Mathematical Sciences: Finite Element Methods for Partial Differential Equations

This project is concerned with the development and analysis of finite element methods for partial differential equations arising from mechanics. The topics of research include: (1) Investigation of the boundary layer of the Reissner-Mindlin plate model, including the development and rigorous justification of an asymptotic expansion of the solution in terms of appropriate powers of the plate thickness for boundary conditions modelling the "soft" simply supported and free plate. (2) The development of higher order uniformly accurate finite element methods for the Reissner-Mindlin model (generalizing a first order method previously developed). In addition, the boundary layer theory developed in (1) will be used to analyze the interior behavior of methods and to develop mesh refinement strategies for improved accuracy. (3) Investigation of the use of nonconforming finite elements for boundary value problems in elasticity. Particular attention is given to the validity of Korn's second inequality for nonconforming elements with traction boundary conditions and to determining the relationship of finite element methods based on equivalent formulations of the equations of linear elasticity. (4) Investigation of the use of nonconforming finite element methods for the approximation of first order linear scalar hyperbolic equations. One application is to the problem of identifying a variable coefficient in an elliptic partial differential equation.