Stable Mesh Decimation

Current mesh reduction techniques, while numerous, all pri-marily 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 adap-tively refine. In this paper, we argue that such a decou-pling 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 decima-tion, 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 computa-tion (e.g. integrals, derivatives, or integral/partial differen-tial equations). We present a methodology to elucidate the geometric sensitivity of functionals via two major functional discretization techniques: Galerkin finite element and dis-crete exterior calculus. A number of examples are given to illustrate the methodology and provide numerical examples to further substantiate our choices. 1