报告题目：On a posteriori error estimates for non-conforming Galerkin methods

报告人：National Institute for Research in Digital Science and Technology (INRIA, France)，董兆楠 研究员

报告时间：2021年6月4日下午15:00-17:00

报告地点：数学与统计学院金融实验室

摘要：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.

报告人简介：Zhaonan (Peter) Dong is currently a permanent researcher (CRCN) at in the National Institute for Research in Digital Science and Technology (INRIA, France) since 10/2020. Before he moved to Paris, he used to be Lecturer at the Cardiff University (UK) from 01/2019 to 09/2020. he was a visiting researcher of the research group lead by Prof. Charalambos Makridakis at the IACM-FORTH (Greece). He was post-doc researcher at the University of Leicester (UK) from 10/2016 to 09/2018.  He obtained his PhD under the supervision of Prof. Emmanuil Georgoulis and Dr. Andrea Cangiani in 10/2016 at the University of Leicester.

His research interest is Numerical Methods for Partial Differential Equations. More specifically: continuous and discontinuous FEM, hp-version FEM, adaptive algorithms, multiscale methods, polygonal discretization methods, solver design. In the past several years, he has obtained one Springer Monograph and several papers accepted and published on leading journals: SIAM J. Numer. Anal.，SIAM J. Sci. Comput.， Math. Comp..