PDE models are often characterised by local features such as solution singularities/layers and domains with complicated boundaries and phase transitions. These special features make the design of accurate numerical solutions challenging, or require a huge amount of computational resources. One way of achieving complexity reduction of the numerical solution for such PDE models is to design novel numerical methods which support general meshes consisting of polygonal/polyhedral elements, such that local features of the model can be resolved efficiently by adaptive choices of such general meshes. In this talk, we will present recent results on a new a posteriori error analysis for the dG method on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. The new a posteriori error analysis generalizes the known results for dG methods to admit an arbitrary number of irregular hanging nodes per element. Moreover, under certain practical mesh assumptions, the new error estimator of dG method was proven to be available to incorporate essentially arbitrarily-shaped elements with an arbitrary number of faces or even curved faces. Finally, we will present the application of the adaptive space-time dG method for solving the Allen-Cahn problem.

Dr. Zhaonan Dong is currently a permanent researcher (CRCN) 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..