David Mora, Alberth Silgado:
A C1 virtual element method for the stationary quasi-geostrophic equations of the ocean
In this present paper, we propose and analyze a $C^1$-conforming virtual element method to solve the so-called one-layer stationary quasi-geostrophic equations (QGE) with applications in the large scale wind-driven ocean circulation, formulated in terms of the stream-function. This problem corresponds to a nonlinear fourth order partial differential equation. The $C^1$ virtual space and the discrete scheme are built in a straightforward way due to the flexibility of the virtual approach. Under the assumption of small data, we prove well-posedness of the discrete problem by using a fixed-point strategy and under standard assumptions on the computational domain, we establish error estimates in $H^2$-norm for the stream-function. Finally, we report four numerical experiments that illustrate the behaviour of the proposed scheme and confirm our theoretical results on different families of polygonal meshes.