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.
This preprint gave rise to the following definitive publication(s):
David MORA, Alberth SILGADO: A C1 virtual element method for the stationary quasi-geostrophic equations of the ocean. Computers & Mathematics with Applications, vol. 116, pp. 212–228, (2022).