Analysis Seminar: Finite element methods and superconvergence

Abstract: We consider various finite element methods for numerically solving PDE's, using several formulations of the Poisson problem as examples. All fall under the framework of Galerkin methods, which provide approximate solutions by solving discrete versions of the problem. Conforming methods may have limitations on account of the continuity requirements of the spaces involved. Discontinuous Galerkin (DG) methods yield additional flexibility but result in greater computational complexity. Hybridizable discontinuous Galerkin (HDG) methods address this issue using additional unknowns, which create a nicer structure for the discrete problems, allowing for an increase in efficiency via static condensation. For HDG methods for the Poisson problem, it has been shown that a property of the discrete spaces called an "M-decomposition" results in desirable properties for the methods, including "superconvergence"-the ability to define a new approximation for one of the unknowns which converges as fast as the difference between two approximations in the discrete space. Current research focuses on whether these results apply in the setting of finite element exterior calculus (FEEC), a framework which encompasses many known methods for certain problems as well as methods which have yet to be studied in detail.