Add support for Vector Laplacian / Second in coupleable

[GiudGiud]

Description

See discussion coupledVectorSecond does not exist at this time. Note that @lindsayad offered a solution, essentially computing d2phi for the vector manually from the component-wise d2phi

Design

More support for Rank3 tensor math in libmesh Add request for the second derivative computation in moosevariable Add routine in coupleable for retrieving reference to second derivative for vector variable

Impact

Simple solution for laplacian computation for vector variables

Discussed in https://github.com/idaholab/moose/discussions/32018

^{Originally posted by mhelel December 2, 2025}

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Question

I want to implement the vector laplacian $\nabla\dot(\nabla\vec{u}) = \nabla^2 \vec{u}$ in its strong form inside a Material that just uses a single vector variable $\vec{u}$ instead of three components. Therefore I need the second spatial derivatives of $\vec{u}$.

I know that if I would just have the components as coupled variables, I could just use the coupledSecond option and If I would use AD, which I don't, it seems that there is a adCoupledVectorSecond option. But since both cases are not applicable I was wondering how to solve this problem. In another discussion one was referred to https://mooseframework.inl.gov/docs/doxygen/moose/classMooseVariableFE.html#a6084c139e379961bfcc3f5ed97d181f4 but I was not able to figure out, how this should help. If this is the right direction, I would appreciate an hint, how to implement this directly.