FINITE ELEMENT APPROXIMATION OF PARTIAL DIFFERENTIAL EQUATIONS

  • Falk, Richard (PI)

Project Details

Description

The first area of proposed research involves the study of compatible discretizations of partial differential equations, i.e., numerical approximation schemes that inherit or mimic fundamental properties of a partial differential equation. The proposed research seeks to capitalize on the idea that some of the same concepts that lead to the well-posedness ofboundary value problems for partial differential equations can be used to develop compatible and stable finite element schemes for their approximation. The plan is to use this idea to develop a simpler and more systematic analysis of finite element methods, which based on past work, should then lead to the design of new and better numerical approximation schemes for a variety of applied problems. The second project proposed is the further development of a new numerical approximation scheme recently developed by the P.I. for the simulation of a parallel-plate Electrowetting on Dielectric Device. Electrowetting is an effect, based on a relationship between electrical and surface tension phenomena, that allows for control of the shape and motion of a liquid-gas interface, through the use of an applied voltage. The liquid surface changes shapes when a voltage is applied in order to minimize the sum of the surface tension energy and electrical energy. The proposed research is to first investigate the new approach computationally to see how the results compare with experiments, and then to establish stability of the method and derive error estimates. Some novel features of the method are that the changing boundary position is now a variable in the formulation and the unknown velocity can be approximated by standard simple finite element spaces. The final project is the design of new simpler quadrilateral and hexahedral finite elements for use in the mixed finite element approximation of partial differential equations. A main issue is that such elements, when defined in the usual way, do not retain the same approximation properties as those defined on squares and cubes. 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 a variety of applications. One central theme is to develop approximation schemes that preserve discrete versions of some of the key properties of the mathematical model, in order to more accurately capture the fundamental features of the underlying process being modeled.
StatusFinished
Effective start/end date8/1/097/31/12

Funding

  • National Science Foundation (National Science Foundation (NSF))

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