# vector PDEs on manifolds

What are the subtleties involved in solving vector PDEs on manifolds? Can someone suggest a reference summarizing the problems involved?

Specifically I want to solve a vector Helmholtz equation with a source field on a curved manifold (e.g. a sphere, or a sphere with distortions). How do I go about solving this? Say using finite elements? Is there any such study available?

EDIT: I must clarify that I am looking for methods to solve vector PDEs on general surfaces which are topologically equivalent to the sphere (e.g., ellipsoid, sphere + spherical harmonic deformations...). The deformations away from the sphere are not infinitesimal but finite.

You should browse the recent publications by Arnold Reusken from Aachen, Germany.

He works on solving PDEs on surfaces. I am sure you can apply the methods developed there to the case where the surface is pregiven and even parametrized.

• Thanks for the link. Most of the papers seem to deal with PDEs for scalar fields on surfaces. My interest was in PDEs for vector fields. Could you please point me to a specific paper in the list that deals with vector PDEs on surfaces? Perhaps I have missed it. – Vijay Murthy Jul 9 '14 at 11:53
• And I have missed your point on vector fields. Sorry, but I don't whether in one of the papers the author refers to vector valued PDEs. – Jan Jul 10 '14 at 11:54

If your surface is a sphere, Helmholtz equation can be solved numerically using vector spherical harmonics. Vector spherical harmonics are eigenfunctions of the Laplace Beltrami operator forming a complete basis. In order to obtain a general overview on this topic, I point you to the book "Lectures on Constructive Approximation" by Michel, Volker. You will find expressions for the differential operators wrtthe basis of Vector spherical harmonics.

If your spherical surface is noisy (e.g. sphere with distortions) but the noise is not too large, you could still assume a sphere an approximate the PDE in vector spherical harmonics. Of course it depends on the application and the targeted accuracy.

For a general surface one would choose a different method depending on the type of surface one has.

• Thanks Nicolas. I'd start with a sphere. But I want to solve the equations on a spherical surface with finite deformations imposed on it, for eg a sphere plus spherical harmonic deformations or on the surface of an ellipsoid. Nevertheless your reference is very useful. – Vijay Murthy Jul 9 '14 at 19:10
• If the underlying geometry is ellipsoidal use this asianscientist.com/books/wp-content/uploads/2013/06/… as a reference. Thus express the vector Helmholtz equation $Ax=f$, the involved operator is $A=(\Delta +k)$, in the Vector Ellipsoidal Harmonics basis. Further project your right hand side $f$ into the same basis and invert the the system $x = A^{-1} f$. – nico Jul 9 '14 at 20:09
• Thanks again. This is very useful. I had never heard of vector ellipsoidal harmonics. – Vijay Murthy Jul 9 '14 at 20:45

I suggest you also take a look at the papers by Andrea Bonito: http://www.math.tamu.edu/~bonito/ He also has a number of papers on solving PDEs on surfaces.

• Thanks for the link. As in the comment to Jan's answer, I have to stress that I am looking to solve for vector PDEs on manifolds. The link has several papers on free surface flows (but the simulation domain is really a 3D domain), and one paper  on solutions of the Laplace-Beltrami operator on surfaces for a scalar field. Can you please point me to a specific paper which looks at solutions at vector PDEs on surfaces? – Vijay Murthy Jul 13 '14 at 14:45
• I can't (I don't know all his papers -- I just happen to have a general overview of what he does because his office is down the hall from mine). But if you ask him yourself, I'm sure he can help. – Wolfgang Bangerth Jul 14 '14 at 0:42