Discrete spaces for div and curl-free fields

The Second Order Finite Element Analysis of Eddy Currents Based on the T-Ω Method

“…As discussed in [15], one cannot construct any vector basis that should be simultaneously curl-free and divergence-free. However, it is still possible to construct a vector basis that is locally divergence-free and globally curl-free.…”

“…Reference [23] introduced an efficient cut-generating algorithm with computational complexity O(N 2 ) where N denotes the number of DOF's in the finite element discretization. The thickcut, which is one layer of tetrahedral elements having their edges passing through cutting surfaces, as introduced in [15,16] has the computational complexity O(N 1.5 ) or less for most practical simulations. Also, thick-cut can be naturally adapted to Hp assignment procedure.…”

“…Note that the dimension of the column vector [χ], which is the degree of freedom (DOF) of the harmonic field component, is the number of loops, so it is independent of the mesh and orders of the basis functions. There are several approaches to model (21), the topological property of the harmonic field component [15][16][17][18][19][20][21][22][23]. Reference [23] introduced an efficient cut-generating algorithm with computational complexity O(N 2 ) where N denotes the number of DOF's in the finite element discretization.…”

High order differential form-based elements for the computation of electromagnetic field

“…The complement of in is called De Rham's th cohomology group. Dimension of depends on the topology of the manifold [11].…”

Construction of a Finite Element Basis of the First de Rham Cohomology Group and Numerical Solution of 3D Magnetostatic Problems

“…These fields are of central importance for numerical electromagnetism in general topological domains (see, e.g., Kotiuga [40], Kettunen et al [37]; see Bossavit [13], Gross and Kotiuga [31]). To make precise one of their most important properties, let us first give a definition: if the only linear combination of a maximal set of loop fields that equals a gradient is the trivial one, we say that those loop fields are linearly cohomologically independent.…”