This paper is concerned with development of an efficient spectral Galerkin algorithm for elliptic problems. The stability problem of parallel disperse flow is employed as an application of described method. The interest in highly effective spectral projection algorithms is raised with development of contemporary extensions of industrial programming languages (C++14, for example), which grants frameworks for constructing high performance and robust parallel programming code. The exploration of flows with special properties is the problem of numerical and mathematical hydrodynamics. The central focus of exploration is to describe properties of linear operators of corresponding dynamical systems. The development of algorithm is based on Legendre Galerkin approach. It is shown, that spectral coefficients of differential operators are explicitly expressed with algebraic formulas. The test of algorithm accuracy is performed. The stability spectrum of disperse Couette-Poiseuille flow is analyzed. The form of sufficiency conditions of stability is established. The metastability regions of disperse flow are observed numerically.