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

TL;DR

This paper introduces high-order hybrid polyhedral methods for accurately approximating div-curl systems in complex three-dimensional domains, with a focus on magnetostatics applications and topologically intricate geometries.

Contribution

It develops and analyzes novel hybrid high-order methods tailored for div-curl systems on non-trivial topologies, extending existing inequalities and providing comprehensive error analysis.

Findings

Methods achieve high-order accuracy on complex geometries.

Numerical assessments confirm robustness on non-simply-connected domains.

Error bounds are established for regular solutions.

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.