Divergence free Virtual Elements for the Stokes problem on polygonal meshes
This work addresses the challenge of enforcing divergence-free conditions in finite element methods for Stokes flow, providing a more accurate and efficient solution for computational fluid dynamics on polygonal meshes.
The authors develop a new family of Virtual Elements for the Stokes problem on polygonal meshes that guarantees pointwise divergence-free discrete velocities, leading to an efficient scheme equivalent to a reduced problem with fewer degrees of freedom. Numerical tests confirm the method's effectiveness.
In the present paper we develop a new family of Virtual Elements for the Stokes problem on polygonal meshes. By a proper choice of the Virtual space of velocities and the associated degrees of freedom, we can guarantee that the final discrete velocity is pointwise divergence-free, and not only in a relaxed (projected) sense, as it happens for more standard elements. Moreover, we show that the discrete problem is immediately equivalent to a reduced problem with less degrees of freedom, thus yielding a very efficient scheme. We provide a rigorous error analysis of the method and several numerical tests, including a comparison with a different Virtual Element choice.