Is it possible to define adaptive solver goal functional at a point/s of interest?

Question

Hi,

For the goal functional when using the adaptive solvers I know of examples using either a goal functional over the whole domain:

M = u*dx

Or over a boundary:

M = u*ds(boundary_number)

If one is interested in just some point or points of interest, couldn't the adaptive calculations be faster to define the functional for one or more points only? Is this possible?

Answer 1

I'm not sure that this is possible with the (limited) adaptive solver in DOLFIN and its awkward interface. Someone may correct me on this.

If the quantity of interest is the numerical approximation at a point, the goal functional is weighted by a Dirac delta function at that point, e.g:

$$ M = \int_{\Omega} u(x) \delta (x - x_0) \; dx $$

See example 7.1 of

http://epubs.siam.org/doi/abs/10.1137/S1064827501389084