The goal of this project is the development and analysis of least-squares methods for partial differential equations (PDEs) arising from applications in fluid and solid mechanics. These systems are naturally nonlinear and numerical simulation is typically difficult and expensive. The least-squares finite element method is a powerful tool for many PDE-based applications in science and engineering. One of the main characteristics of a least-squares formulation is that it transforms a given set of equations into a loosely coupled system of scalar equations that can be treated easily by multilevel finite element methods. The finite element spaces for the individual unknowns may be chosen independently, based on simplicity and availability or from the physics of the underlying problem. The linear systems of equations resulting from well-posed least-squares discretizations are always self-adjoint and positive definite, and variational multigrid methods generally provide a robust, scalable solver. In addition, the associated functional itself provides a natural sharp local error estimator, which can be used for effective adaptive mesh refinement. Coupling nested iteration and Newton linearization with a least-squares discretization and multigrid iterative solver constitutes a robust, comprehensive solution strategy for difficult nonlinear problems.
This project represents a focused study of least-squares methods for several linear and nonlinear elasticity and fluid flow problems, extending to applications in viscoelasticity. Some research in these areas has been successful and preliminary results are encouraging, but much remains to be done. This project includes both general solution methodologies for problems with strong nonlinearities and specific least-squares formulations for models that have not been analyzed in a least-squares framework. The applications considered in this project include complex and specialized systems important in areas including engineering, physics, aerodynamics, atmospheric sciences, geology, and biomechanics. One target application, for example, is a specific incompressible, non-Newtonain flow which arises in the modeling of blood flow. Here, a viscoelastic model that takes into account the elastic nature of the suspended red blood cells is to be considered. In these and many other areas of interest, computer simulation of complex phenomena are currently limited by efficiency of numerical methods. The basic techniques developed here will enhance the broader area of scientific computation by adding to the collective understanding of how to analyze and solve complex problems.
