Raviart-Thomas Staggered DG methods on polygonal meshes
报告人：Eun-Jae Park（Yonsei University）
时间：2025-12-26 16:00-17:00
地点：智华楼109
Abstract:
The Staggered Discontinuous Galerkin (SDG) method is a class of finite element methods designed to solve partial differential equations while preserving local conservation properties and handling complex geometries. It employs a staggered mesh structure in which scalar and vector variables are discretized on distinct, yet overlapping, primal and dual meshes. This arrangement facilitates a natural enforcement of conservation laws and enables element-wise postprocessing for superconvergent approximations. The SDG framework supports high-order accuracy and geometric flexibility, making it well-suited for problems involving unstructured or polytopal meshes. Moreover, by decoupling the discretization of variables, the method enhances stability and allows for efficient hybridization, yielding compact global systems and connections to other modern finite element approaches such as HHO, weak Galerkin, and virtual element methods.
In this talk, we present a new family of polygonal SDG methods utilizing Raviart-Thomas mixed finite element spaces. Formulated in a mixed setting, the method approximates the primal and dual variables using a locally $H^1$-conforming finite element space and a locally $\vH(\div)$-conforming Raviart--Thomas finite element space, respectively. Unlike in classical mixed FEM, the standard $RT_k \times P_k$ pair is not inf–sup stable in the SDG framework due to the staggered primal–dual mesh structure. To restore stability, the primal space is enriched with bubble type functions on dual elements. The inf-sup stability and optimal convergence are proved. Next, with a simple modification of the loading term we are able to obtain globally $\vH(\div)$-conforming velocity fields. The theoretical results are verified by numerical experiments. Some recent work on the eigenvalue problem will be discussed.
