{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Building our operators: the Face Divergence" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The divergence is the integral of a flux through a closed surface as that enclosed volume shrinks to a point. Since we have discretized and no longer have continuous functions, we cannot fully take the limit to a point; instead, we approximate it around some (finite!) volume: *a cell*. The flux out of the surface ($\\vec{j} \\cdot \\vec{n}$) is actually how we discretized $\\vec{j}$ onto our mesh (i.e. $\\bf{j}$) except that the face normal points out of the cell (rather than in the axes direction). After fixing the direction of the face normal (multiplying by $\\pm 1$), we only need to calculate the face areas and cell volume to create the discrete divergence matrix.\n", "\n", "\n", "\n", "