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 Antonietti et al., (2018) and Zhao et al., (2018). 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 enriching operatorEh defined from the nonconforming space into its H2-conforming counterpart. With the help of these tools we prove optimal error estimates for the stream-function in broken H2-, H1- and L2-norms 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.