New techniques in characteristic finite element methods for flow problems

Colorado State University, Fort Collins CO

Investigators

Abstract

This proposal is awarded using funds made available by the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The focus of the proposed research is to design, analyze, and implement accurate and efficient numerical methods for flow problems. As is well known, incompressibility and convection-dominance in flow problems pose significant challenges to numerical simulators. In this research project, the PI and his collaborators will develop numerical schemes based on the locally divergence-free (LDF) finite elements and characteristic tracking to overcome the aforementioned two major difficulties. The LDF finite elements have strong potentials for both h-adaptivity (due to the independence of the basis functions to element geometry) and p-adaptivity (due to the hierarchical structure of the basis functions). Discrete Helmholtz decomposition and hybridization will be utilized to decouple resulted discrete linear systems and reduce computational costs. Higher order characteristic methods are developed for long-term simulations with increased accuracy. These new techniques will be incorporated in the discontinuous Galerkin framework to develop fast and robust solvers for Stokes, Navier-Stokes, convection-diffusion-reaction, and magnetohydrodynamics equations. The success of the proposed research will have direct impact on the efficiency and robustness of a large class of numerical simulators for real world problems such as groundwater contamination remediation, intracellular protein trafficking, nuclear waste disposal, oil recovery, weather forecast, and wildfire control. The new mathematical methods and software modules developed in this project will provide reliable tools for a wide range of scientific computing tasks. This project will also provide hands-on training opportunities for graduate students to gain a wide knowledge base and sophisticated skills to apply mathematics and computer simulations to a variety of applications in sciences and engineering.