Recent Advances in Stabilization-Free Polygonal Methods
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This talk focuses on recent advances in the study of polygonal methods for solving partial differential equations, named stabilization-free. These approaches design discrete bilinear forms that preserve the structure of the problem operator. In recent years, polygonal methods – such as Virtual Element Methods, Hybrid High-Order Methods and others – have attracted significant attention for their robustness and their ability to handle highly complex geometries. Traditionally, these methods include in their discrete bilinear forms an arbitrary non-polynomial term, known as a stabilization term. This term has been extensively studied in the literature, as it may introduce difficulties in the presence of anisotropies, nonlinearities, the design of a posteriori error estimators, and in eigenvalue problems. Building on recent developments in Virtual Element Methods and Hybrid High-Order Methods, this talk aims to highlight new approaches that eliminate the need for the stabilization term. The proposed framework introduces consistent and stable discrete bilinear forms based on higher-order polynomial projections and investigates the properties of these projections that ensure the theoretical stability of the method.
