## Section: New Results

*A posteriori* error
analysis of sensitivities

Participants : Roland Becker, Daniela Capatina, Robert Luce, David Trujillo.

Most practical applications involve parameters $q={\left({q}_{i}\right)}_{1\le i\le N}$ of different origins: physical (viscosity, heat conduction), modeling (computational domain, boundary conditions) and numerical (mesh, stabilization parameters, stopping criteria, values of a turbulence model). Numerical simulations can provide information related to the (first order) sensitivity of a quantity of physical interest $I\left(q\right)$ with respect to different parameters: $\partial I/\partial {q}_{i}$. Their computation can help to validate the physical model, to explain unexpected behaviour and also to guide efforts to improve both the physical and the computational models.

*A posteriori* error estimates for the functional itself, for
fixed values of the parameters $q$, are well-known, cf. for
example [47] where a goal-oriented error
control is achieved by introducing an adjoint problem. Our goal is
to provide a general framework for the *a posteriori* error
estimation of sensitivities $\partial I/\partial {q}_{i}-\partial {I}_{h}/\partial {q}_{i}$, which has not been given yet in the
literature.

So far, we have applied the proposed method to the computation of the Nusselt number measuring the efficiency of a cooling process, described in the project Optimal. A cold liquid is injected in a annular domain through several inlets in order to cool a heated interior stator.

First numerical results, including adaptation with respect to the
functional and to the sensitivity, have been carried out with the
library Concha. They have been presented
in [33] ,
[30] . In
Figure 10 one may see the computed
temperature and velocity field, while the *a posteriori* error
estimator for the sensitivity of the Nusselt number with respect to
the inflow speed at the right-hand side inlet, as well as the
adapted mesh, are given in Figure 11 .

In the future, several important aspects related to the adaptive method are still to be investigated such as design of an appropriate adaptive algorithm, proof of its convergence and optimality etc.