Introduction

This section contains an overview of general approach to formulating the physics that is solved in Allsolve.

Vector calculus identities

Divergence of a curl

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

Leibniz rule for nabla operator

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

Per partes theorem for scalars

Let $U$ and $V$ be a scalar functions, $\Omega$ a set with boundary $\Gamma$ . Then let $\boldsymbol{n}$ be a unit normal vector pointing outwards. Then

\begin{align} \int_{\Omega}\; U\nabla V\; d \Omega = \int_{\Gamma}\; UV\boldsymbol{n}\; d \Gamma - \int_{\Omega}\; V\nabla U\; d \Omega \end{align}

Per partes theorem for vectors

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

\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 \end{align}

Divergence theorem

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

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

Strong and weak formulation

Common example of PDE (partial differential equation) is the Poisson’s equation.

\begin{align} \Delta u &= f\; \text{on}\ \Omega \end{align}

This is the strong formulation of Poisson’s equation. The strong formulation refers to a precise mathematical representation of a problem with very strict requirements for a solution. The weak formulation is an alternative mathematical representation that relaxes the regularity requirements for solutions. The weak formulation is computationally preferable as it provides the foundation for numerical methods such as the Finite Element Method (FEM). Both Divergence theorem and Per partes theorem (Integration by parts) can be applied on the strong formulation. Using these theorems we derive the weak formulation from the strong formulation. To begin, we choose a test function $u^{\prime}$ . We then multiply both sides of the differential equations by $u^{\prime}$ and integrate over $\Omega$ .

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

For the first term of the left side we apply Per partes theorem.

In this case $U = u^{\prime}$ , $V = \nabla u$ and $\boldsymbol{n}$ is the unit normal vector.

\begin{align} \int_{\Omega}\; \Delta u\; u^{\prime}\; d \Omega = \int_{\Omega}\; (\nabla \cdot \nabla u)\; u^{\prime}\; d \Omega = \int_{\Gamma}\; (\nabla u \cdot \boldsymbol{n})\; u^{\prime}\; d \Gamma - \int_{\Omega}\; \nabla u \cdot \nabla u^{\prime}\; d \Omega \end{align}

Overall we get a weak formulation of (3).

\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 \end{align}

Now, we have less strict regularity requirements for $u$ and $f$ . Certainly we know that solution to (3) is also a solution to (7) and every weak solution smooth enough is a solution to (3).