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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?

Thanks in advance!

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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.

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  • $\begingroup$ thanks for your reply. so you mean, you solve for $u$ and then approximate $\nabla u$ with say finite element? $\endgroup$ – ljl Oct 16 '14 at 15:07
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    $\begingroup$ @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). $\endgroup$ – user_of_math Oct 16 '14 at 16:47

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