Mathematical Sciences: Differential Equations of Mechanics and their Numerical Solution

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The investigator will study systems of partial differential equations which describe the behavior of elastic materials. Particular emphasis will be on the Reissner.Mindlin theory and related models for the bending of elastic plates and shells. The investigations will include a qualitative analysis of the solutions of the plate equations, design and analysis of numerical schemes for their solution, and both analytical and computational studies of the relation between solutions of the Reissner.Mindlin equations and the equations of three dimensional elasticity. Specific goals include an analysis of the boundary layer for the Reissner.Mindlin model near free and simply.supported edges, the implementation of a recently developed optimal order finite element method for this model which is uniform with respect to the plate thickness, and the design and analysis of uniformly optimal order finite element methods of higher degree. In addition the investigator will continue work on boundary element methods for the numerical solution of time.dependent partial differential equations, with an emphasis on applying recent theoretical advances to practical computation, and will study applications of the boundary element approach to exact control problems for such time.dependent equations.