I'm looking for a simple expression for the vector Laplacian $\nabla^2\mathbf{A}$ in orthogonal curvilinear coordinates.
Actually, I don't require the whole thing, just the part of $\mathbf{u}_i\cdot\nabla^2(\mathbf{A})$ that depends on $A_i = \mathbf{u}_i\cdot\mathbf{A}$
For example, I note that in spherical coordinates the radial component of the vector Laplacian of $\mathbf{A}=A_r\mathbf{u}_r+A_\theta\mathbf{u}_\theta+A_\phi\mathbf{u}_\phi$ that only depends on $A_r$ is $(\nabla^2-\frac{2}{r^2})A_r$.
And in cylindrical coordinates the radial component of the vector Laplacian of $\mathbf{A}=A_r\mathbf{u}_r+A_\phi\mathbf{u}_\phi+A_z\mathbf{u}_z$ that only depends on $A_r$ is $(\nabla^2-\frac{1}{r^2})A_r$.
Is there a simple expression that applies in general? Perhaps the sum of $\nabla^2A_i$ and another term of the form $f(r_1,r_2)A_i$ where ${r_1,r_2}$ are the local radii of curvature of the iso-$u_i$ surface?
I might mention, that in the end I'm looking for an expression that I can apply to a vector function that lives on the nodes of a volume mesh bounded by a facetized surface. I can compute spatial derivatives numerically. However, neither the volume nor the surface is described by a smooth parameterization, so expressions that require geometric characteristics that derive from derivatives of a parameterization, such as the metric or scale factors, may not be so useful to me. But any help appreciated.
~~~ Supplemental information ~~~
After some online research I have come to the conclusion that
$\hat{\mathbf{n}}\cdot\nabla^2(A_n \hat{\mathbf{n}})=\nabla^2 A_n-(4H^2-2K)A_n$
is a generally valid identity, where H represents the mean curvature and K represents the Gaussian curvature (and $4H^2-2K = \kappa^2_1+\kappa^2_2$, where $\kappa_1$ and $\kappa_2$ are the principal curvatures). This result is consistent with the cylinder and sphere examples stated above.
Based on the cylinder and sphere examples, I infer that the following identity is also valid:
$\hat{\mathbf{n}}\cdot\nabla^2(A_n \hat{\mathbf{n}})=(\nabla^2_\parallel+\partial_n^2 + (\kappa_1+\kappa_2)\partial_n -(\kappa^2_1+\kappa^2_2))A_n$
In this expression $\nabla^2_\parallel$ is the surface Laplacian and the signs of the $\kappa_i$ are chosen such that the principal curvatures are non-negative for the sphere and cylinder cases.
If correct, this would satisfy my objective of trying to separate out the tangential and normal derivatives of $A_n$ from $\hat{\mathbf{n}}\cdot\nabla^2(A_n \hat{\mathbf{n}})$.
Can anyone confirm my assessment by demonstration or literature citation?
jjo