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

研究启动奖为历史悠久的黑人学院和大学的初级和职业生涯中期教师提供支持，这些教师正在建立新的研究计划或重新定向和重建现有的研究计划。预计该奖项有助于进一步提高教师的研究能力和有效性，改善她所在机构的研究和教学，并让本科生参与研究经验。授予阿肯色大学派恩布拉夫分校的奖项可能会在许多领域产生更广泛的影响。本项目将致力于用L平方投影法建立二阶椭圆型问题非协调有限元超收敛的数学模型。该项目将加强学院本科生在数学方面的研究经验和培训。此外，还将开发一门有限元方法课程，并将其作为专题课程提供。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，从而使新的近似比现有的有限元解更接近精确解。其目的是：对于具有齐次本质Dirichlet和自然Neumann边界条件的二阶椭圆型问题，利用不同的单元空间得到NCFEM超收敛的数学理论；编写计算机程序进行数值逼近以支持理论结果；以及利用L平方投影方法研究二阶椭圆型问题协调有限元方法超收敛的现有理论结果。最后，将用拉普拉斯方程和泊松方程的真实世界数据来检验数学理论，这些方程用于模拟热传导、通过多孔介质的渗流、理想流体的无旋流以及其他应用。

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.