I have a probably dumb question with the finite element method. Consider a 2D Poisson equation: $\nabla (a(x)\nabla u(x)) = f(x)$ with suitable boundary conditions. Suppose we solve the problem with finite element. In my problem, I am particular interested in the gradient $\nabla u$. I am wondering if there is some easier way to find the gradient. Can it by some by product of the finite element method for solving the equation?
The boundary conditions will typically get in the way of solving directly for ∇u(x). For instance, you might have a Dirichlet boundary condition specifying u(x) on the boundary. So you'll have to do your FEA as usual, and then determine $\nabla u$ a posteriori.
• thanks for your reply. so you mean, you solve for $u$ and then approximate $\nabla u$ with say finite element? – ljl Oct 16 '14 at 15:07
• @ljl You would need finite element to solve for $u$ at the nodes. Finding $\nabla u$ is a post-processing operation, not FEA per se. Typically, spatial derivative quantities are evaluated at the element Gauss quadrature points (where they are most accurate). – user_of_math Oct 16 '14 at 16:47