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The purpose of this paper is to study algebraic structures that underlie the geometric approaches. The structures and their properties are analyzed to address how to systematically pose a class of boundary value problems in a pair of interlocked complexes.

The work utilizes concepts of algebraic topology to have a solid framework for the analysis. The algebraic structures constitute a set of requirements and guidelines that are adhered to in the analysis.

A precise notion of “relative dual complex”, and certain necessary requirements for discrete Hodge‐operators are found.

The paper includes a set of prerequisites, especially for discrete Hodge‐operators. The prerequisites aid, for example, in verifying new computational methods and algorithms.

The paper gives an overall view of the algebraic structures and their role in the geometric approaches. The paper establishes a set of prerequisites that are inherent in the geometric approaches.

To describe and extend existing concepts of discrete electromagnetism in a unified formalism; to give examples for the usefulness of the presented ideas for our theoretical work, especially with regard to energy.

After a concise introduction to the mathematical concepts of discrete electromagnetism, we introduce continuous de Rham currents and give their discrete counterpart. We define operators acting upon discrete currents, and apply the theory to electromagnetism.

de Rham current theory yields a mathematical framework for the discussion of discrete electromagnetic problems: The focus is on energy‐balance equations; a discrete Lagrangian can be defined for various modeling problems; the Galerkin approach fits nicely into the proposed formalism; boundary terms in discrete formulations are an implicit feature to the theory.

In this paper, we use the interpolation of discrete fields by Whitney forms on a simplicial cell complex. The resulting discrete formulation is identical to a Galerkin finite‐element method. Other numerical techniques that do not resort to Whitney‐form interpolation can equally be discussed in de Rham‐current terminology.

Rather than a novel numerical technique, the paper presents a unified mathematical framework for the discussion of different practical approaches. We advocate a canonical treatment of energy‐related quantities and of boundary terms in discrete formulations.

The purpose of this paper is to show the applicability of a discrete Hodge operator in the context of the De Rham cohomology to bicomplex-valued electromagnetic wave propagation problems. It was applied in the finite element method (FEM) to get a higher accuracy through conformal discretization. Therewith, merely the primal mesh is needed to discretize the full system of Maxwell equations.

At the beginning, the theoretical background is presented. The bicomplex number system is used as a geometrical algebra to describe three-dimensional electromagnetic problems. Because we treat rotational field problems, Whitney edge elements are chosen in the FEM to realize a conformal discretization. Next, numerical simulations regarding practical wave propagation problems are performed and compared with the common FEM approach using the Helmholtz equation.

Different field problems of three-dimensional electromagnetic wave propagation are treated to present the merits and shortcomings of the method, which calculates the electric and magnetic field at the same spatial location on a primal mesh. A significant improvement in accuracy is achieved, whereas fewer essential boundary conditions are necessary. Furthermore, no numerical dispersion is observed.

A novel Hodge operator, which acts on bicomplex-valued cotangential spaces, is constructed and discretized as an edge-based finite element matrix. The interpretation of the proposed geometrical algebra in the language of the De Rham cohomology leads to a more comprehensive viewpoint than the classical treatment in FEM. The presented paper may motivate researchers to interpret the form of number system as a degree of freedom when modeling physical effects. Several relationships between physical quantities might be inherently implemented in such an algebra.

To introduce a Whitney‐element based coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM); to discuss the algebraic properties of the resulting system and propose solver strategies.

The FEM is interpreted in the framework of the theory of discrete electromagnetism (DEM). The BEM formulation is given in a DEM‐compatible notation. This allows for a physical interpretation of the algebraic properties of the resulting BEM‐FEM system matrix. To these ends we give a concise introduction to the mathematical concepts of DEM.

Although the BEM‐FEM system matrix is not symmetric, its kernel is equivalent to the kernel of its transpose. This surprising finding allows for the use of two solution techniques: regularization or an adapted GMRES solver.

The programming of the proposed techniques is a work in progress. The numerical results to support the presented theory are limited to a small number of test cases.

The paper will help to improve the understanding of the topological and geometrical implications in the algebraic structure of the BEM‐FEM coupling.

Several original concepts are presented: a new interpretation of the FEM boundary term leads to an intuitive understanding of the coupling of BEM and FEM. The adapted GMRES solver allows for an accurate solution of a singular, unsymetric system with a right‐hand side that is not in the image of the matrix. The issue of a grid‐transfer matrix is briefly mentioned.

In this paper, a hybrid formulation for solving time harmonic eddy current problems in terms of magnetic field h is considered. In particular, we discuss some properties of the implicit boundary condition on the discrete level and the computation of the integral operator exploited in this context. An iterative technique is confirmed to be efficient in solving the arising, partly dense, complex linear system of equations. Furthermore, some test results, including timings for linear solvers are presented.

The purpose of this paper is to review the mutual coupling of electromagnetic fields in the magnetic vector potential formulation with electric circuits in terms of (modified) nodal and loop analyses. It aims for an unified and generic notation.

The coupled formulation is derived rigorously using the concept of winding functions. Strong and weak coupling approaches are proposed and examples are given. Discretization methods of the partial differential equations and in particular the winding functions are discussed. Reasons for instabilities in the numerical time domain simulation of the coupled formulation are presented using results from differential-algebraic-index analysis.

This paper establishes a unified notation for different conductor models, e.g. solid, stranded and foil conductors and shows their structural equivalence. The structural information explains numerical instabilities in the case of current excitation.

The presentation of winding functions allows to generically describe the coupling, embed the circuit equations into the de Rham complex and visualize them by Tonti diagrams. This is of value for scientists interested in differential geometry and engineers that work in the field of numerical simulation of field-circuit coupled problems.

The purpose of this paper is to introduce a perturbation method for the A‐χ geometric formulation to solve eddy‐current problems and apply it to the feasibility design of a non‐destructive evaluation device suitable to detect long‐longitudinal volumetric flaws in hot steel bars.

The effect of the flaw is accurately and efficiently computed by solving an eddy‐current problem over an hexahedral grid which gives directly the perturbation due to the flaw with respect to the unperturbed configuration.

The perturbation method, reducing the cancelation error, produces accurate results also for small variations between the solutions obtained in the perturbed and unperturbed configurations. This is especially required when the tool is used as a forward solver for an inverse problem. The method yields also to a considerable speedup: the mesh used in the perturbed problem can in fact be reduced at a small fraction of the initial mesh, considering only a limited region surrounding the flaw in which the mesh can be refined. Moreover, the full three‐dimensional unperturbed problem does not need to be solved, since the source term for computing the perturbation is evaluated by solving a two‐dimensional flawless configuration having revolution symmetry.

A perturbation method for the A‐χ geometric formulation to solve eddy‐current problems has been introduced. The advantages of the perturbation method for non‐destructive testing applications have been described.

The purpose of this paper is to present a geometric approach to the problem of dimensional reduction. To derive (3 + 1) D formulations of 4D field problems in the relativistic theory of electromagnetism, as well as 2D formulations of 3D field problems with continuous symmetries.

The framework of differential‐form calculus on manifolds is used. The formalism can thus be applied in arbitrary dimension, and with Minkowskian or Euclidean metrics alike.

The splitting of operators leads to dimensionally reduced versions of Maxwell's equations and constitutive laws. In the metric‐incompatible case, the decomposition of the Hodge operator yields additional terms that can be treated like a magnetization and polarization of empty space. With this concept, the authors are able to solve Schiff's paradox without use of coordinates.

The present formalism can be used to generate concise formulations of complex field problems. The differential‐form formulation can be readily translated into the language of discrete fields and operators, and is thus amenable to numerical field calculation.

The approach is an evolution of recent work, striving for a generalization of different approaches, and deliberately avoiding a mix of paradigms.

The purpose of this paper is to describe and analyse a class of finite element fractional step methods for solving the incompressible Navier-Stokes equations. The objective is not to reproduce the extensive contributions on the subject, but to report on long-term experience with and provide a unified overview of a particular approach: the characteristic-based split method. Three procedures, the semi-implicit, quasi-implicit and fully explicit, are studied and compared.

This work provides a thorough assessment of the accuracy and efficiency of these schemes, both for a first and second order pressure split.

In transient problems, the quasi-implicit form significantly outperforms the fully explicit approach. The second order (pressure) fractional step method displays significant convergence and accuracy benefits when the quasi-implicit projection method is employed. The fully explicit method, utilising artificial compressibility and a pseudo time stepping procedure, requires no second order fractional split to achieve second order or higher accuracy. While the fully explicit form is efficient for steady state problems, due to its ability to handle local time stepping, the quasi-implicit is the best choice for transient flow calculations with time independent boundary conditions. The semi-implicit form, with its stability restrictions, is the least favoured of all the three forms for incompressible flow calculations.

A comprehensive comparison between three versions of the CBS method is provided for the first time.

The purpose of this paper is to develop Triple Finite Volume Method (tFVM), the author discretizes incompressible Navier-Stokes equation by tFVM, which leads to a special linear system of saddle point problem, and most computational efforts for solving the linear system are invested on the linear solver GMRES.

In this paper, by recently developed preconditioner Hermitian/Skew-Hermitian Separation (HSS) and the parallel implementation of GMRES, the author develops a quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

Computational results show that, the quick solver HSS-pGMRES-tFVM significantly increases the solution speed for saddle point problem from incompressible Navier-Stokes equation than the conventional solvers.

Altogether, the contribution of this paper is that the author developed the quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.