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

研究启动奖为历史上黑人学院和大学的初级和中级职业教师提供支持，他们正在建立新的研究项目或重新定向和重建现有的研究项目。预计该奖项有助于进一步提高教师的研究能力和效率，改善她所在机构的研究和教学，并使本科生参与研究经验。 授予位于派恩海崖的阿肯色州大学的奖项在许多领域都有潜在的更广泛的影响。本计画将以L-平方投影法建构二阶椭圆型问题之有限元素法超收敛之数学模型。该项目将加强该机构数学本科生的研究经验和培训。此外，在有限元方法课程将开发和提供作为一个主题课程。该项目将使用L平方投影方法来提高现有有限元解的收敛速度，使新的近似值比现有有限元解更接近精确解。目标是：获得NCFEM的超收敛的数学理论，使用各种元素空间的二阶椭圆问题的齐次本质Dirichlet和自然Neumann边界条件;编写计算机程序来执行数值逼近，以支持理论结果;利用L-有限元方法研究了二阶椭圆型问题协调有限元方法的超收敛性，平方投影法最后，将测试数学理论与真实的世界数据的拉普拉斯和泊松方程，这是用于模拟热传导，渗流通过多孔介质，无旋流动的理想流体，和其他应用。

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.