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We study a recovery problem for an unknown boundary coefficient relating to one material characteristic in an eddy current field. The field equations are represented in terms of the potential field method ( $$\varvec{T} - \psi $$ method) and can be solved numerically by the nodal finite element method. We introduce a measurement as an additional condition and prove the existence and uniqueness of the weak solution. Further, we present an iteration algorithm for the recovery problem and validate its efficiency by two numerical experiments.

A fully discrete A-ϕ finite element scheme for a nonlinear model of type-II superconductors is proposed and analyzed. The nonlinearity is due to a field dependent conductivity with the regularized power-law form. The challenge of this model is the error estimate for the nonlinear term under the time derivative. Applying the backward Euler method in time discretisation, the well-posedness of the approximation problem is given based on the theory of monotone operators. The fully discrete system is derived by standard finite element method. The error estimate is suboptimal in time and space.

In this paper we present a fully discrete A-ø finite element method to solve Maxwell’s equations with a nonlinear degenerate boundary condition, which represents a generalization of the classical Silver-Müller condition for a non-perfect conductor. The relationship between the normal components of the electric field E and the magnetic field H obeys a power-law nonlinearity of the type H x n = n x (|E x n|α-1E x n) with α ∈ (0,1]. We prove the existence and uniqueness of the solutions of the proposed A-ø scheme and derive the error estimates. Finally, we present some numerical experiments to verify the theoretical result.

We investigate the A- Â? finite element method with the global coarse grids and the local fine grids (composite grids) for solving a time-dependent eddy current problem, which can improve the accuracy of the coarse grid solutions in some subdomains of interest in the case when properly increasing computational costs. Meanwhile, to decrease calculation complexity and avoid dealing with a saddle-point problem from the traditional A- Â? scheme, we design an iteration which combines the composite grid method with classic steepest descent such that the solutions of the unknowns A and Â? for the global coarse grids domain are decoupled in two separate equations. We prove it converges with a bounded rate independent of the mesh sizes.

The aim of this paper is to develop a fully discrete (T,ψ)-ψe finite element decoupled scheme to solve time-dependent eddy current problems with multiply-connected conductors. By making ‘cuts’ and setting jumps of ψe across the cuts in nonconductive domain, the uniqueness of ψe is guaranteed. Distinguished from the traditional T- ψ method, our decoupled scheme solves the potentials T and ψ- ψe separately in two different simple equation systems, which avoids solving a saddle-point equation system and leads to a remarkable reduction in computational efforts. The energy-norm error estimate of the fully discrete decoupled scheme is provided. Finally, the scheme is applied to solve two benchmark problems—TEAM Workshop Problems 7 and IEEJ model. Copyright © 2013 John Wiley & Sons, Ltd.

The aim of this paper is to propose improved T − ψ finite element schemes for eddy current problems in the three-dimensional bounded domain with a simply-connected conductor. In order to utilize nodal finite elements in space discretization, we decompose the magnetic field into summation of a vector potential and the gradient of a scalar potential in the conductor; while in the nonconducting domain, we only deal with the gradient of the scalar potential. As distinguished from the traditional coupled scheme with both vector and scalar potentials solved in a discretizing equation system, the proposed decoupled scheme is presented to solve them in two separate equation systems, which avoids solving a saddle-point equation system like the traditional coupled scheme and leads to an important saving in computational effort. The simulation results and the data comparison of TEAM Workshop Benchmark Problem 7 between the coupled and decoupled schemes show the validity and efficiency of the decoupled one.

A new H-splitting decoupled scheme by using Finite Element Method and Natural Boundary Reduction is suggested to approximate a transient eddy current problem over an unbounded domain containing a simply connected conductor. It is based on solving a vector potential only in the conducting domain and a scalar potential over the whole space. As distinguished from the traditional fully discrete coupled scheme in which both vector and scalar unknowns are solved in an equation system at every time step, our computing scheme deals with them in two different equation systems respectively and thus decreases the total storage amount of non-zero entries of coefficient matrix and computational costs. In this paper, two error estimates of our scheme in finite time are obtained.

Fully discrete potential-based finite element methods called $${\mathbf{A}-\phi}$$methods are used to solve a transient eddy current problem in a three-dimensional convex bounded polyhedron. Using $${\mathbf{A}-\phi}$$methods, fully discrete coupled and decoupled numerical schemes are developed. The existence and uniqueness of solutions for these schemes together with the energy-norm error estimates are provided. To verify the validity of both schemes, some computer simulations are performed for the model from TEAM Workshop Problem 7.

In this paper, we propose novel finite element A - θ approaches to realize the approximation of a three-dimensional eddy current problem in a bounded domain containing a simply-connected conductor region where the eddy current is formed. These algorithms are based on solving a vector potential in the whole domain by using the edge finite-element method and a local scalar potential only in the conductor region by using the nodal finite-element method. Further, error estimates for coupled and decoupled A - θ schemes are given.

A fully discrete potential-based finite-element method which is called the A–φ approach is used to approximate time-dependent Maxwell’s equations in a three-dimensional bounded and convex domain. By appending the Coulomb gauge in the form of a penalty function to the magnetic vector potential, the existence and uniqueness of approximated solutions are ensured. Further, an optimal energy-norm error estimate is given and some computer simulation tests are performed to demonstrate the validity of our schemes.

A new decoupled finite element method based on solving a vector and a scalar from the splitting of the electric field is presented to approximate time-dependent eddy current equations with discontinuous coefficients in a three-dimensional polyhedral domain. By introducing a Lagrangian multiplier, this method realizes an optimal control for the interface problem. A non-matching finite element grid on the interface is considered and an optimal energy-norm error estimate in finite time is obtained.

In this paper, we suggest some new algorithms to realize approximating magnetic field equations derived from the basic eddy current model in the bounded domain. These algorithms are based on solving two different equation systems inside and outside the conductor in which eddy current is formed. In addition, we extend relative results to the corresponding unbounded-domain problem by the coupling of finite elements and natural boundary elements and the domain decomposition method based on the natural boundary reduction respectively. Meanwhile some error estimates are given.

The aim of this paper is to present a new T-ψ finite element decoupled scheme with Crank-Nicholson difference format for time-dependent eddy current problem, which avoids solving a saddle-point equations system and is more economical than the traditional scheme. The holes of multiply connected conductors are stemmed by making "cuts" and jumps of the function value ψ across the "cuts" are enforced. The construction of the finite element space with discontinuities at the "cuts" are given. The superior performance of the decoupled scheme is demonstrated by numerical experiment of the benchmark problem-TEAM Workshop Problem 7.

In this paper we investigate the potential-based finite element algorithms for eddy-current problems. In our models the source coil carries a pulse excitation current generating a transient field. As distinguished from the traditional coupled algorithm whose discrete equation system including both vector and scalar potential unknowns is solved at every time-step, a novel decoupled algorithm is presented here. Because the vector and scalar unknowns are calculated at two different equation systems respectively, it decreases the storage amount of non-zero entries of coefficient matrix and computational costs, and achieves the better results under the alike CPU time. Some computer simulation results of eddy-current densities for two eddy-current models (IEEJ model and TEAM Workshop Problem 7) are demonstrated to verify the feasibility and efficiency of our algorithm.

In this paper, based on the natural boundary reduction advanced by Feng and Yu, we couple the finite element approach with the natural boundary element method to study the weak solvability and Galerkin approximation of a class of nonlinear exterior boundary value problems. The analysis is mainly based on the variational formulation with constraints. We prove the error estimate of the finite element solution and obtain the asymptotic rate of convergence. Finally, we also give a numerical example.

A new decoupled finite element method is suggested to approximate time-dependent eddy current equations in a three-dimensional polyhedral domain. This method is based on solving a vector and a scalar from the splitting of the electric field by using edge and nodal finite elements. An optimal energy-norm error estimate in finite time is obtained by introducing a projection operator.