project and complete point data onto boundary

Question

I'd like to solve the heat equation on a domain $\Omega$ with Dirichlet boundary conditions. Instead of hard Dirichlet boundary data, I've got experimental data $\theta_i$ measured at points $(x_i, y_i)$ close to the boundary of $\Omega$.

What would think is a sensible way of translating this data into Dirichlet boundary data, and how would you implement it in FEniCS?

Answer 1

Define mesh wit vertices $(x_i, y_i)$, and set DOFs of CG1 function on it. This mesh does not need to be the same as your mesh for $\Omega$ unless you need it in parallel. (In parallel this Dirichlet function can be loaded as a serial to every process using serial-meshes branch of DOLFIN.)

There may be a problem that your data may cover only mesh of topological dimension 1 but geometric dimension is 2. I hope you can always extend this somehow to tdim 2 using a translation adding vertices $(x_i, y_i) + \mathbf{r}$ with $\mathbf{r}$ constant dependent on arrangement of your experimental points and Dirichlet boundary. For example having $\Omega = (0, 1)\times(0, 1)$ and $(x_i, y_i) = (0, 0), (0.1, 0), (0.2, 0), \ldots, (1, 0)$ you can extend your data to points $(x_i, y_i+1) = (0, 1), (0.1, 1), (0.2, 1), \ldots, (1, 1)$ so that you have enough vertices to define mesh for this Dirichlet data.