Preprint 2022-15
Dibyendu Adak, David Mora, Alberth Silgado:
A Morley-type virtual element approximation for a wind-driven ocean circulation model on polygonal meshes
Abstract:
In this work, we propose and analyze a Morley-type virtual element method to approximate the Stommel-Munk model in stream-function form. The discretization is based on the fully nonconforming virtual element approach presented in [5,47]. The analysis restricts to simply connected polygonal domains, not necessarily convex. Under standard assumptions on the computational domain we derive some inverse estimates, norm equivalence and approximation properties for an {\it enriching operator} $E_h$ defined from the nonconforming space into its $H^2$-conforming counterpart. With the help of these tools we prove optimal error estimates for the stream-function in broken $H^2$-, $H^1$- and $L^2$-norms {\it under minimal regularity} condition on the weak solution. Employing postprocessing formulas and adequate polynomial projections we compute from the discrete stream-function further fields of interest, such as: the velocity and vorticity. Moreover, for these postprocessed variables we establish error estimates. Finally, we report practical numerical experiments on different families of polygonal meshes.
