A mixed formulation for 3D magnetostatic problems: theoretical analysis and face-edge finite element approximation

Abstract

A mixed field-based variational formulation for the solution of threedimensional magnetostatic problems is presented and analyzed. This method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship, while constraints represented by Maxwell's equations are imposed by means of Lagrange multipliers. In this way, both the magnetic field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements, and boundary conditions can be imposed in a natural way. Moreover, this method is more suitable than classical approaches for the approximation of problems featuring strong discontinuities of the magnetic permeability, as is usually the case. A finite element discretization involving face and edge elements is also proposed, performing stability analysis and giving error estimates.

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

“…The same remark holds for the least squares approach of Chang and Gunzburger [16] and the even more expensive mixed methods of Kikuchi [39], Perugia [51], and Alotto and Perugia [4].…”

Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation

“…It follows from (17) that (16) is a variational analogue of conservation law (10). In other words, the solution of variational problem (14) satisfies conservation law (5) in a weak sense.…”

“…Due to large dimension of the kernel, application of standard preconditioners usually does not give a reduction in the number of iterations or even may lead to divergence of the iterative process [1,9,10].…”

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

“…We limit our presentation to the elements of the lowest degree, just for the sake of simplicity. We refer to Reference [6] for a complete analysis, covering the case of higher degree elements.…”

“…The proof of existence and uniqueness of the discrete solution, and its linear convergence to the solution of problem (6) is given in Reference [6].…”