A Three-Dimensional Hybrid High-Order Method for Magnetostatics

Abstract

We introduce a three-dimensional Hybrid High-Order method for magnetostatic problems. The proposed method is easy to implement, supports general polyhedral meshes, and allows for arbitrary orders of approximation.

Virtual elements for Maxwell's equations

“…curl E I = (curl E) I for a given vector-valued field E ∈ H s (curl, Ω), 1/2 < s ≤ 1, where on the leftand right-hand sides we consider the edge and face interpolations of E and curl(E) that are respectively defined in (19) and (27);…”

“…As in Remark 1, the edge interpolant of ∇p in the sense of ( 19) is such that (∇p) I = ∇p I . Therefore, the edge interpolant of v in the sense of (19) satisfies…”

“…Let E I denote the interpolation of E in the edge space V edge h ; cf. (19). Then, for a given E ∈ H(curl, Ω), Π curl h (E) is the solution to the following mixed variational problem:…”

“…Important applications involve, for instance, the analysis and design of microwave devices [21], cavity resonators [30,38,41], coaxial cables and waveguides [42], antennas and high-power amplifiers [26,28,39,40], electromagnetic scattering [29,33] Due to the complex geometries that are often faced in many applicative areas of electromagnetism, the additional flexibility of general polytopal grids is an important asset, not only in generating an efficient mesh to partition the domain of interest, but also in handling/gluing/adapting existing meshes. Among the other polytopal technologies, in the realm of electromagnetism one can find (in a non-exhaustive list) Polygonal finite elements [24], Mimetic Finite Differences [31], Hybrid High-Order methods [19], and Discrete Exact Sequences [23].…”

Hybrid High-Order Methods for the Acoustic Wave Equation in the Time Domain

“…In the present work, we focus on the acoustic wave equation. The extension to elastodynamic waves is relatively straightforward and is reported in [10], whereas the extension to electromagnetics could, e.g., be based on the ideas developed in [15] for magnetostatics.…”