Discrete Maximal Parabolic Regularity for Time Discontinuous Galerkin Methods with Applications

University Of Connecticut, Storrs CT

Investigators

Abstract

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.