Robust Numerical Methods and Fast Iterative Solvers for Flow Problems

Efficient numerical algorithms for solving partial differential equations play increasingly important role in computational sciences and engineering. The present proposal integrates a research project and an education plan in the area of scientific computing and its applications. It is planned to develop and analyze efficient and robust computational tools for solving convection-dominated diffusion problems arising in groundwater pollution and seawater intrusion and other fluid flow problems. Further, simultaneous advances of the numerical solution to PDEs will be established in two fronts: (i) for developing stable and highly accurate solution, and (ii) for computing discrete solutions in a minimal computational time by using fast iterative algorithms. The emphasis will be on high-order numerical schemes such as compact difference methods; Discontinuous Galerkin schemes and also on stabilized Galerkin finite element methods for convection-diffusion and incompressible flow problems. It is intended to develop a posteriori estimators for adaptive procedures in order to compute numerical solutions more efficiently. It is further proposed to design efficient geometric multigrid algorithms and sparse preconditioning techniques to achieve high degree of robustness. An important component of this project is to use the knowledge gained from the theory and computational experiments to study problems related to transport phenomena and underground aquifers like flows in karstic aquifers and seawater intrusion. The project team will customize the designed solvers for a few specific scientific and industrial applications as well as for educational purposes.