Superconvergent post-processing of some newly developed numerical methods with weak derivatives

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There has recently been rapid development in computational mathematics due to the demands from science and engineering. Many new methods and algorithms for approximating solutions of partial differential equations have been proposed and analyzed. Compared to traditional methods such as finite element, finite difference, and finite volume methods, these newly developed methods are still in their infancy, especially with respect to post-processing techniques. Encouraged by the success of a special post-processing technique called Polynomial Preserving Recovery (PPR has been adopted by the commercial software COMSOL Multiphysics since 2008) designed by the PI and his students for finite element methods, this project is intended to develop post-processing techniques for some other newly developed numerical methods. The proposed project will not only design algorithms, but also establish a mathematical foundation for post-processing techniques under a unified framework. The proposed research has direct application to other scientific disciplines such as classical mechanics, molecular dynamics, hydrodynamics, electrodynamics, plasma physics, relativity, and astronomy. The success of the project will impact science and engineering practice as well as theoretical mathematical development. The goal of this project is to develop some robust and high accuracy post-processing algorithms and related mathematical theory for some newly developed numerical methods such as Weak Galerkin methods, Virtual Element methods, Hybridizable Discontinuous Galerkin methods, etc. Research efforts will be devoted to developing problem and method independent gradient recovery techniques for the aforementioned numerical methods. The implementation and theory of the Polynomial Preserving Recovery (PPR) for continuous finite element methods will be utilized and further developed and combined with recently developed algorithms and theory for the aforementioned numerical methods.