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Adaptive Mesh Refinement for arbitrary initial Triangulations

Lars Diening, Lukas Gehring, Johannes Storn

TL;DR

It is shown that Maubach’s routine with this initialization always terminates and generates meshes that preserve shape regularity and satisfy the closure estimate needed for optimal convergence of adaptive schemes.

Abstract

We introduce a simple initialization of the Maubach bisection routine for adaptive mesh refinement which applies to any conforming initial triangulation and terminates in linear time with respect to the number of initial vertices. We show that Maubach's routine with this initialization generates meshes that preserve shape regularity and satisfy the closure estimate needed for optimal convergence of adaptive schemes. Our ansatz allows for the intrinsic use of existing implementations.

Adaptive Mesh Refinement for arbitrary initial Triangulations

TL;DR

It is shown that Maubach’s routine with this initialization always terminates and generates meshes that preserve shape regularity and satisfy the closure estimate needed for optimal convergence of adaptive schemes.

Abstract

We introduce a simple initialization of the Maubach bisection routine for adaptive mesh refinement which applies to any conforming initial triangulation and terminates in linear time with respect to the number of initial vertices. We show that Maubach's routine with this initialization generates meshes that preserve shape regularity and satisfy the closure estimate needed for optimal convergence of adaptive schemes. Our ansatz allows for the intrinsic use of existing implementations.
Paper Structure (10 sections, 17 theorems, 54 equations, 11 figures, 1 table, 4 algorithms)

This paper contains 10 sections, 17 theorems, 54 equations, 11 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Novel Initialization
  3. Theoretical Investigation of the novel Initialization
  4. Initial Coloring with $N = n$
  5. Initial Coloring with $N \geq n$
  6. Verification of Theorems \ref{['thm:basic-properties']} and \ref{['thm:closuregcolored']}
  7. Numerical Experiments
  8. Experiment 1 (Properties of the algorithm)
  9. Experiment 2 (Comparison)
  10. Shape regularity

Key Result

lemma 1

Let the coloring $\frc$ result from Algorithm algo:coloring. Then the maximal degree bounds the largest color in the sense that

Figures (11)

  • Figure 1: Colorable triangulation (left) and a non-colorable triangulation (right) for $n=2$
  • Figure 2: Initial triangulation with bisection edges (marked by arrows) where the refinement of the simplex on the left causes the refinement of all simplices.
  • Figure 3: $4$-coloring of the non-colorable triangulation of Figure \ref{['fig:noncolorable']} and the coloring of the virtual extension to $\setR^3$, see Remark \ref{['rem:virtual-extension']}.
  • Figure 4: Refinements of of $\mathcal{T}_0$ (last picture) and refinements of the virtual extension $\mathcal{T}_0^+$ (first three pictures) induced by the displayed refinement point $(1,7/16,3/16)$ as explained in Remark \ref{['rem:problem-virtual']}. The picture is rotated such that the plane $\set{x_1=1}$ is in the front.
  • Figure 5: A triangulation which can be tagged satisfying the initial conditions by Binev--Dahmen--DeVore and Stevenson with good shape regularity and closure estimate, but its coloring needs many colors for obtaining the new edges illustrated by the dotted lines.
  • ...and 6 more figures

Theorems & Definitions (43)

  • definition 1: $N{+}1$-Coloring
  • definition 2: Maubach initialization
  • remark 1: Alternative sorting
  • remark 2: Traxler
  • definition 3: Triangulation
  • lemma 1: Largest color
  • proof
  • theorem 1: Basic properties
  • theorem 2: Closure estimate
  • remark 3: Colorability
  • ...and 33 more