Mathematical aspect of the stream-function equations of compressible turbomachinery flows and their finite element approximations using optimal control

In this paper we establish the partial differential equation and properly pose conditions for the stream function on an arbitrary stream surface in a turbomachine using tensor analysis and a semi-geodesic coordinate system. We provide computational examples using the finite element method and also the error estimate of Galerkin's finite element approximation which depends on the Mach number.