Discrete Maximal Parabolic Regularity for Time Discontinuous Galerkin Methods with Applications

抛物问题涉及到理论数学和应用数学的许多领域，也是许多环境和现实问题的模型，如污水排放口的最佳位置，污染源的位置，心脏细胞中钙波的建模等。通常，所得到的方程非常复杂，不能解析处理，必须用数值方法求解。这种近似的分析通常是困难和技术性的，需要时间和空间离散化方法的专业知识。对于连续问题的重要性，最大抛物正则性是公认的，并有一些应用，例如非线性问题，最优控制问题，并一般问题的尖锐的结果是必需的。在连续的情况下，离散的最大抛物正则性只是最近才引起数值分析界的注意，其潜力尚未完全实现。这样的结果，例如，减少瞬态问题的分析，以固定的，这通常是少得多的技术与许多结果已经在文献中。例如，这将有利于一些研究人员，特别是谁是在他们的职业生涯的开始，而不是专家在时间discretizationmethods.In这项研究的主要研究者将扩展离散最大抛物正则性理论的不连续Galerkin时间计划在几个方向。研究计划包括以下项目。第一个项目扩展了已知的结果，非对称自治椭圆算子，如瞬态对流反应扩散问题，包括对流主导的情况下。这些问题是经典的，多年来一直是研究的中心。作为这些结果的应用，PI打算回答一些开放的问题，例如，是否稳定的方法需要稳定的参数，取决于时间步长。在第二个项目中，PI打算将我们以前的非自治问题的结果扩展到更一般的规范，这对一些应用程序很重要，例如，拟线性抛物方程和最优控制问题。最后，PI计划研究更一般的抛物系统，如瞬态Stokes和Navier-Stokes问题，这对流体流动问题非常重要，并在一般Lebesgue空间规范中获得完全离散的最佳近似型结果。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Parabolic problems touch many areas of pure and applied mathematics and serve as a model of many environmental and real-life problems, such as optimal location of wastewater outfalls, location of the pollution sources, modeling of calcium waves in a heart cell and etc. Usually the resulting equations are very complicated to be treated analytically and must be solved by numerical methods. The analysis of such approximations is usually hard and technical and requires expertise in time and space discretization methods. For continuous problems the importance of the maximal parabolic regularity is well-recognized and has a number of applications, for example to nonlinear problems, optimal control problems and generally to problems where sharp results are required. In contrast to the continuous case, the discrete maximal parabolic regularity only recently came to the attention of the numerical analysis community and its potential is not yet fully realized. Such results, for example, reduce the analysis of transient problems to stationery ones, which are usually much less technical with many results already available in the literature. This for example would benefit a number of researchers, especially who are at the beginning of their careers and are not experts in time discretization methods.In this research the principal investigator will extend the theory of discrete maximal parabolic regularity for discontinuous Galerkin time schemes in several directions. The research plan includes the following projects. The first project extends the known results to non-symmetric autonomous elliptic operators, such as transient advection-reaction-diffusion problems, including the advection-dominated case. Such problems are classical and have been at the center of research for many years. As an application of such results, the PI intends to answer some of the open questions, for example whether the stabilized methods require stabilization parameters that depend on the time steps. In the second project, the PI intends to extend our previous results for non-autonomous problems to more general norms, that are important for a number of applications, for instance, quasilinear parabolic equations and optimal control problems. Finally, the PI plans to investigate more general parabolic systems, such as transient Stokes and Navier-Stokes problems, which are very important for the fluid flow problems and to obtain fully discrete best approximation type result in general Lebesgue space norms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.