Nodal configurations and Voronoi tessellations for triangular spectral elements
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Date
2005-10-07T23:09:49Z
Authors
Roth, Michael James
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Abstract
By combining the high-order accuracy of spectral expansions with the locality and
geometric flexibility of finite elements, spectral elements are an attractive option for
the next generation of numerical climate models. Crucial to their construction is the
configuration of nodes in an element — casual placement leads to polynomial fits
exhibiting Runge phenomena manifested by wild spatial oscillations. I provide highorder
triangular elements suitable for incorporation into existing spectral element
codes. Constructed from a variety of measures of optimality, these nodes possess the
best interpolation error norms discovered to date.
Motivated by the need to accurately determine these error norms, I present an
optimization method suitable for finding extrema in a triangle. It marries a branch
and bound algorithm to a quadtree smoothing scheme. The resulting scheme is both
robust and efficient, promising general applicability.
In order to qualitatively evaluate these nodal distributions, I introduce the concept
of a Lagrangian Voronoi tessellation. This partitioning of the triangle illustrates the
regions over which each node dominates. I argue that distant and disconnected regions
are undesirable as they exhibit a non-physical influence.
Finally, I have discovered a link between point distributions in the simplex and
on the hypersphere. Through a simple transformation, a distance metric is defined
permitting the construction of Voronoi diagrams and the calculation of mesh norms.
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Keywords
numerical climate models, spectral elements, Lagrangian Voronoi tessellation