Question

I've been working on a new method for 2-dimensional FEM on Riemannian Manifolds that involves using geodesic Triangles instead of approximating them in an embedded form using "traditional" triangles.

I'm quite far along, and up until now I wasn't even sure if my Method would only work on an abstract level or if it can actually be applied. So far, it works for trivial cases ($\mathbb{R}^2, S^2$,...), which is "good enough" at this point.

Now my problem is this: I've been searching high and low (using Zentralblatt MATH) for any research related to mine in any way. So far, all I could find is a single paper (http://vs24.kobv.de/documents-matheon/580/6086_sander_geodesic_fe.pdf).

I find it hard to believe that nothing else exists on this matter.

My questions are:

• Does anyone know any research done in this direction?
• Or is the Idea "useless" (as in, it wouldn't improve the result)?
• Or have I simply been searching wrong?

Any feedback is appreciated

Answer 1

I do not believe this is exactly what you seek, but there is literature in computer graphics that starts with a mesh of "traditional triangles" as you describe them, and "remeshes" the surface with geodesic triangles. It may be that the three papers from the last six years listed below will give you additional leads. To provide a sense of the work, here is a figure from the Peyré-Cohen paper:
           

"Fast exact and approximate geodesics on meshes" Vitaly Surazhsky, Tatiana Surazhsky, Danil Kirsanov, Steven J Gortler, Hugues Hoppe (SIGGRAPH 2005, Volume 24, Issue 3).

"Geodesic Remeshing Using Front Propagation," Gabriel Peyré, Laurent D. Cohen (International Journal of Computer Vision, Volume 69 Issue 1, August 2006).

"A continuum theory for unstructured mesh generation in two dimensions," Guy Bunin (Computer Aided Geometric Design, Volume 25, Issue 1, January 2008, Pages 14-40).

Answer 2

Interesting question! There's recent work on isogeometric analysis (eg. Hughes et al in CMAME 2005), who seek to combine FEM with NURBS. This may be a useful literature to look at, for potential applications of your own ideas.

In essence, if one is to study PDE on surfaces, then an intrinsic (geodesic) mesh may confer advantages in terms of accuracy.

Answer 3

There are many papers on Riemannian meshing. See Meshing Research Corner and International Meshing Roundtable for some references.