Applicability of Green's Theorem in 2D Axisymmetric Cylindrical Coordinates for Poisson's Equation

General
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Hello everyone,

I am currently exploring the solution of Poisson’s equation in a 2D axisymmetric cylindrical coordinate system and have encountered a conceptual challenge regarding the applicability of Green’s theorem.

Traditionally, in Cartesian coordinates, we transform the volume integral of the Laplacian using Green’s theorem as follows:

image

transforms to:
image

This step is crucial as it allows us to convert a second-order differential equation into a system involving only first derivatives, which are easier to handle in finite element formulations.

However, I am unsure how this transformation holds up under the cylindrical coordinate system, especially considering the radial and axial components and their effect on the differential operators and integration measures. The specific form of the Laplacian in cylindrical coordinates is:
image

I would appreciate any insights or references on how to apply Green’s theorem in this context. Does the standard transformation apply directly, or are there modifications required to accommodate the unique aspects of the cylindrical geometry?

Thank you in advance for your help!

2

Hi nikoleeaa,

I think the transformation you are talking about is rather an application of Green’s first identity (see e.g. https://en.wikipedia.org/wiki/Green%27s_identities), which is derived from the divergence theorem, and not an application of Green’s theorem. The divergence theorem relates the total divergence inside a closed domain to the total flux across the boundaries of the domain, thus it does not depend on your choice of coordinate system.

Operating with cylindrical coordinates instead of cartesian coordinates would certainly alter the definitions of the gradient, divergence, and Laplacian operators, but the transformation from the strong form to the weak form using Green’s first identity should hold in any case. That is, the equations you present would be exactly the same, but were we to write out the gradient terms the equations would look different, because of the different expressions for the del operator in the different coordinate systems.

Let me know if this was of help.

Best,
Halvor

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Hello Halvor,

Thank you very much for your detailed explanation and for clarifying the distinction between Green’s first identity and Green’s theorem.

I appreciate your confirmation that the transformation from the strong form to the weak form using Green’s first identity holds regardless of the coordinate system. However, as you correctly noted, the specific expressions for gradient, divergence, and the Laplacian do indeed differ in cylindrical coordinates compared to Cartesian coordinates.

Thanks again for your help!

Best regards,
Niko

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For reference, maybe you could look at ch2. and ch.3 here: