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In this work, following the discrete de Rham (DDR) approach, we develop adiscrete counterpart of a two-dimensional de Rham complex with enhancedregularity. The proposed construction supports general polygonal meshes andarbitrary approximation orders. We establish exactness on a contractible domainfor both the versions of the complex with and without boundary conditions and,for the former, prove a complete set of Poincar\'e-type inequalities. Thediscrete complex is then used to derive a novel discretisation method for aquad-rot problem which, unlike other schemes in the literature, does notrequire the forcing term to be prepared. We carry out complete stability andconvergence analyses for the proposed scheme and provide numerical validationof the results.

An arbitrary-order discrete rot-rot complex on polygonal meshes with application to a quad-rot problem