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@zachmprince It's not the most intuitive interface in the world, but possibly NumericVector::add_vector_transpose could work for your purposes.
/**
* Computes \f$ \vec{u} \leftarrow \vec{u} + A^T \vec{v} \f$,
* i.e. adds the product of the transpose of a \p SparseMatrix \p A
* and a \p NumericVector \p v to \p this.
*/
virtual void add_vector_transpose (const NumericVector<T> & v,
const SparseMatrix<T> & A) = 0;
idaholab/moose#24209 has to cast to a petsc matrix and then manually call the corresponding PETSc API. We should be able to abstract that in libMesh
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