Talk by M. Arrutselvi
Posted on: 27 Jan 2026
Title: Solving PDEs on Arbitrary Meshes Without Explicit Basis Functions: The Virtual Element Method
Speaker: M. Arrutselvi, NBHM Postdoctoral fellow, IIT Madras
Date: 3 Feb 2026
Time: 4 PM IST
Venue: Zoom
Abstract:
Many classical discretization techniques for partial differential equations rely on explicitly constructed finite-dimensional approximation spaces defined on simplicial or tensor-product meshes. While highly successful, this paradigm becomes restrictive when one seeks robust and stable methods on general polygonal or polyhedral meshes arising from adaptivity, mesh agglomeration, or complex geometries.
The Virtual Element Method (VEM) offers a principled framework for the discretization of partial differential equations on arbitrary meshes without re- quiring explicit knowledge of basis functions inside mesh elements. Instead, the method is formulated through carefully chosen degrees of freedom, computable projection operators, and discrete bilinear forms designed to satisfy polynomial consistency and stability properties.
This talk presents a conceptual introduction to the Virtual Element Method for elliptic problems, emphasizing the underlying approximation and stability mechanisms rather than implementation details. Starting from a model problem, we explain how consistency is enforced through polynomial projections and how stabilization controls the non-polynomial components of the virtual space. We also discuss convergence properties, mesh-regularity considerations, and the flexibility of the framework in comparison with classical finite element approaches. The talk concludes with a brief overview of applications and current research directions in virtual element technology.