The Morley-type virtual element method for the Navier-Stokes equations in stream-function form on general meshes

TL;DR

This paper introduces a Morley-type virtual element method for the Navier-Stokes equations in stream-function form on general meshes, providing rigorous analysis, optimal error estimates, and effective pressure recovery techniques.

Contribution

It develops a new Morley-type virtual element scheme with a novel enriching operator, enabling optimal error estimates and pressure recovery on general polygonal meshes.

Findings

Optimal error estimates in multiple norms.

Effective pressure recovery algorithm.

Validated performance through benchmark tests.

Abstract

The nonconforming Morley-type virtual element method for the incompressible Navier-Stokes equations formulated in terms of the stream-function on simply connected polygonal domains (not necessarily convex) is designed. A rigorous analysis by using a new enriching operator is developed. More precisely, by employing such operator, we provide novel discrete Sobolev embeddings, which allow to establish the well-posedness of the discrete scheme and obtain optimal error estimates in broken $H^2$-, $H^1$- and $L^2$-norms under minimal regularity condition on the weak solution. The velocity and vorticity fields are recovered via a postprocessing formulas. Furthermore, a new algorithm for pressure recovery based on a Stokes complex sequence is presented. Optimal error estimates are obtained for all the postprocessed variables. Finally, the theoretical error bounds and the good performance of the method are validated through several benchmark tests.