Finite Element Approximation of Partial Differential Equations

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The first area of proposed research involves the study of compatiblediscretization schemes for partial differential equations, an approach thatattempts to produce numerical approximations that inherit or mimicfundamental properties of a partial differential equation, such asconservation and symmetries. Several problems, modeling phenomena inelasticity and fluids, will be studied from this point of view. The newapproach taken is based on the construction of piecewise polynomial exactelasticity sequences, which are closely related to the development of stablemixed finite element schemes. Other work in this area includes a new approach to the construction of hierarchical bases for scalar and vector-valued finite element spaces in arbitrary space dimensions, and exact sequence properties of rectangular and quadrilateral finite elements and their applications to thestability of mixed finite element approximation. The second area of study is the approximation properties of several types of finite element spaces defined on irregular hexahedral elements obtained by trilinear mappings from areference cube. Such spaces are used to approximate three-dimensional vectorfunctions and arise naturally in many applications, including the approximation of Maxwell's equations and the use of mixed and least squares finite elementmethods for second order elliptic equations. Although it is often implicitlyassumed that approximation results known for regular hexahedrons extend tothese spaces, in fact this is not the case. The research is to determineprecisely what is needed for optimal order approximation and constructfamilies of finite element spaces that have this property. The third area of research is to study convergence rates for discontinuous Galerkin methods for linear hyperbolic problems. Although optimal order convergence rates areoften seen in practice, the theory guarantees such rates only for uniformmeshes, while a lower rate is known to be the best possible on speciallyconstructed meshes. The proposed research is to classify the type of meshesfor which the optimal convergence rate is achieved.Mathematical modeling of physical and biological processes using partialdifferential equations has become the standard method of studying a host ofimportant problems. Such models capture in a concise and precise way thefundamental features of the process being modeled. Unfortunately, theresulting equations rarely have solutions that can be expressed by simplemathematical formulas. Hence, the development of reliable and efficientnumerical approximation schemes are necessary to make this method into apractical approach and is central to progress in many areas of science andengineering. Part of this development involves the investigation of thetheoretical underpinnings of numerical methods. Such investigation can leadto a greater understanding of existing methods and to the development of newmethods with desirable properties. Thus, such study has the potential toimprove the accuracy of, or even make possible, essential computer simulations performed by scientists and engineers. This project is concerned with thestudy of numerical methods for approximating equations modeling phenomena inelasticity and fluid flow. One central theme is to develop approximationschemes that preserve discrete versions of some of the fundamental properties of the mathematical model, in order to more accurately capture the fundamental features of the underlying process being modeled.