Edge Finite Element Research Articles

Superconvergence recovery of cubic edge elements for Maxwell’s equations

In this article, a new recovery method is designed and analyzed for curl conforming elements on cuboid mesh. The proposed recovery method fully exploits the potential of symmetry to obtain the superconvergence of recovered edge finite element solution in discrete ℓ2 norm. The idea is to identify a symmetry sub-domain such that a polynomial of the same degree is recovered by local L2 projection, based on which the recovered values follow. Combined with extrapolation and the least square fitting for the linear edge elements and quadratic edge elements, respectively, we obtain global superconvergence for recovered quantities in L2 norm. Numerical experiments are provided to validate our theoretical findings.

Analysis of Variational Formulations and Low-regularity Solutions for Time-harmonic Electromagnetic Problems in Complex Anisotropic Media

We consider the time-harmonic Maxwell's equations with physical parameters, namely the electric permittivity and the magnetic permeability, that are complex, possibly non-Hermitian, tensor fields. Both tensor fields verify a general ellipticity condition. In this work, the well-posedness of formulations for the Dirichlet and Neumann problems (i.e. with a boundary condition on the electric field or its curl, respectively) is proven using well-suited functional spaces and Helmholtz decompositions. For both problems, the a priori regularity of the solution and the solution's curl is analysed. The regularity results are obtained by splitting the fields and using shift theorems for second-order divergence elliptic operators. Finally, the discretization of the formulations with a H(curl)-conforming approximation based on edge finite elements is considered. An a priori error estimate is derived and verified thanks to numerical results with an elementary benchmark.

Superconvergent recovery of edge finite element approximation for Maxwell’s equations

A new recovery method of edge elements for Maxwell’s equations is proposed to reduce the recovery problem into one dimensional settings by using the local symmetry projection. The recovered method is applied to both Nédélec interpolation and edge finite element approximation. Superconvergence results are established for both the postprocessed Nédélec interpolation and the recovered edge finite element approximation in a discrete norm. Numerical examples are presented to illustrate our theoretical analysis.

MODELING OF ELECTROSTATIC PROBLEM VIA A SUBPROBLEM FINITE ELEMENT METHOD

This work deals with the modeling of electrostatic problem by means of a subproblem finite element method. The aim of this paper is to present naturally the distributions of the electric scalar potential and the electric field in regions existing the various voltages or difference of voltages. The method permits to simultaneously use both node and edge elements in function spaces of the studied problem. Indeed, the distributed fields (e.g. electric scalar potential and electric field) are presented in the same function space and a week formulation written for both nodal and edge finite elements.

GEMM3D: An Edge Finite Element program for 3D modeling of electromagnetic fields and sensitivities for geophysical applications

This paper presents the 3D modeling program GEMM3D (Geophysical ElectroMagnetic Modeling 3D), a Fortran program for modeling frequency domain electromagnetic responses of geophysical sources in three-dimensional models. The program is an implementation of the Edge Finite Element method, in an electric field formulation, to simulate any electromagnetic source of interest, at all frequencies that are used in the geophysical methods in the quasi-static regime. Its current version includes routines for plane waves and simulation of magnetotelluric data; electrical and magnetic dipoles; circular loops. Other sources and configurations may easily be included with the addition of routines for simulating primary (layered earth) fields. The program includes routines for calculating sensitivities using the Adjoint State method, which makes efficient use of direct solvers for factoring once and solving for multiple systems with the same coefficient matrix. The paper details the methods, the implementation, and the usage of the program. Validation tests are presented for different sources. The program is freely available for use in research and teaching.

Convergence analysis of adaptive edge finite element method for variable coefficient time-harmonic Maxwell’s equations

In this paper, our main goal is to study the convergence analysis of adaptive edge finite element method (AEFEM) based on arbitrary order Nédélec edge elements for the variable-coefficient time-harmonic Maxwell’s equations, i.e., we prove that the AEFEM gives a contraction for the sum of the energy error and the error estimator, between two consecutive adaptive loops provided the initial mesh is fine enough. First, we give the variational problem of the variable-coefficient time-harmonic Maxwell’s equations and the posteriori error estimator of the residual type. Then we establish the quasi-orthogonality, the global upper bound of the error, the compressibility of the error estimator, and prove the convergence result. Finally, our numerical results verify that the error estimator is valid.

Using edge elements for modeling of 3-D magnetodynamic problem via a subproblem method

Introduction: The mathematical modeling of electromagnetic problems in electrical devices are often presented by Maxwell’s equations and constitutive material laws. These equations are partial differential equations linked to fields and their sources. In order to solve these equations and simulate the distribution of magnetic fields and eddy current losses of electromagnetic problems, a subproblem method for modeling a 3-D magnetodymic problem with the b-conformal formulation is proposed.
 Methods: In this paper, the subproblem method with using edge finite elements is proposed for coupling subproblems via several steps to treat and deal with some troubles regarding to electromagnetic problems that gets quite difficulties when directly applying a finite element method. In the strategy subproblem method, it allows a complete problem to define into several subproblems with adapted dimensions. Each subproblem can be solved on its independent domain and mesh without performing in whole domain or mesh. This easily supports meshing and decreases computing time.
 Results: The obtained results, the subproblem method with edge elements indicates magnetic flux densities and the eddy current losses in the conducting region. The computed results is also compared with the measured results done by other authors. This can be shown that there is a very good agreement.
 Conclusion: The validated method has been successfully applied to a practical test problem (TEAM Problem 7).

Convergence of an AEFEM for time-harmonic Maxwell equations with variable coefficients

An Adaptive Edge Finite Element Method (AEFEM) for three-dimensional indefinite time-harmonic Maxwell equations with variable coefficients is proved to be convergent and arbitrary order Nédélec edge elements are considered. A posteriori upper bound, quasi-orthogonality and the contraction of the error estimator are provided to prove the contraction of the sum of the energy error and the scaled error estimator between two consecutive adaptive loops. Numerical experiments are presented to support the theoretical results.

On the Approximation of Electromagnetic Fields by Edge Finite Elements. Part 3: Sensitivity to Coefficients

In bounded domains, the regularity of the solutions to boundary value problems depends on the gometry, and on the coefficients that enter into the definition of the model. This is in particular the case for the time-harmonic Maxwell equations, whose solutions are the electromagnetic fields. In this paper, emphasis is put on the electric field. We study the regularity in terms of the fractional order Sobolev spaces $H^s$, $s\in[0,1]$. Precisely, our first goal is to determine the regularity of the electric field and of its curl, that is to find some regularity exponent $\tau\in(0,1)$, such that they both belong to $H^s$, for all $s\in[0,\tau)$. After that, one can derive error estimates. Here, the error is defined as the difference between the exact field and its approximation, where the latter is built with N\'ed\'elec's first family of finite elements. In addition to the regularity exponent, one needs to derive a stability constant that relates the norm of the error to the norm of the data: this is our second goal. We provide explicit expressions for both the regularity exponent and the stability constant with respect to the coefficients. We also discuss the accuracy of these expressions, and we provide some numerical illustrations.

An adaptive edge finite element method for the Maxwell's equations in metamaterials

<p style='text-indent:20px;'>In this paper, we study an adaptive edge finite element method for time-harmonic Maxwell's equations in metamaterials. <i>A-posteriori</i> error estimators based on the recovery type and residual type are proposed, respectively. Based on our <i>a-posteriori</i> error estimators, the adaptive edge finite element method is designed and applied to simulate the backward wave propagation, electromagnetic splitter, rotator, concentrator and cloak devices. Numerical examples are presented to illustrate the reliability and efficiency of the proposed <i>a-posteriori</i> error estimations for the adaptive method.</p>

Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations

This paper concerns the numerical resolution of a data completion problem for the time-harmonic Maxwell equations in the electric field. The aim is to recover the missing data on the inaccessible part of the boundary of a bounded domain from measured data on the accessible part. The non-iterative quasi-reversibility method is studied and different mixed variational formulations are proposed. Well-posedness, convergence and regularity results are proved. Discretization is performed by means of edge finite elements. Various two- and three-dimensional numerical simulations attest the efficiency of the method, in particular for noisy data.

Superconvergent Recovery of Rectangular Edge Finite Element Approximation by Local Symmetry Projection

A new recovery method of rectangular edge finite element approximation for Maxwell’s equations is proposed by using the local symmetry projection. The recovery method is applied to the Nedelec interpolation to obtain the superconvergence of postprocessed Nedelec interpolation. Combining with the superclose result between the Nedelec interpolation and edge finite element approximation, it is shown that the postprocessed edge finite element solution superconverges to the exact solution. Numerical examples are presented to illustrate our theoretical analysis.

Scalable solvers for complex electromagnetics problems

In this work, we present scalable balancing domain decomposition by constraints methods for linear systems arising from arbitrary order edge finite element discretizations of multi-material and heterogeneous 3D problems. In order to enforce the continuity across subdomains of the method, we use a partition of the interface objects (edges and faces) into sub-objects determined by the variation of the physical coefficients of the problem. For multi-material problems, a constant coefficient condition is enough to define this sub-partition of the objects. For arbitrarily heterogeneous problems, a relaxed version of the method is defined, where we only require that the maximal contrast of the physical coefficient in each object is smaller than a predefined threshold. Besides, the addition of perturbation terms to the preconditioner is empirically shown to be effective in order to deal with the case where the two coefficients of the model problem jump simultaneously across the interface. The new method, in contrast to existing approaches for problems in curl-conforming spaces does not require spectral information whilst providing robustness with regard to coefficient jumps and heterogeneous materials. A detailed set of numerical experiments, which includes the application of the preconditioner to 3D realistic cases, shows excellent weak scalability properties of the implementation of the proposed algorithms.

On a general implementation of h- and p-adaptive curl-conforming finite elements

Edge (or Nédélec) finite elements are theoretically sound and widely used by the computational electromagnetics community. However, its implementation, especially for high order methods, is not trivial, since it involves many technicalities that are not properly described in the literature. To fill this gap, we provide a comprehensive description of a general implementation of edge elements of first kind within the scientific software project FEMPAR . We cover into detail how to implement arbitrary order (i.e., p-adaptive) elements on hexahedral and tetrahedral meshes. First, we set the three classical ingredients of the finite element definition by Ciarlet, both in the reference and the physical space: cell topologies, polynomial spaces and moments. With these ingredients, shape functions are automatically implemented by defining a judiciously chosen polynomial pre-basis that spans the local finite element space combined with a change of basis to automatically obtain a canonical basis with respect to the moments at hand. Next, we discuss global finite element spaces putting emphasis on the construction of global shape functions through oriented meshes, appropriate geometrical mappings, and equivalence classes of moments, in order to preserve the inter-element continuity of tangential components of the magnetic field. Finally, we extend the proposed methodology to generate global curl-conforming spaces on non-conforming hierarchically refined (i.e., h-adaptive) meshes with arbitrary order finite elements. Numerical results include experimental convergence rates to test the proposed implementation.

Adaptive Finite Element Method for the Maxwell Eigenvalue Problem

In this paper we prove the optimal convergence of a standard adaptive scheme based on edge finite elements for the approximation of the solutions of the eigenvalue problem associated with Maxwell's...

PETGEM: A parallel code for 3D CSEM forward modeling using edge finite elements

We present the capabilities and results of the Parallel Edge-based Tool for Geophysical Electromagnetic modeling (PETGEM), as well as the physical and numerical foundations upon which it has been developed. PETGEM is an open-source and distributed parallel Python code for fast and highly accurate modeling of 3D marine controlled-source electromagnetic (3D CSEM) problems. We employ the Nédélec Edge Finite Element Method (EFEM) which offers a good trade-off between accuracy and number of degrees of freedom, while naturally supporting unstructured tetrahedral meshes. We have particularised this new modeling tool to the 3D CSEM problem for infinitesimal point dipoles asumming arbitrarily isotropic media for low-frequencies approximations. In order to avoid source-singularities, PETGEM solves the frequency-domain Maxwell's equations of the secondary electric field, and the primary electric field is calculated analytically for homogeneous background media. We assess the PETGEM accuracy using classical tests with known analytical solutions as well as recent published data of real life geological scenarios. This assessment proves that this new modeling tool reproduces expected accurate solutions in the former tests, and its flexibility on realistic 3D electromagnetic problems. Furthermore, an automatic mesh adaptation strategy for a given frequency and specific source position is presented. We also include a scalability study based on fundamental metrics for high-performance computing (HPC) architectures.

Three-dimensional Land FD-CSEM Regularized Inversion Based on Edge Finite-element Method

There is a desire to obtain rapid and stable inversion results and clearly reconstruct subsurface resistivity structure in frequency domain (FD) electromagnetics. Three-dimensional modeling of land FD controlled-source electromagnetic (CSEM) data is vital to improve the understanding of electromagnetic responses collected in increasingly complex geologic settings. Three-dimensional inversion of land FD-CSEM data is a mathematically non-unique problem with instability, due to the noise contained in the data and its inherent incompleteness. The main difference between our method and those from previous work is that the edge finite-element approach is applied to solve the three-dimensional FD-CSEM generated by a horizontal electric dipole source. Firstly, we formulate the edge finite-element equation through the Galerkin method, based on the Helmholtz equation of the electric fields. Secondly, in order to check the validity of the modeling code, we compare the numerical results with the analytical solutions for a homogeneous half-space model. For further tests, we calculate the electromagnetic responses for another two models with more practical structures. Finally, the three-dimensional inversion is carried out based on a regularization method with smoothness-constraints to obtain stable solutions.

Fast Fourier transform-based solution of 2D and 3D magnetization problems in type-II superconductivity

We consider the fast Fourier transform (FFT) based numerical method for thin film magnetization problems (Vestgården and Johansen 2012 Supercond. Sci. Technol. 25 104001), compare it with the finite element methods, and evaluate its accuracy. Proposed modifications of this method implementation ensure stable convergence of iterations and enhance its efficiency. A new method, also based on the FFT, is developed for 3D bulk magnetization problems. This method is based on a magnetic field formulation, different from the popular h-formulation of eddy current problems typically employed with the edge finite elements. The method is simple, easy to implement, and can be used with a general current–voltage relation; its efficiency is illustrated by numerical simulations.

Lowest order Virtual Element approximation of magnetostatic problems

We give here a simplified presentation of the lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic field H on each edge, and the vertex values of the Lagrange multiplier p (used to enforce the solenoidality of the magnetic induction B=μH). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called “first kind Nédélec” elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions.

Three-dimensional land FD-CSEM forward modeling using edge finite-element method

A modeling tool for simulating three-dimensional land frequency-domain controlled-source electromagnetic surveys, based on a finite-element discretization of the Helmholtz equation for the electric fields, has been developed. The main difference between our modeling method and those previous works is edge finite-element approach applied to solving the three-dimensional land frequency-domain electromagnetic responses generated by horizontal electric dipole source. Firstly, the edge finite-element equation is formulated through the Galerkin method based on Helmholtz equation of the electric fields. Secondly, in order to check the validity of the modeling code, the numerical results are compared with the analytical solutions for a homogeneous half-space model. Finally, other three models are simulated with three-dimensional electromagnetic responses. The results indicate that the method can be applied for solving three-dimensional electromagnetic responses. The algorithm has been demonstrated, which can be effective to modeling the complex geo-electrical structures. This efficient algorithm will help to study the distribution laws of 3-D land frequency-domain controlled-source electromagnetic responses and to setup basis for research of three-dimensional inversion.