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Advanced Numerical Methods in the Mathematical Sciences

Andrew Gillette, University of Arizona
What is a Good Linear Finite Element... on a Generic Polytope?


  • Andrew Gillette, University of Arizona
  • Alexander Rand, CD-adapco


The notion of what constitutes a "good" linear finite element on geometries other than simplices and cubes remains largely unexplored. We use harmonic coordinates as a means to investigate this question, arriving at a few key conclusions. On convex polygons, harmonic coordinates in general provide no improvement over standard interpolation on the constrained Delaunay triangulation of the polygon. On non-convex polygons, however, harmonic coordinates can provide optimal interpolation estimates even when all triangulations fail to do so. We also present the extension and implication of these results to finite elements on non-convex polyhedra in 3D.