Trace gas sensors are currently used in many applications from leak detection to national security and may some day help with disease diagnosis. These sensors are modelled by a coupled system of complex elliptic partial differential equations for pressure and temperature. Solutions are approximated using the finite element method which yields a skew-Hermitian dominant discretization for which classical algebraic preconditioners quickly degrade. In this paper we study various block-structured preconditioners that give a low iteration count.