $$$\psi$$$ and integrated, yielding

$$$$\left(L\nabla(\kappa\nabla\eta),\psi\right) = \left(\nabla \cdot \underbrace{(\kappa\nabla\eta)}_{\equiv \mathbf{a}},\underbrace{L\psi}_{\equiv b}\right)$$$$

we moved the $$$L$$$ over to the right and identify a vector term $$$\mathbf{a}$$$ and a scalar term $$$b$$$. Then we use the third equality in the Product rule section to obtain

$$$$-\mathbf{a}\cdot \nabla b + \nabla (\mathbf{a}b).$$$$

The last term is converted into a surface integral using the Divergence theorem, tielding

$$$$- \left( \kappa\nabla\eta,\nabla(L\psi) \right) + \left< L\kappa \nabla\eta \cdot \vec n, \psi\right>$$$$