A discrete Weber inequality on three-dimensional hybrid spaces with application to the HHO approximation of magnetostatics

TL;DR

This paper establishes a discrete Weber inequality for 3D hybrid spaces and introduces two high-order methods for magnetostatics, applicable on general polyhedral meshes, with theoretical analysis and validation.

Contribution

It presents a novel discrete Weber inequality and develops two high-order hybrid methods for magnetostatics on polyhedral meshes, with rigorous analysis and validation.

Findings

The methods achieve high-order accuracy.

The discrete Weber inequality ensures stability and convergence.

Validation confirms effectiveness on test cases.

Abstract

We prove a discrete version of the first Weber inequality on three-dimensional hybrid spaces spanned by vectors of polynomials attached to the elements and faces of a polyhedral mesh. We then introduce two Hybrid High-Order methods for the approximation of the magnetostatics model, in both its (first-order) field and (second-order) vector potential formulations. These methods are applicable on general polyhedral meshes, and allow for arbitrary orders of approximation. Leveraging the previously established discrete Weber inequality, we perform a comprehensive analysis of the two methods. We finally validate them on a set of test-cases.