Compute heat flux from Galerkin solution

I have a Galerkin solution for a heat equation

$$u_t = \Delta u + f$$

with Dirichlet conditions $$u=0, \qquad x \in \partial\Omega$$

The time discretization is done using a BDF scheme. How can I accurately and efficiently compute the heat flux on the boundary $$\sigma = \frac{\partial u}{\partial n}, \qquad x \in \partial\Omega$$ from the Galerkin solution $u_h$.

• How accurately do you need it? What kind of elements are you using? Do you need values at the nodes, or will values at the cell-faces do? Apr 23, 2013 at 17:23
• I am looking at 1-d and 2-d cases. In 1-d I am using linear elements. In 2-d I am using linear elements on triangles. I would like the flux to be also second order accurate like the solution. I have heard of some post-processing techniques but I cannot find this in any of the textbooks and I dont know what are the papers to refer to. I need the flux at the nodes. Apr 24, 2013 at 4:33

• That may be well defined in 1d, but in 2d the solution is in $H^1$ and the point value of a derivative is not defined. It also wouldn't be a flux (the heat flux through a point is zero). Apr 24, 2013 at 12:43