Graddiv

Graddiv, short for gradient of the divergence, is a differential operator that acts on a vector field v by first taking its divergence (a scalar field) and then taking the gradient of that scalar. It is written as grad(div v) or ∇(∇·v). In Cartesian coordinates, if v = (v1, v2, v3), then grad(div v) has components ∂/∂x (∂v1/∂x + ∂v2/∂y + ∂v3/∂z), ∂/∂y (∂v1/∂x + ∂v2/∂y + ∂v3/∂z), ∂/∂z (∂v1/∂x + ∂v2/∂y + ∂v3/∂z).

Graddiv is related to the vector Laplacian through the identity ∇²v = ∇(∇·v) − ∇×(∇×v). Thus graddiv is the first term in this decomposition and represents the portion of the Laplacian arising from spatial variations of the divergence of the field.

Properties and scope: Graddiv is a linear differential operator of order two, mapping vector fields to vector fields. Its exact form depends on the coordinate system, and while it is well-defined within a chosen framework, it is not an invariant scalar operator. It is distinct from applying the scalar Laplacian to each component of v, though related through the vector identity above.

Applications: Graddiv appears in the analysis of fluid dynamics and elasticity, particularly in equations involving the vector Laplacian and Poisson-type problems. In numerical methods for incompressible flows, graddiv terms are used in stabilization and in formulations that aim to enforce or encourage divergence-free velocity fields. It also features in vector Poisson problems and in contexts where the spatial variation of divergence influences the solution.

See also: divergence, gradient, curl, vector Laplacian, Helmholtz decomposition, finite element methods, incompressible flow.