Research Initiation Award: Superconvergence of Finite Element Approximations for the Second Order Elliptic Problems by L^2 Projection Methods

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Research Initiation Awards provide support for junior and mid-career faculty at Historically Black Colleges and Universities who are building new research programs or redirecting and rebuilding existing research programs. It is expected that the award helps to further the faculty member's research capability and effectiveness, improves research and teaching at her home institution, and involves undergraduate students in research experiences. The award to the University of Arkansas at Pine Bluff has potential broader impact in a number of areas. The project will focus on constructing a mathematical model on superconvergence of the nonconforming finite element method (NCFEM) for second-order elliptic problems by L-squared projection methods. This project will enhance the research experience and training of undergraduate students in mathematics at the institution. Additionally, a course in Finite Element Methods will be developed and offered as a topics course. The project will use L-squared projection methods to improve the convergence rate of an existing finite element solution so that the new approximation is closer to the exact solution than the existing finite element solution. The objectives are: to obtain mathematical theories for the superconvergence of NCFEM using various element spaces for the second order elliptic problems with the homogeneous essential Dirichlet and natural Neumann boundary conditions; to write computer programs to perform numerical approximations to support the theoretical results; and to investigate existing theoretical results for the superconvergence of the conforming finite element method for second-order elliptic problems by the L-squared projection method. Finally, mathematical theories will be tested with real world data for the Laplace and Poisson equations, which are used in modeling heat conduction, seepage through porous media, irrotational flow of ideal fluids, and other applications.