Mathematics

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 with an arbitrary number of tiny faces. The new a posteriori error analysis first appeared in the literature and 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 the 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 a posteriori error estimator of the space-time dG method for solving the Allen-Cahn problem.