Discrete Maximal Parabolic Regularity for Time Discontinuous Galerkin Methods with Applications

抛物问题涉及纯数学和应用数学的许多领域，并作为许多环境和现实生活问题的模型，如废水出口的最佳位置，污染源的位置，心脏细胞中钙波的建模等。通常得到的方程非常复杂，无法用解析方法处理，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方法方面的专业知识。对于连续问题，极大抛物正则性的重要性是公认的，并且有许多应用，例如非线性问题，最优控制问题以及通常需要明确结果的问题。与连续情况相比，离散最大抛物正则性直到最近才引起数值分析界的注意，其潜力尚未充分实现。例如，这样的结果将对瞬态问题的分析简化为简单的分析，这种分析通常技术性较低，因为文献中已有许多结果。例如，这将使许多研究人员受益，特别是那些刚刚开始他们的职业生涯并且不是时间离散方法专家的研究人员。在本研究中，主要研究者将不连续伽辽金时间格式的离散极大抛物正则性理论在几个方向上进行推广。研究计划包括以下项目。第一个项目将已知的结果扩展到非对称自治椭圆算子，如瞬态平流-反应-扩散问题，包括平流主导的情况。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打算回答一些悬而未决的问题，例如，稳定方法是否需要依赖于时间步长的稳定参数。在第二个项目中，PI打算将我们之前的非自治问题的结果扩展到更一般的规范，这对许多应用很重要，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空间范数下获得完全离散的最佳逼近型结果。该奖项反映了美国国家科学基金会的法定使命，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.