Mathematical Sciences: Finite Element Methods For Incompressible, Viscous Flows

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Layton The investigator studies solution algorithms for viscous, incompressible flows at high Reynolds number. This problem includes the interrelated difficulties of: boundary and interior layers, dominating and sensitive nonlinearities, highly nonsymmetric and possibly indefinite linear systems and the incompressibility constraint. Solution algorithms are studied for each of these difficulties from one aspect to another. (For example, it would not be satisfactory to simplify the linear system at the expense of increasing the difficult of resolving the nonlinearity.) The general approach is (1) higher order methods (2) point adaptivity and unstructured meshes, (3) stabilized finite element discretizations, (4) stabilized, multi-level, multi-step Newton methods for the nonlinearity, (5) robust (= uniform in Re), parallel, iterative solvers for the linear systems, and (6) full mathematical support for all algorithmic developments. Fluid flow problems at high Reynolds number arise in many technological and scientific applications, such as convection in the melted region in the solidification of materials, transport and dispersion of pollutants in air and groundwater and simulations of climatic changes. Since exact solution of these equations is impossible, computer based simulation of fluid flow problems is essential in accurately predicting, and ultimately controlling the quantities which are of interest. Accurate and reliable simulation of high Reynolds number flow problems is a very challenging scientific problem which is studied in this research. In large scale applications this involves the study of algorithms which are highly parallel.