Finite Element Approximation of Partial Differential Equations

The first area of proposed research involves the study of compatible discretization schemes for partial differential equations, an approach that attempts to produce numerical approximations that inherit or mimic fundamental properties of a partial differential equation, such as conservation and symmetries. Several problems, modeling phenomena in elasticity and fluids, will be studied from this point of view. The new approach taken is based on the construction of piecewise polynomial exact elasticity sequences, which are closely related to the development of stable mixed 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 the stability 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 a reference cube. Such spaces are used to approximate three-dimensional vector functions and arise naturally in many applications, including the approximation of Maxwell's equations and the use of mixed and least squares finite element methods for second order elliptic equations. Although it is often implicitly assumed that approximation results known for regular hexahedrons extend to these spaces, in fact this is not the case. The research is to determine precisely what is needed for optimal order approximation and construct families 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 are often seen in practice, the theory guarantees such rates only for uniform meshes, while a lower rate is known to be the best possible on specially constructed meshes. The proposed research is to classify the type of meshes for which the optimal convergence rate is achieved. Mathematical modeling of physical and biological processes using partial differential equations has become the standard method of studying a host of important problems. Such models capture in a concise and precise way the fundamental features of the process being modeled. Unfortunately, the resulting equations rarely have solutions that can be expressed by simple mathematical formulas. Hence, the development of reliable and efficient numerical approximation schemes are necessary to make this method into a practical approach and is central to progress in many areas of science and engineering. Part of this development involves the investigation of the theoretical underpinnings of numerical methods. Such investigation can lead to a greater understanding of existing methods and to the development of new methods with desirable properties. Thus, such study has the potential to improve the accuracy of, or even make possible, essential computer simulations performed by scientists and engineers. This project is concerned with the study of numerical methods for approximating equations modeling phenomena in elasticity and fluid flow. One central theme is to develop approximation schemes 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.