Seventh Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 17 (2009), pp. 133-148. 

Title: A multilevel adaptive mesh generation scheme using Kd-trees

Authors: Alfonso Limon (Claremont Graduate Univ., CA, USA)
         Hedley Morris (Claremont Graduate Univ., CA, USA)
 
Abstract: 
 We introduce a mesh refinement strategy for PDE based simulations
 that benefits from a multilevel decomposition. Using Harten's MRA
 in terms of Schroder-Pander linear multiresolution analysis
 [20], we are able to bound discontinuities in
 $\mathbb{R}$. This MRA is extended to $\mathbb{R}^n$ in terms
 of n-orthogonal linear transforms and utilized to identify cells
 that contain a codimension-one discontinuity. These refinement
 cells become leaf nodes in a balanced Kd-tree such that a
 local dyadic MRA is produced in $\mathbb{R}^n$, while maintaining
 a minimal computational footprint. The nodes in the tree form an
 adaptive mesh whose density increases in the vicinity of a discontinuity.

Published April 15, 2009.
Math Subject Classifications: 35R05, 65N50.
Key Words: Adaptive grid refinement; Wavelet refined mesh; quadtree grids;
           multilevel decomposition; codimension-one discontinuities.