Difference between PDEs defined on 2D manifolds in R^3 and those in 2D in R^2

mathematics
1

Hi,

I was wondering if there is any difference between the governing equations for a 2D PDE defined over a “flat” 2D region, versus the same PDE defined over a 2D manifold in 3D space (geometric dimension 3, topological dimension 2). Does the same PDE hold, just with differential operators defined on the “tangent” space coordinates? Does this change the UFL implementation?

Sorry if this question is too general; I’m not familiar with this topic so would appreciate any help or sources where I can better understand it.

Thanks,
Jon

2

Hi Jon, according to here, “The solver code is identical to the case where thegeometrical equals the topological dimension” for Poisson equation on Mobius strip on page 4.
Please see this paper for more details.

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3

Yes, basically the difference is that grad will be expressed in the global X,Y,Z frame in 3D. If your manifold has an arbitrary location in this 3D space, you will need to project it into the local tangent frame to obtain the in-plane gradient that you would have from a 2D formulation.

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4

Okay I think I understand. Thanks for the answers everyone!