Abstract
In this work, a new Virtual Element Method (VEM) of arbitrary order k ≥ 2 for the time dependent Navier–Stokes equations in stream-function form is proposed and analyzed. Using suitable projection operators, the bilinear and trilinear terms are discretized by only using the proposed degrees of freedom associated with the virtual space. Under certain assumptions on the computational domain, error estimations are derived and shown that the method is optimally convergent in both space and time variables. Finally, to justify the theoretical analysis, four benchmark examples are examined numerically.
Highlights
We study a Virtual Element Method (VEM) for a fourth order nonlinear problem arising in the numerical discretization of the Navier–Stokes problem
The VEM, introduced in [6], is a generalization of the Finite Element Method (FEM) which is characterized by the capability of dealing with very general polygonal/polyhedral meshes, and it permits to construct in a straightforward way highly regular discrete spaces
In [25] a C1 finite element method based on the Argyris element has been proposed for the stationary quasi-geostrophic equations, which corresponds to an extension of a stream-function formulation for the Navier–Stokes problem
Summary
We study a Virtual Element Method (VEM) for a fourth order nonlinear problem arising in the numerical discretization of the Navier–Stokes problem. In [25] a C1 finite element method based on the Argyris element has been proposed for the stationary quasi-geostrophic equations, which corresponds to an extension of a stream-function formulation for the Navier–Stokes problem. We will propose a new VEM discretization to solve the time dependent Navier– Stokes problem written in terms of the stream-function variable. The advantages of the present method are: the C1 conforming virtual space can be built with a straightforward construction due to the flexibility of the VEM and it provides a very competitive alternative to solve the time dependent Navier–Stokes problem on polygonal meshes.
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