David Aristoff's homepage Title: On orienting the edges of quadrilateral and hexahedral finite element meshes.

Abstract: When working with either higher order finite elements, or using Nedelec elements on quadrilaterals in 2d, one needs to define a coordinate system on each edge. On the other hand, a coordinate system is already defined on each cell by virtue of the mapping from the reference cell. This raises the question whether it is possible to somehow align the coordinate systems of cells and edges in such a way that they "naturally" fit as this can make finite element implementations significantly simpler. In this talk, I will show that this question is best formulated in terms of graph theory, and will provide a proof that such a "natural fit" always exists that is based on the (global) geometry of curves in the plane. I will also show an algorithm that finds such edge and cell orientations in optimal O(N) complexity. Finding the right approach to proving such statements in 2d also allows to generalize to hexahedral meshes in 3d. I will show that not all hexahedral meshes allow for "natural fit" edge orientations (and why). There are, however, special classes of hexahedral meshes for which this is always possible.