  Current mesh reduction techniques, while numerous, all primarily reduce mesh
size by successive element deletion (e.g. edge collapses) with the goal of
geometric and topological feature preservation. The choice of geometric error
used to guide the reduction process is chosen independent of the function the
end user aims to calculate, analyze, or adaptively refine. In this paper, we
argue that such a decoupling of structure from function modeling is often
unwise as small changes in geometry may cause large changes in the associated
function. A stable approach to mesh decimation, therefore, ought to be guided
primarily by an analysis of functional sensitivity, a property dependent on
both the particular application and the equations used for computation (e.g.
integrals, derivatives, or integral/partial differential equations). We present
a methodology to elucidate the geometric sensitivity of functionals via two
major functional discretization techniques: Galerkin finite element and
discrete exterior calculus. A number of examples are given to illustrate the
methodology and provide numerical examples to further substantiate our choices.
