Does current Mathematica capability allow solving PDEs (e.g. reaction diffusion equation) over surface of a mesh or surfaces

I came across this nice paper (https://arxiv.org/pdf/1605.01583.pdf) where the authors simulated a reaction diffusion system over the surface of a gecko, primarily to understand how various patterns on lizard surfaces emerge. The image below shows real geckos and their simulation counterparts.

The author of the article told me that his code is not ready to be shared publicly and is unsure when or if it will ever be ready - which is such a pity as it happens quite often. Nevertheless, he told me that he has used Trilinos library (https://trilinos.github.io/) and Deal.ii (https://www.dealii.org/) package to perform the simulations and that both libraries have received frequent updates since he published the article. To all the Mathematica aficionados who love solving PDEs numerically, I am wondering whether the current capabilities of Mathematica allow solving PDEs of such kind (reaction-diffusion, heat equation) over surface of a mesh. From what I know, NDSolve complains when you feed it a mesh that has RegionDimension of 2 (essentially a surface) while the RegionEmbeddingDimension is 3.

It would be wonderful if such problems can be tackled with the burgeoning Finite Elements capability in Mathematica. My question, is it possible?

Note: one can easily download a similar geometry or surface mesh from https://www.thingiverse.com/thing:1363148

• Currently (V13.0) this is not possible out of the box, there is a feature request for that, found here you can up-vote that if you like. The link also has some examples on how to do it manually. Jan 5 at 14:13
• Interesting question. Could you give a simple example for "NDSolve complains ..."? Jan 5 at 14:13
• @user21 will post this link there. No doubt it will be a useful addition to the PDE solving capabilities thanks ! Jan 7 at 8:38
• @UlrichNeumann for example you can try to solve a poisson equation over a FilledTorus and simple Torus in version 13.0 or for that matter even solving equation over a Sphere vs Ball Jan 7 at 8:40
• @AliHashmi Thanks, even simpler in 2D: poisson equation in recangle. But I don't know the equivalent problem described on the boundary... Jan 7 at 8:47