Algorithms for Discrete and Stochastic Partial Differential Equations

University Of Maryland, College Park, College Park MD

Investigators

Abstract

This project concerns the development and analysis of efficient numerical algorithms for problems arising in computational modeling, with emphasis on two main topics: algorithms for systems of equations arising from the stochastic finite element method, and algorithms for algebraic systems arising in models of fluid dynamics. The first of these addresses the fact that models of physical phenomena often contain parameters or equation coefficients whose precise properties are not well understood. Examples include permeability properties of media in which quantities (e.g., pollutants in groundwater) are flowing or diffusing, and boundary conditions (e.g., along the ocean bottom). In the stochastic finite element method, the random aspects of problems are handled in a manner analogous to the introduction of new spatial dimensions. This methodology appears to have the potential to be more efficient than Monte-Carlo methods, provided efficient algorithms are available for the algebraic systems that are generated after discretization. Our aim is to study the algorithmic issues that arise from this approach. For the second project, we will develop and study efficient algorithms for solving systems of equations arising in models of incompressible flow, principally, methods for eigenvalue problems derived from linear stability analysis of steady solutions, and multigrid algorithms for the discrete convection-diffusion equation. These are fundamental problems arising throughout fluid dynamics, and their efficient solution is critical for development of effective computational models. The general aim of this project is to enhance the utility and effectiveness of mathematical modeling for understanding scientific and engineering phenomena. There are useful models for many disparate physical processes, including blood flows, dispersal of environmental pollutants, performance of aerospace vehicles, and atmospheric and oceanographic phenomena. Understanding such processes through purely experimental techniques is prohibitively expensive or impossible, whereas the use of modeling and together with algorithmic solution introduces a basic understanding of the physics by providing approximations to quantities such as flow rates and pressures. Accurate solutions are only available, however, if reliable and fast solution algorithms can be used. Moreover, it is often the case that certain aspects of models, such as the geologic properties of transporting media or the velocities of flows along boundaries, are not known with certainty. Our goal for this work is to develop fast solution algorithms for mathematical models and to ensure that the solution strategies are able to handle uncertainty and to produce reliable statistical information about solutions at low computational cost.