Mathematical Sciences: Construction of Accurate, Robust and Efficient Numerical Techniques for Partial Differential Equations

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Bramble 9626567 The main objectives of this project are the construction and study of finite element approximations to steady-state and transient problems of mechanics and engineering, including convection-diffusion-reaction problems, Stokes and Navier-Stokes equations, equations of linear elasticity, plate bending problems, and the construction, analysis, and testing of fast methods for solving the resulting algebraic systems. The emphasis is on the development of new accurate, robust and efficient computational techniques. Approximation strategies involving standard Galerkin finite element methods, novel least-squares methods, and mixed finite element methods are studied especially from the standpoint of their accuracy and stability. The solution techniques that are developed emphasize preconditioning via domain decomposition or multigrid/multilevel methods. The broad objective of this program is to use the full power of rigorous mathematics both to analyze the behavior of numerical methods currently used in scientific computation and to develop new algorithms exhibiting improved accuracy, stability and performance. More efficient and scalable algorithms can have a significant impact on the development of effective simulators for understanding physical processes of national importance. For example, computer modeling of ground water processes involve the solution of large complex systems of nonlinear partial differential equations. The results of these simulations provide important predictive information that can influence decisions with regard to ground water remediation strategies and national conservation programs. Due to the complexity in the geometry and the physical processes, meaningful simulations overwhelm today's most powerful parallel computers and hence the need for more effective numerical techniques. The techniques developed are applicable as well to problems such as simulation of smart materials, heat and mass transfer, and high performance computing.