A Scalable High-Order Discontinuous Finite Element Framework for Partial Differential Equations: with Application to Geophysical Fluid Flows

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Partial differential equations (PDEs) are pervasive in engineering and science, and their numerical solutions are of paramount importance in understanding complex, natural, engineered, and societal systems. Though the past decades have seen tremendous advances in both theories and computational algorithms for PDEs, scalable numerical methods for complex, coupled, and multiphysics systems that fully exploit the extreme-scale computing systems remain challenging. This becomes the major impediment for future exascale systems unless dramatic changes in mathematical methods and computational algorithms take place. However, not much work has been done for this critical component of computational mathematics. Thus, there is a critical need to develop advanced mathematical discretizations that can enable scientific applications to harness the potential of extreme-scale computing in order to continue the pace of scientific discoveries and to promote the progress of science. The PI will develop a high-order discontinuous finite element (FE) framework, including the weak Galerkin and the hybridized discontinuous Galerkin methods, and its scalable solver for geophysical fluid dynamic applications. In particular, the PI will design and rigorously analyze an abstract high-order discontinuous FE framework for a large class of PDEs including elliptic, parabolic, and hyperbolic types. The PI will also develop, analyze, and implement scalable solvers. Both hydrostatic and non-hydrostatic models will be developed and served as test beds for the developments. The advanced computational and mathematical methods developed in this project will potentially impact the computation of geophysical fluid dynamics. The project will contribute a competitive extreme-scale discretization for the dynamical cores of future high-resolution earth system models.