Zhaonan Dong: Thursday 5 November at 11:00 via zoom (meeting ID: 961 6781 3449 and code: g4q44P)
Non-conforming Galerkin methods are very popular for the stable and accurate numerical approximation of challenging PDE problems. The term “non-conforming” refers to approximations that do not respect the continuity properties of the PDE solutions. Nonetheless, to arrive at rigorous error control via a posteriori error estimates, non-conforming methods pose a number of challenges. I will present a novel methodology for proving a posteriori error estimates for the “extreme” class (in terms of non-conformity) of discontinuous Galerkin methods in various settings. For instance, we prove a posteriori error estimates for the recent family of Galerkin methods employing the general shaped polygonal and polyhedral elements, solving an open problem in the literature. Furthermore, with the help this new idea, we prove new a posteriori error bounds for various $hp$-version non-conforming FEMs for fourth-order elliptic problems; these results also solve a number of open questions in the literature, yet they arise relatively easily within the new reconstruction framework of proof. These results open a door to design new reliable adaptive algorithms for solving the problems in thin plate theories of elasticity, phase-field modeling and mathematical biology.