  The supersingular locus of the $\mathrm{GU}(1,n-1)$ Shimura variety at a
ramified prime $p$ is stratified by Coxeter varieties attached to finite
symplectic groups. In this paper, we compute the $\ell$-adic cohomology of the
Zariski closure of any such stratum. These are known as closed Bruhat-Tits
strata. We prove that the cohomology groups of odd degree vanish, and those of
even degree are explicitely determined as representations of the symplectic
group with a Frobenius action. Each closed Bruhat-Tits stratum is linearly
stratified by Coxeter varieties attached to smaller symplectic groups. Thanks
to results of Lusztig who computed the cohomology of Coxeter varieties for
classical groups, we make use of the spectral sequence associated to this
stratification and describe explicitely all the terms at infinity. We point out
that the closed Bruhat-Tits strata have isolated singularities when the
dimension is greater than 1. Our analysis requires discussing the smoothness of
the blow-up at the singular points, as well as comparing the ordinary
$\ell$-adic cohomology with intersection cohomology. A by-product of our
computations is that these two cohomologies actually coincide, so that
surprisingly the presence of singularities does not interfere with the
cohomology.
