Preconditioning Techniques for Algebraic Equations Arising from Partial Differential Equations

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ElmanThis project concerns the development, analysis and testing of techniquesof numerical linear and nonlinear algebra for solving systems of equationsarising from discretization of a collection of partial differential equations.The emphasis will be on preconditioning strategies and their combination with Krylov subspace iterative methods for linear and nonlinear systems.These ideas will be applied to several examples of differential equations,primarily coming from models of fluid dynamics. Our focus will be on two types of problems, the saddle point problems produced by direct discretization of the primitive formulation of the incompressible Navier-Stokes equations, and the convection-diffusion equation. These have been chosen because they are used in numerous engineering models of fluid flow, and because they containchallenging mathematical and computational qualities that make them difficult to solve. In particular, accurate discretization in two and three dimensions leads to large sparse systems of algebraic equations that are nonsymmetric,and for the first problem, nonlinear and indefinite. In addition, they have associated with them fundamental parameters such as Reynolds numbers and discretization mesh widths that, for the solution of problems of practical interest, cause many algorithms to converge slowly and require large-scale systems to attain accuracy. Our goals are to extend and analyze certain preconditioning techniques designed for saddle point problems. Plans include demonstration of the effectiveness of these preconditioners for general mixed finite element discretizations, development of rigorous mathematical techniquesfor analyzing convergence, demonstration of their usefulness in realistic settings involving general meshes and domains, and incorporation of fastalgorithms for the convection-diffusion equation into the solution process.The purpose of developing computational algorithms is to enable the efficient numerical solution of differential equations used in mathematical modelling. The use of such models is becoming an increasingly importantcomponent in manufacturing and design (for example, of aerospace vehicles,automobiles, cooling devices for nuclear devices) and in enhancing ourunderstanding of critical natural phenomena (for example, blood flows anddispersal of environmental pollutants). Understanding these phenomenathrough purely experimental techniques is prohibitively expensive orimpossible, whereas the use of mathematical models introduces a basic understanding of the physics by providing approximations to quantitiessuch as flow rates and pressures. This leads to the identification of promising design features, such as wing shape in airplane manufacturing. (For example, many of the design decisions for the Boeing 777 jet were made using computational studies.) Accurate solution of the mathematical models is only feasible, however, if reliable and fast solution algorithms are available. The goal of this project is to develop such algorithms.