Theoretical Considerations for Stabilization-Free Virtual Elements for 3D problems
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The aim of this talk is to present recent advances in the development of a stabilization-free Virtual Element Method for three-dimensional problems. Virtual Element Methods (VEM) are polytopal numerical methods designed to solve partial differential equations on domains with highly complex geometries. The presence of an additional stabilization term in the discrete bilinear forms -- typical of standard VEMs -- may introduce certain difficulties in applied or engineering-oriented problems. Although such terms are required to guarantee stability, they do not preserve the structure of the continuous operator and often need to be tuned depending on the problem considered. In this talk, we present a novel three-dimensional stabilization-free nodal VEM formulation, designed for specific classes of polyhedral meshes and based on the use of higher-order polynomial spaces. We investigate the properties of the underlying polynomial space that are needed to construct a stable method, and we provide a theoretical proof of its stability. The proposed approach is then illustrated in problems of interest in computational mechanics.
