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This contribution covers the variational‐based modeling of non‐dissipative magneto‐mechanical systems using a vector potential approach and the thorough analysis and discussion of corresponding conforming finite element methods. Since the construction of divergence‐free finite element spaces explicitly enforcing the Coulomb gauge poses some major challenges, we propose some primal and mixed variational principles that ensure well posedness of the problem and allow to seek the vector potential in unconstrained function spaces. The performance of these methods is assessed in two comparative benchmark studies. The focus of both studies lies on the accurate approximation of field quantities in systems with material discontinuities and re‐entrant corners. [ABSTRACT FROM AUTHOR]

Efficient and accurate computation of H(curl) interface problems is of great importance in many electromagnetic applications. Unfitted mesh methods are especially attractive in three-dimensional (3D) computation as they can circumvent generating complex 3D interface-fitted meshes. However, many unfitted mesh methods rely on nonconforming approximation spaces, which may cause a loss of accuracy for solving Maxwell-type equations, and the widely used penalty techniques in the literature may not help in recovering the optimal convergence. In this article, we provide a remedy by developing Nédélec-type immersed finite element (IFE) spaces with a Petrov--Galerkin scheme that is able to produce optimal-convergent solutions. To establish a systematic framework, we analyze all the H¹, H(curl), and H(div) IFE spaces and form a discrete de Rham complex. Based on these fundamental results, we further develop a fast solver using a modified Hiptmair--Xu preconditioner which works for both the generalized minimal residual (GMRES) and conjugate gradient (CG) methods for solving the nonsymmetric linear algebraic system. The approximation capabilities of the proposed IFE spaces will be also established. [ABSTRACT FROM AUTHOR]

In this paper, a generalized parametric iterative method (GPIM) is proposed to solve the nonlinear discrete formulation of the magnetohydrodynamic (MHD) equations. We exploit the idea of decoupling to solve the strongly coupled MHD system, expecting to save storage space and to accelerate the convergence. The noticeable advantage of the proposed algorithm is that it is not necessary to solve the saddle point system for each iteration step and converge geometrically with the contract number. Numerical experiments are given to verify the validity and correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is the second of two articles, in which we aim to prove the convergence of the Hiptmair--Xu (HX) preconditioner (originally proposed by [R. Hiptmair and J. Xu, SIAM J. Numer. Anal., 45 (2007), pp. 2483--2509]) for H(curl)-elliptic boundary value problems with jump coefficients. In this paper, based on the auxiliary results obtained in our first article [Q. Hu, SIAM J. Numer. Anal., 59 (2021), pp. 2500--2535], we establish two new regular decompositions for lowdimensional edge finite element functions in three dimensions, which, under suitable assumptions on the distribution of the coefficients, are stable up to a polylogarithmic factor of the meshwidth with respect to weighted norms defined by the coefficients. Using these regular decompositions, we analyze the convergence of the HX preconditioner for the case of strongly discontinuous coefficients. We show that the HX preconditioner is asymptotically optimal up to a polylogarithmic factor in the meshwidth and is not severely affected by large jumps of the coefficients across the interface between two neighboring subdomains. [ABSTRACT FROM AUTHOR]

Finite element methods developed for unfitted meshes have been widely applied to various interface problems. However, many of them resort to non-conforming spaces for approximation, which is a critical obstacle for the extension to $\textbf{H}(\text{curl})$ equations. This essential issue stems from the underlying Sobolev space $\textbf{H}^s(\text{curl};\,\Omega)$ , and even the widely used penalty methodology may not yield the optimal convergence rate. One promising approach to circumvent this issue is to use a conforming test function space, which motivates us to develop a Petrov–Galerkin immersed finite element (PG-IFE) method for $\textbf{H}(\text{curl})$ -elliptic interface problems. We establish the Nédélec-type IFE spaces and develop some important properties including their edge degrees of freedom, an exact sequence relating to the $H^1$ IFE space and optimal approximation capabilities. We analyse the inf-sup condition under certain assumptions and show the optimal convergence rate, which is also validated by numerical experiments. [ABSTRACT FROM AUTHOR]

Modeling the unusual mechanical properties of metamaterials is a challenging topic for the mechanics community and enriched continuum theories are promising computational tools for such materials. The so-called relaxed micromorphic model has shown many advantages in this field. In this contribution, we present significant aspects related to the relaxed micromorphic model realization with the finite element method (FEM). The variational problem is derived and different FEM-formulations for the two-dimensional case are presented. These are a nodal standard formulation H 1 (B) × H 1 (B) and a nodal-edge formulation H 1 (B) × H (curl , B) , where the latter employs the Nédélec space. In this framework, the implementation of higher-order Nédélec elements is not trivial and requires some technicalities which are demonstrated. We discuss the computational convergence behavior of Lagrange-type and tangential-conforming finite element discretizations. Moreover, we analyze the characteristic length effect on the different components of the model and reveal how the size-effect property is captured via this characteristic length parameter. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze a new mixed finite element method for a stationary magneto-hydrodynamic (MHD) model. The method is based on the utilization of a new dual-mixed formulation recently introduced for the Navier-Stokes problem, which is coupled with a classical primal formulation for the Maxwell equations. The latter implies that the velocity and a pseudostress tensor relating the velocity gradient with the convective term for the hydrodynamic equations, together with the magnetic field and a Lagrange multiplier related with the divergence-free property of the magnetic field, become the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the aforementioned pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k for the associated Lagrange multiplier. The analysis of the continuous and discrete problems are carried out by means of the Lax–Milgram lemma, the Banach–Nečas–Babuška and Banach fixed-point theorems, under a sufficiently small data assumption. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. We also develop an a priori error analysis and show that the proposed finite element method is optimal convergent. Finally, some numerical results illustrating the good performance of the method are provided. [ABSTRACT FROM AUTHOR]

In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces of [ H 1 ] 3 × 3 , [ H (curl) ] 3 and H (sym Curl) for a linear abstract variational problem connected to the relaxed micromorphic model. The formulation of the corresponding finite elements is introduced, followed by numerical benchmarks and our conclusions. The relaxed micromorphic continuum model reduces the continuity assumptions of the classical micromorphic model by replacing the full gradient of the microdistortion in the free energy functional with the Curl. This results in a larger solution space for the microdistortion, namely [ H (curl) ] 3 in place of the classical [ H 1 ] 3 × 3 . The continuity conditions on the microdistortion can be further weakened by taking only the symmetric part of the Curl. As shown in recent works, the new appropriate space for the microdistortion is then H (sym Curl) . The newly introduced space gives rise to a new differential complex for the relaxed micromorphic continuum theory. [ABSTRACT FROM AUTHOR]

The Hilbert spaces H (curl) and H (div) are employed in various variational problems formulated in the context of the de Rham complex in order to guarantee well-posedness. Seeing as the well-posedness follows automatically from the continuous setting to the discrete setting in the presence of commuting interpolants as per Fortin's criterion, the construction of conforming subspaces becomes a crucial step in the formulation of stable numerical schemes. This work aims to introduce a novel, simple method of directly constructing semi-continuous vectorial base functions on the reference element via template vectors associated with the geometric polytopes of the element and an underlying H 1 -conforming polynomial subspace. The base functions are then mapped from the reference element to the element in the physical domain via consistent Piola transformations. The method is defined in such a way, that the underlying H 1 -conforming subspace can be chosen independently, thus allowing for constructions of arbitrary polynomial order. We prove a linearly independent construction of Nédélec elements of the first and second type, Brezzi–Douglas–Marini elements, and Raviart–Thomas elements on triangulations and tetrahedralizations. The application of the method is demonstrated with two examples in the relaxed micromorphic model. • A new method of constructing semi-continuous vectorial finite elements via polytopes. • Proofs of linear independence and conformity for hierarchical- and partition of unity-bases. • Templates for Nédélec elements of the first and second type on simplices. • Templates for Brezzi–Douglas–Marini and Raviart–Thomas elements on simplices. • Split in the basis between kernel and non-kernel base functions on triangles. [ABSTRACT FROM AUTHOR]

A mixed finite element method is proposed for the Maxwell eigenproblem under the general setting. The method is based on a modification of the Kikuchi mixed formulation in terms of the electric field and the multiplier, with a mesh-dependent Gauss law of the electric field enforced in the formulation. The electric field is discretized by discontinuous elements, and the multiplier is always discretized by the lowest-order continuous nodal element (e.g., the linear element). The method renders four key features: the discrete de Rham complex exact sequence is not required and is replaced by a gradient inclusion condition of a low-order scalar element; i.e., the finite element space of the electric field includes the gradient of an auxiliary scalar H¹-conforming finite element space of low order; the discrete compactness property holds; the strong convergence of the Gauss law is ensured globally for the finite element solution; the method converges nearly optimally for both singular and smooth solutions. With these features, we develop a general analysis to prove that whether or not the discrete eigenmodes are spurious-free and spectral-correct attributes essentially to the first-order approximation property in the H(curl ; Ω) norm. As a direct application, except three lowest-order elements that do not have the first-order approximation property on nonaffine meshes, the first-kind Nédélec elements on nonaffine quadrilateral and hexahedral meshes and the second-kind Nédélec elements on affine and nonaffine quadrilateral and hexahedral meshes, including their discontinuous versions, are spurious-free and spectral-correct in the new mixed method, while these Nédélec elements generate spurious and incorrect discrete eigenmodes in the classical methods. [ABSTRACT FROM AUTHOR]

This paper is the first in a series of two articles, aiming to prove the convergence of the HX preconditioner originally proposed by Hiptmair and Xu [SIAM J. Numer. Anal., 45 (2007), pp. 2483-2509] for Maxwell's equations with jump coefficients. In this paper, we establish several extensions of the discrete regular decomposition for edge finite element functions defined in threedimensional domains. The functions defined by the new discrete regular decompositions can inherit zero degrees of freedom of the considered edge finite element function on some faces and edges of polyhedral domains as well as of some non-Lipschitz domains and possess nearly optimal stability with only a logarithmic factor. [ABSTRACT FROM AUTHOR]

One approach for the simulation of metamaterials is to extend an associated continuum theory concerning its kinematic equations, and the relaxed micromorphic continuum represents such a model. It incorporates the Curl of the nonsymmetric microdistortion in the free energy function. This suggests the existence of solutions not belonging to H 1 , such that standard nodal H 1 -finite elements yield unsatisfactory convergence rates and might be incapable of finding the exact solution. Our approach is to use base functions stemming from both Hilbert spaces H 1 and H (curl) , demonstrating the central role of such combinations for this class of problems. For simplicity, a reduced two-dimensional relaxed micromorphic continuum describing antiplane shear is introduced, preserving the main computational traits of the three-dimensional version. This model is then used for the formulation and a multi step investigation of a viable finite element solution, encompassing examinations of existence and uniqueness of both standard and mixed formulations and their respective convergence rates. [ABSTRACT FROM AUTHOR]

In this work, a flexible discretization technique for the approximate solution of electromagnetic field problems with rotating parts, based on non-conforming (NC) interfaces of Nitsche-type, is investigated. This approach enables the use of edge elements of the first and second kind for the discretization of the magnetic vector potential without posing severe restrictions on gaps and overlaps of elements on both sides of the rotating interface. It improves the computational efficiency, compared to other NC interface formulations because no additional unknowns are introduced, and edge elements of different polynomial order can be coupled with this approach. The applicability is shown in two numerical examples, involving a cylindrical NC interface between a stationary and a rotating domain. [ABSTRACT FROM AUTHOR]

In this article, a methodology to incorporate non-conforming interfaces between several conforming mesh regions is presented for Maxwell’s curl–curl problem. The derivation starts from a general interior penalty discontinuous Galerkin formulation of the curl–curl problem and eliminates all interior jumps in the conforming parts but retains them across non-conforming interfaces. Therefore, it is possible to think of this Nitsche approach for interfaces as a specialization of discontinuous Galerkin on meshes, which are conforming nearly everywhere. The applicability of this approach is demonstrated in two numerical examples, including parameter jumps at the interface. A convergence study is performed for h-refinement, including the investigation of the penalization- (Nitsche-) parameter. [ABSTRACT FROM AUTHOR]

The relaxed micromorphic model is a generalized continuum model that is well-posed in the space X = [ H 1 ] 3 × [ H (curl) ] 3 . Consequently, finite element formulations of the model rely on H 1 -conforming subspaces and Nédélec elements for discrete solutions of the corresponding variational problem. This work applies the recently introduced polytopal template methodology for the construction of Nédélec elements. This is done in conjunction with Bernstein–Bézier polynomials and dual numbers in order to compute hp-FEM solutions of the model. Bernstein–Bézier polynomials allow for optimal complexity in the assembly procedure due to their natural factorization into univariate Bernstein base functions. In this work, this characteristic is further augmented by the use of dual numbers in order to compute their values and their derivatives simultaneously. The application of the polytopal template methodology for the construction of the Nédélec base functions allows them to directly inherit the optimal complexity of the underlying Bernstein–Bézier basis. We introduce the Bernstein–Bézier basis along with its factorization to univariate Bernstein base functions, the principle of automatic differentiation via dual numbers and a detailed construction of Nédélec elements based on Bernstein–Bézier polynomials with the polytopal template methodology. This is complemented with a corresponding technique to embed Dirichlet boundary conditions, with emphasis on the consistent coupling condition. The performance of the elements is shown in examples of the relaxed micromorphic model. • Kinematical reduction of the relaxed micromorphic model to antiplane shear. • Bernstein–Bézier polynomial basis and its factorization via the Duffy transformation. • Forward automatic differentiation via dual numbers for enhanced optimal complexity. • Higher order Nédélec elements based on Bernstein polynomials via polytopal templates. • Discrete consistent coupling condition for the relaxed micromorphic model. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze three interior penalty DG methods for the stationary incompressible magnetohydrodynamics (MHD) equations. We use div -conforming Brezzi–Douglas–Marini (BDM) elements to discretize the velocity field and therefore the approximation of the velocity field is exactly divergence-free. The magnetic field is approximated by curl -conforming Nédélec elements. In the mixed formulation, the zero mean-valued constraint of the pressure is enforced via a Lagrange multiplier technique. The stability and convergence of the proposed methods are proved in a general Lipschitz domain. Finally, a series of numerical experiments are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The classical Cauchy continuum theory is suitable to model highly homogeneous materials. However, many materials, such as porous media or metamaterials, exhibit a pronounced microstructure. As a result, the classical continuum theory cannot capture their mechanical behaviour without fully resolving the underlying microstructure. In terms of finite element computations, this can be done by modelling the entire body, including every interior cell. The relaxed micromorphic continuum offers an alternative method by instead enriching the kinematics of the mathematical model. The theory introduces a microdistortion field, encompassing nine extra degrees of freedom for each material point. The corresponding elastic energy functional contains the gradient of the displacement field, the microdistortion field and its Curl (the micro-dislocation). Therefore, the natural spaces of the fields are [ H 1 ] 3 for the displacement and [ H (curl) ] 3 for the microdistortion, leading to unusual finite element formulations. In this work we describe the construction of appropriate finite elements using Nédélec and Raviart–Thomas subspaces, encompassing solutions to the orientation problem and the discrete consistent coupling condition. Further, we explore the numerical behaviour of the relaxed micromorphic model for both a primal and a mixed formulation. The focus of our benchmarks lies in the influence of the characteristic length L c and the correlation to the classical Cauchy continuum theory. • Existence and uniqueness results for primal and mixed formulations of the relaxed micromorphic model. • A priori convergence estimates. • A detailed description of the construction of the appropriate finite elements on tetrahedra. • An intrinsic solution to the orientation problem of vectorial elements. • A specialized scheme for an exact discrete consistent coupling condition. [ABSTRACT FROM AUTHOR]

We examine the construction of p-hierarchical local a posteriori error estimators for time-harmonic electromagnetic problems using edge-based finite elements and boundary elements for hexahedral and tetrahedral meshes in ℝ3. The error estimators rely on stable subspace decompositions of Nédélec elements in H(curl, Ω) and Raviart–Thomas elements in . [ABSTRACT FROM AUTHOR]

Abstract: In this note, we provide a conveniently implementable basis for simplicial Nédélec spaces of any order in any space dimension. The main feature of the basis is that it is expressed solely in terms of the barycentric coordinates of the simplex. [Copyright &y& Elsevier]

A main issue in nonstationary, compressible magnetohydrodynamic (MHD) simulations is controlling the divergence of the magnetic flux. This paper presents a general procedure showing how to modify the intercell fluxes in a conservative MHD finite volume code such that the scheme becomes locally divergence preserving. That is, a certain discrete divergence operator vanishes exactly during the entire simulation, which results in the suppression of any divergence error. The procedure applies to arbitrary finite volume schemes provided they are based on intercell fluxes. We deduce the necessary modifications for numerical methods based on rectangles and triangles and present numerical experiments with the new schemes. The theoretical justifition of the schemes is given in two independent ways. One way starts with the discrete divergence operator that has to be preserved and modifies the fluxes accordingly. The second way uses a finite element reconstruction via Nedelec elements. Both methods lead to equivalent numerical methods. [ABSTRACT FROM AUTHOR]