# Stable Mesh Decimation

## TL;DR

This paper introduces a mesh decimation method that considers functional sensitivity, ensuring geometric simplification preserves the accuracy of application-specific functions, unlike traditional geometry-focused approaches.

## Contribution

It proposes a novel mesh decimation approach guided by functional sensitivity analysis using Galerkin finite element and discrete exterior calculus techniques.

## Key findings

- Functional sensitivity analysis improves mesh reduction quality.
- Numerical examples demonstrate the effectiveness of the proposed method.
- The approach better preserves application-specific functions during mesh decimation.

## Abstract

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.