Gauging in Whitney spaces

A field-based finite element method for magnetostatics derived from an error minimization approach

“…By testing Equation (6a), with B * ∈ H B (div 0 ; ; ux), owing to the orthogonality of decomposition (14), we get U 2 = 0 in L 2 ( ) 3 3 , that is B − H = 0 a.e. in , therefore we can conclude that the constitutive law also is exactly satisÿed by the solution of (6).…”

“…The usual approach for the numerical solution of three-dimensional magnetostatic problems, that all commercial codes propose, involves the use of potentials [1; 2], which are mathematically equivalent to magnetic ÿelds [3]. However, the derived techniques show some computational drawbacks (possible numerical cancellation, di culties in choosing appropriate potentials in non-trivial topologies, weak enforcement of some physical laws and fulÿllment of the continuity properties for just one of the two ÿelds in the numerical solution), which cannot be easily solved unless a radically di erent formulation is adopted.…”

Global formulation of 3D magnetostatics using flux and gauged potentials

“…The problem of uniqueness and gauging of magnetic vector potential has been addressed by many researchers, see for instance References [6,[9][10][11][12] in the frame of variational formulation. Here this problem is recast in a discrete way linking the minimal set of independent unknowns to the minimal set of variables needed to define the solution.…”

A Discreate Calculus with Applications of High-Order Discretizations to Boundary-Value Problems

“…There was a workshop held at San Diego State University in 2003 and in 2004 there was an IMA "Hot Topics" Workshop: Compatible Spatial Discretizations for Partial Differential Equations. There is an interest in creating discrete analogs of the vector calculus [1,[14][15][16][17][18]22] including the supportoperators methods [35,37], differential forms [3,5], and algebraic topology [2,11,13,20,23]. Additionally, some authors have studied a range of discretization methods from a mimetic points of view [4,9,10,19,23,24,29,30,36].…”