Seminar in Numerical Analysis: Zhaonan Dong (Inria Paris)

A posteriori error analysis for discontinuous Galerkin on polytopic meshes

PDE models are often characterized by local features such as solution singularities, boundary layers, domains with complicated boundaries, and phase transitions. These unique characteristics make designing accurate numerical solutions challenging or demand substantial computational resources. One effective strategy is to develop novel numerical methods that support general meshes composed of polygonal or polyhedral elements, enabling adaptive refinement that efficiently captures local features.

In this talk, we present recent results on a new a posteriori error analysis for the discontinuous Galerkin (dG) method applied to general computational meshes consisting of polygonal/polyhedral (polytopic) elements with an arbitrary number of faces. This analysis, which first appeared in the literature, generalizes known dG methods by allowing an arbitrary number of irregular hanging nodes per element. Moreover, under practical mesh assumptions, the new error estimator accommodates nearly any element shape—even with curved faces. We will also briefly discuss the a posteriori error estimator for the space-time dG method in solving the Allen–Cahn problem, as well as the hp-a posteriori error estimator for the DG method in tackling fourth-order PDEs.

For further information about the seminar, please visit this webpage.

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