Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 17 (2009), pp. 133-148.

A multilevel adaptive mesh generation scheme using Kd-trees

Alfonso Limon, Hedley Morris

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:
Key Words: Adaptive grid refinement; Wavelet refined mesh; quadtree grids; multilevel decomposition; codimension-one discontinuities.

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Alfonso Limon
School of Mathematical Sciences
Claremont Graduate University, CA 91711, USA
email: alfonso.limon@cgu.edu
Hedley Morris
School of Mathematical Sciences
Claremont Graduate University, CA 91711, USA
email: hedley.morris@cgu.edu

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