RUI: Analysis and Numerical Solutions for Stochastic Stokes Equations

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The PI and colleagues study the numerical simulation of an incompressible The PI and colleagues study the numerical simulation of an incompressible viscous fluid, obeying the Stokes equations. The research will study the effect of an additional stochastic forcing term, random viscosity and random boundary values . Methods of constructing families of solutions will be used and compared. The PI is interested in the relationship between uncertainty in the input (forcing term) to the uncertainty in the output (solution values used to estimate expected values.) The research has three major components: 1. Monte Carlo methods enhanced with sensitivity derivatives, for the stochastic Stokes equations with random input parameters; 2. finite element approximation of stochastic Stokes equations with white noise forcing terms; 3. the stochastic spectral finite element method with polynomial chaos spaces, with random coefficients and boundary conditions. Computer calculations can seem deceptively precise. But scientists have discovered that there is a significant amount of uncertainty in all physical systems; not just when something is measured, but even when an attempt is made to describe how the system changes. No computer calculation can possibly consider every slight variation in the measurements and dynamics of a system under study. And yet it is known that, at least in some cases, small amounts of uncertainty can lead to significant, and even disastrous, errors in the computed results. The proposed research is one example of the effort to understand, quantify and control the effect of uncertainties, in this case, for a standard physical computation involving the flow of a fluid. An important bonus of this research is the involvement of a group of largely minority students, who will receive the support, guidance and training necessary to begin scientific careers.