Jacobian matrices on unstructured grids: underlying map?

Question

Suppose I have an unstructured polygonal mesh system like so:

Each node $x$ has Cartesian coordinates $(x_1,x_2)$, so for a given node can form matrices like this:

$$ J(x,y,z) = \left(\begin{array}{cc} y_1-x_1 & z_1-x_1 \\ y_2-x_2 & z_2-x_2 \end{array}\right) = (y-x,z-x) $$

If my mesh were structured, i.e. a mapped Cartesian grid so that $(x_1,x_2) = (x_1(\xi,\eta),x_2(\xi,\eta))$, then these matrices would be approximations to the Jacobian matrix of the map $(\xi,\eta)\mapsto x$.

My question is, for an unstructured mesh like the one above, what are my $J(x,y,z)$ approximations of? Is there a well-defined, underlying "continuum" mapping for an unstructured grid?

Answer 1

For quadrilaterals and triangles, one standard thing to do is to use the isoparametric map. That is, you use the finite element basis you are about to use to approximate the solution to map your physical-space element to a "reference element" from the $\xi, \eta$ to $x, y$. You obviously have to add interior and edge nodes to do this for higher-order elements. Another common approach is to use the piecewise (bi-)linears for everything regardless of the actual finite-element order. If your boundaries are straight, the two approaches are equivalent for triangles. The typical reference elements are $[-1,1]\times[-1,1]$ and a right unit isosceles triangle in the upper right quadrant.

I presume you could do something similar for higher-degree polygons, but I've never done that.