Mathematical Sciences: High-Order Mixed Elements on Tetrahedral Meshes for the Stokes Problem

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Zhang 9625907 The investigator studies high order mixed finite element methods on tetrahedral meshes for the 3D stationary Stokes problems, and the iterative solvers for the resulting discrete linear systems of equations. Preliminary work done by the investigator finds a family of stable mixed elements on some macro-element type grids. This result indicates the possibility of finding the minimal polynomial degree of continuous piecewise polynomials for the velocity where one degree less piecewise polynomials are employed to approximate the pressure and the mixed elements are stable; this has been an open problem for more than a decade. The investigator studies various augmented Lagrangian methods (including the well-known Uzawa method and the iterative penalty method) where the effective multigrid method or/and domain decomposition method are combined to achieve the optimal order of computation. The difficulty in the design of both the mixed elements and the iterative methods is to treat the incompressible condition. The investigator develops computer algorithms for some well-known equations in incompressible fluid dynamics. Problems of incompressible computational fluid dynamics arise throughout industry and science. There are difficult challenges in developing these methods involving limits of computer memory and speed, as well as accuracy and robustness of the method itself. The algorithm is expected to have applications to areas such as airfoil design and to manufacturing processes involving fluid flow in complex shapes, for example, the chemical industry.