Formulations and theorems

Allsolve solves simulations using mathematical formulations that model the underlying physics. This page introduces the general approach to formulating physics — from strong form to weak form — together with the vector calculus identities and integral theorems referenced by every physics derivation on this site.

Vector calculus identities

Divergence of a curl

\begin{align*} \nabla \cdot (\nabla \times \boldsymbol{A}) = 0 \tag{1} \end{align*}

Leibniz rule for a nabla operator

\begin{align*} \nabla \cdot (a \boldsymbol{b}) = \nabla a \cdot \boldsymbol{b} + a(\nabla \cdot \boldsymbol{b}) \tag{2} \end{align*}

Divergence of a cross product

\begin{align*} \nabla \cdot (\boldsymbol{a} \times \boldsymbol{b}) = (\nabla \times \boldsymbol{a}) \cdot \boldsymbol{b} - \boldsymbol{a} \cdot (\nabla \times \boldsymbol{b}) \tag{3} \end{align*}

Scalar triple product rule

\begin{align*} \boldsymbol{a} \cdot (\boldsymbol{b} \times \boldsymbol{c}) = \boldsymbol{c} \cdot (\boldsymbol{a} \times \boldsymbol{b}) = \boldsymbol{b} \cdot (\boldsymbol{c} \times \boldsymbol{a}) \tag{4} \end{align*}

Time derivative of a curl

\begin{align*} \frac{\partial}{\partial t} (\nabla \times \boldsymbol{A}) = \nabla \times \frac{\partial \boldsymbol{A}}{\partial t} \tag{5} \end{align*}

Integration by parts

Let $\boldsymbol{A}$ be a vector field and $f$ a scalar function defined on a domain $\Omega$ with boundary $\Gamma$ . Then let $\boldsymbol{n}$ be a unit normal vector pointing outwards on $\Gamma$ . Now

\begin{align*} \int_{\Omega}\; (\boldsymbol{A} \cdot \nabla f)\; d \Omega = \int_{\Gamma}\; f\; (\boldsymbol{A} \cdot \boldsymbol{n})\; d \Gamma - \int_{\Omega}\; f\; (\nabla \cdot \boldsymbol{A})\; d \Omega. \tag{6} \end{align*}

Divergence theorem

Let $\Omega$ be a set with boundary $\Gamma$ , let $\boldsymbol{n}$ be a unit normal vector pointing outwards on $\Gamma$ and let $\boldsymbol{T}$ be a vector field or a second-order tensor. Now

\begin{align*} \int_{\Omega} \nabla \cdot \boldsymbol{T}\; d \Omega = \int_{\Gamma}\; \boldsymbol{T} \cdot \boldsymbol{n}\; d \Gamma. \tag{7} \end{align*}

Strong and weak formulation

A common example of a partial differential equation (PDE) is Poisson’s equation,

\begin{align*} \Delta u &= f\; \text{on}\ \Omega. \tag{8} \end{align*}

This is the strong formulation — a precise mathematical statement with strict regularity requirements for a solution.

The weak formulation relaxes those requirements. It is computationally preferable because it provides the foundation for numerical methods such as the finite element method (FEM).

To derive the weak formulation of Poisson’s equation, multiply both sides of the strong formulation by a test function $u^{\prime}$ and integrate over $\Omega$ :

\begin{align*} \int_{\Omega}\; (\Delta uu^{\prime} - fu^{\prime})\; d \Omega = 0 \tag{9} \end{align*}

Apply the Leibniz rule for a nabla operator, then use the Divergence theorem on the divergence term:

\begin{align*} \int_{\Omega}\; \Delta u\; u^{\prime}\; d \Omega &= \int_{\Omega}\; \nabla \cdot (u^{\prime} \nabla u)\; d \Omega - \int_{\Omega}\; \nabla u \cdot \nabla u^{\prime}\; d \Omega \tag{10} \\[5pt] &= \int_{\Gamma}\; (\nabla u \cdot \boldsymbol{n})\; u^{\prime}\; d \Gamma - \int_{\Omega}\; \nabla u \cdot \nabla u^{\prime}\; d \Omega \tag{11} \end{align*}

The resulting weak formulation is

\begin{align*} \int_{\Gamma}\; (\nabla u\ \cdot \boldsymbol{n})\; u^{\prime}\; d \Gamma - \int_{\Omega}\; (\nabla u \cdot \nabla u^{\prime} + f\; u^{\prime})\; d \Omega = 0. \tag{12} \end{align*}

The regularity requirements for $u$ and $f$ are now less strict. A solution to $\text{(8)}$ is also a solution to $\text{(12)}$ , and every solution to $\text{(12)}$ that is smooth enough satisfies $\text{(8)}$ as well.