On the Bisection Method for Triangles

On Faster Convergence of the Bisection Method

Let AABC be a triangle with vertices A, B, and C. It is "bisected" as follows: choose a/the longest side (say AB) of AABC, let D be the midpoint of AB, then replace AABC by two triangles aA.DC and ADBC. Let Ag| be a given triangle. Bisect Aqj into two triangles A¡¡ and Aj2. Next bisect each Aj(-, /= 1, 2, forming four new triangles A2j-, i = 1,2, 3,4. Continue thus, forming an infinite sequence...

Complexity of the bisection method

The bisection method is the consecutive bisection of a triangle by the median of the longest side. In this paper we prove a subexponential asymptotic upper bound for the number of similarity classes of triangles generated on a mesh obtained by iterative bisection, which previously was known only to be finite. The relevant parameter is γ/σ, where γ is the biggest and σ is the smallest angle of t...

The Bisection Method: Which Root?

A New Spectral Method on Triangles

We propose in this note a spectral method on triangles based on a new rectangle-to-triangle mapping, which leads to more reasonable grid distributions and efficient implementations than the usual approaches based on the collapsed transform. We present the detailed implementation for spectral approximations on a triangle and discuss the extension to spectral-element methods and three dimensions.

A tensor extension of the bisection method

where the involved functions f and g are continuous (not necessarily differentiable) functions. The method is a tensor extension of the bisection method. Classical generalizations of bisection method are based on the topological degree, are called exclusion algorithms and are intended for separating all solutions of a system of equation in a given region [20,11,21,23]. More recent research line...