A multiple-scale Pascal polynomial for 2D Stokes and inverse Cauchy–Stokes problems

• We solve direct and inverse Cauchy–Stokes problems.
• We propose a third–first order system and a third–third order system.
• A multiple-scale Pascal polynomial method is developed.
• The scales are determined by the collocation points.
• The multiple-scale Pascal polynomial expansion method is accurate and stable against large noise.

The polynomial expansion method is a useful tool for solving both the direct and inverse Stokes problems, which together with the pointwise collocation technique is easy to derive the algebraic equations for satisfying the Stokes differential equations and the specified boundary conditions. In this paper we propose two novel numerical algorithms, based on a third–first order system and a third–third order system, to solve the direct and the inverse Cauchy problems in Stokes flows by developing a multiple-scale Pascal polynomial method, of which the scales are determined a priori by the collocation points. To assess the performance through numerical experiments, we find that the multiple-scale Pascal polynomial expansion method (MSPEM) is accurate and stable against large noise.