First-Order System Least Squares for Partial Differential Equations

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9619792 Cai The investigator studies the numerical solution of partial differential equations (including convection-diffusion, Helmholtz, incompressible Stokes and Navier-Stokes, and elasticity equations) by a least-squares formulation for an equivalent first-order system. First-Order System Least Squares (FOSLS) formulates the original problem as an equivalent minimization problem by applying a least-squares principle to an equivalent first-order system. Hence, it represents a general methodology that can produce a variety of algorithms, depending on such choices as the first-order system and the least-squares norm, and that can lead to formulations that have substantially different numerical properties (accuracy, adaptivity, and complexity). The project seeks a proper first-order system and a proper least-squares norm for formulating the original problem so well that the numerical process (discretization and multigrid solution) becomes straightforward and optimal. Certain finite element methods for FOSLS minimization problems for second-order elliptic and incompressible Stokes and Navier-Stokes equations are of optimal accuracy in each variable (including new variables). Standard multigrid methods applied to the resulting discrete equations have optimal complexity (i.e., computational cost for full multigrid solution is proportional to the number of unknowns.). This project continues efforts on problems with discontinuous coefficients, high Reynolds number flows, and linear elasticity with general boundary conditions. This project aims to develop a new computational method for systems of parial differential equations that model the physical phenomena of fluid flow. The approach modifies existing mathematical formulations for these processes, and leads to more efficient and robust computational techniques than those currently in use. These new algorithms enable computer simulations of a large class of problems in science and engineering and mi nimize the need for costly experimental measurements.