Greedy bisection generates optimally adapted triangulations

TL;DR

This paper analyzes a greedy algorithm for creating data-adapted anisotropic triangulations that optimize local approximation errors, showing it produces near-optimal aspect ratios and convergence for convex functions.

Contribution

It proves that the greedy bisection algorithm yields triangles with optimal aspect ratios based on the Hessian and achieves asymptotically optimal convergence bounds for convex functions.

Findings

Triangles tend to adopt an optimal aspect ratio dictated by the local Hessian.

The algorithm produces nested triangulations with minimized local approximation errors.

For convex functions, the triangulations satisfy known asymptotically optimal convergence bounds.

Abstract

We study the properties of a simple greedy algorithm for the generation of data-adapted anisotropic triangulations. Given a function f, the algorithm produces nested triangulations and corresponding piecewise polynomial approximations of f. The refinement procedure picks the triangle which maximizes the local Lp approximation error, and bisect it in a direction which is chosen so to minimize this error at the next step. We study the approximation error in the Lp norm when the algorithm is applied to C2 functions with piecewise linear approximations. We prove that as the algorithm progresses, the triangles tend to adopt an optimal aspect ratio which is dictated by the local hessian of f. For convex functions, we also prove that the adaptive triangulations satisfy a convergence bound which is known to be asymptotically optimal among all possible triangulations.