SINGULAR PERTURBATION APPROACH TO MAXIMUM PRINCIPLE FORMULATION OF VISCOUS INCOMPRESSIBLE FLUID FLOW

Abstract

A method for the solution of viscous incompressible flow based on the Pontryagin's maximum principle is presented. By minimizing the cost function that ensures the continuity condition, an explicit control law for the pressure is derived with the help of an adjoint variable satisfying the adjoint differential equation and certain terminal conditions. Employing the singular perturbation method, the first-order equation is found to give the well-known pressure stabilization technique in the mixed finite element method. The implementation of the present method is presented in a simple example that shows that the approximate solution indeed approaches uniformly to the exact solution for large time.