divgrad

Divgrad is shorthand in vector calculus for the divergence of the gradient of a scalar field. If f is a scalar function, its gradient ∇f is a vector field, and applying the divergence to that vector field gives ∇·(∇f). This composition is commonly denoted as divgrad f and is equal to the Laplacian of f, written Δf.

In n-dimensional Euclidean space, the Laplacian is Δf = ∑_{i=1}^n ∂²f/∂x_i². In two dimensions, this becomes Δf = f_xx + f_yy; in three dimensions, Δf = f_xx + f_yy + f_zz. The Laplacian is a scalar differential operator and is central to many problems in physics and engineering.

Applications and properties: The divgrad operator appears in Poisson's equation Δu = g, the diffusion and heat equations, and many potential problems. It is an elliptic operator, and its behavior depends on boundary conditions such as Dirichlet or Neumann. On curved spaces, the generalization is the Laplace-Beltrami operator, defined as div grad with respect to a given metric, reducing to the standard Laplacian in flat space.

Numerical use: In finite difference methods, divgrad yields a discrete approximation to the Laplacian via a stencil. In finite element methods, the weak form of divgrad u involves integrating grad u · grad v, emphasizing energy-based formulations.

Example: For f(x,y) = x^2 + y^2, ∇f = (2x, 2y) and ∇·(∇f) = ∂(2x)/∂x + ∂(2y)/∂y = 4, so Δf = 4.