Solution of a regularized problem for a stationary magnetic field in a nonhomogeneous conducting medium by a finite element method

Kremer, I. A.; Urev, M. V.

doi:10.1134/s1995423910010040

Cited by 3 publications

(8 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where the matrices , and have the form ∑ ∑ u u u u u The proof is similar to that of Theorem 3 in [5] and is omitted from this paper. The theorem guarantees that SLAE (4.1) can be solved using the conjugate gradient method with various preconditioners.…”

Section: System Of Linear Equationsmentioning

confidence: 91%

“…The convergence rate of this method is independent of the step size . It was also shown in [5] that the matrix can be inverted in a time com parable to that required for the inversions of . This approach was implemented in the MODEM 3D code intended for interpreting three dimensional electromagnetic soundings of the Earth (see [19]).…”

Section: System Of Linear Equationsmentioning

confidence: 95%

“…The theorem guarantees that SLAE (4.1) can be solved using the conjugate gradient method with various preconditioners. An opti mal preconditioner for the regularized problem was presented in [5]. The convergence rate of this method is independent of the step size .…”

Section: System Of Linear Equationsmentioning

confidence: 99%

“…Here, the vectors , and the matrices , are given by Since is a dense matrix, the following representation is used in practical computations (see [5]):…”

Section: System Of Linear Equationsmentioning

confidence: 99%

“…In [5] a vector finite element method (FEM) was used to numerically solve a stationary regularized equation and numerical results were presented. In this paper, we analyze a discrete scheme as applied to the numerical solution of a regularized generalized formulation of nonstationary problem (0.4) studied in [6].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Regularization method for solving the quasi-stationary Maxwell equations in an inhomogeneous conducting medium

Ivanov¹,

Kremer²,

Urev

2012

Comput. Math. and Math. Phys.

Self Cite

View full text Add to dashboard Cite

Nedelec vector finite elements are used for the numerical solution of a regularized version of the quasi stationary Maxwell equations written in terms of a scalar and a vector magnetic potential with special calibration taking into account the conductivity of the medium. An optimal energy esti mate for the error of the approximate solution in Lipschitz polyhedral domains is established. Numer ical results are presented that demonstrate the stability of the method.

show abstract

Section: System Of Linear Equationsmentioning

confidence: 91%

Section: System Of Linear Equationsmentioning

confidence: 95%

Section: System Of Linear Equationsmentioning

confidence: 99%

“…Here, the vectors , and the matrices , are given by Since is a dense matrix, the following representation is used in practical computations (see [5]):…”

Section: System Of Linear Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Regularization method for solving the quasi-stationary Maxwell equations in an inhomogeneous conducting medium

Ivanov¹,

Kremer²,

Urev

2012

Comput. Math. and Math. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

Solving the Pure Neumann Problem by a Mixed Finite Element Method

Ivanov

Kremer

Laevsky

2022

Numer. Analys. Appl.

View full text Add to dashboard Cite

Solution of problems of harmonic electromagnetic field simulation in regularized and mixed formulations

Butyugin

Ильин

2014

Russian Journal of Numerical Analysis and Mathematical Modelling

View full text Add to dashboard Cite

Classic mixed and regularized variational formulations are considered in the paper for a complex vector Helmholtz equation. A comparative analysis is presented for these formulations and solvability issues are discussed. Preconditioned formulations of problems proposed here can be used for further solution of nite element analogues of continuous problems by iterative methods in Krylov's subspaces.The problem of simulation of electromagnetic elds harmonic in time (in the frequency domain) is urgent for many electrophysical applications, including electronics and electrical engineering, various antenna devices and microwave instruments. The Maxwell system of equations is reduced in this case to complex vector Helmholtz equations with respect to electric or magnetic elds E, H. In addition, there are approaches which involve the formulation of problems for auxiliary values, for example, for scalar and vector potentials. Many papers are focused on numerical solution of such problems with the use of nite element or nite volume methods, see, e.g., [6,11,12,14,16,19,22,26] and the references therein.The main di culties in the construction of e cient algorithms relate to the requirement of high-accuracy and divergence-free grid approximations of electromagnetic elds, because otherwise parasitic charges (having no physical sense) may appear [5,6]. Moreover, solving sparse systems of linear algebraic equations (SLAE) with non-Hermitian sign-inde nite matrices of a high order arising in this case is a hard computational problem even for modern supercomputers.This paper is focused on comparative analysis of two following mathematical formulations: the 'classic' Helmholtz equation for the electric eld E and a similar formulation with a Lagrangian multiplier. Section 1 contains a description of variational representations of the three-dimensional boundary value problems for a complex Helmholtz equation considered here. In Section 2 we consider mixed and regularized formulations of the problem and discuss their solvability. Section 3 presents the constructions of feasible preconditioning operators for the mixed and regularized formulations of the problem. In conclusion we discuss the results obtained in the paper. Formulations of boundary value problems for the Helmholtz equationLet the vectors of the intensity of the electric and magnetic elds have the form ℜ(Eae −iωt ), ℜ(Hae −iωt ), where t is the time, ω is the angular frequency, i is the imaginary unit, ℜ(x) is the real part of x, Ea and Ha are the complex amplitudes dependent on spatial coordinates only. In this case the system of Maxwell equations describing electromagnetic elds in the frequency domain in the absence of external volume electric charges and permeance is written in the following form:∇ × Ea = iωµHa, ∇ · (εrEa) = ∇ × Ha = −iωεEa + J, ∇ · (µrHa) = .(1.1) Brought to you by | Florida International University Libraries Authenticated Download Date | 5/31/15 11:37 AM

show abstract

Solution of a regularized problem for a stationary magnetic field in a nonhomogeneous conducting medium by a finite element method

Cited by 3 publications

References 10 publications

Regularization method for solving the quasi-stationary Maxwell equations in an inhomogeneous conducting medium

Regularization method for solving the quasi-stationary Maxwell equations in an inhomogeneous conducting medium

Solving the Pure Neumann Problem by a Mixed Finite Element Method

Solution of problems of harmonic electromagnetic field simulation in regularized and mixed formulations

Contact Info

Product

Resources

About