Hybrid high-order approximations of div-curl systems on domains with general topology

Abstract

We devise and analyze hybrid polyhedral methods of arbitrary order for the approximation of div-curl systems on three-dimensional domains featuring non-trivial topology. The div-curl systems we are interested in stem from magnetostatics, and can either be first-order (field formulation) or second-order (vector potential formulation). The well-posedness of the resulting discrete problems essentially hinges on recently established, topologically generic, hybrid versions of the (first and second) Weber inequalities. Our error analysis covers the case of regular solutions. Leveraging (co)homology computation techniques from the literature, we perform an in-depth numerical assessment of our approach, covering, in particular, the case of non-simply-connected domains.