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We are concerned with a special class of discretizations of general linear transmission problems stated in the calculus of differential forms and posed on R n . In the spirit of domain decomposition, we partition R n = Ω ∪ Γ ∪ Ω + , Ω a bounded Lipschitz polyhedron, Γ : = ∂ Ω , and Ω + unbounded. In Ω , we employ a mesh-based discrete co-chain model for differential forms, which includes schemes like finite element exterior calculus and discrete exterior calculus. In Ω + , we rely on a meshless Trefftz-Galerkin approach, i.e., we use special solutions of the homogeneous PDE as trial and test functions. Our key contribution is a unified way to couple the different discretizations across Γ . Based on the theory of discrete Hodge operators, we derive the resulting linear system of equations. As a concrete application, we discuss an eddy-current problem in frequency domain, for which we also give numerical results., (© The Author(s) 2022.)

We couple the mixed variational problem for the generalized Hodge-Helmholtz or Hodge-Laplace equation posed on a bounded 3D Lipschitz domain with the first-kind boundary integral equations arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderón projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Gårding inequality (T-coercivity). The resulting system of equations describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a "rough" surface. The low-frequency robustness of the potential formulation of Maxwell's equations makes this model a promising starting point for Galerkin discretization., (© The Author(s) 2022.)

Geophysicists today face the challenge of quickly and reliably interpreting extensive controlled-source electromagnetic (CSEM) data sets to map subsurface conductivity structures within realistic geological environments. An ideal 3-D CSEM inversion algorithm using tetrahedral grids should be capable of distinguishing different resolution requirements between forward modelling and inversion grids, have an optimal parallel strategy that fully exploits the inherent independence of CSEM data sets while also possessing the capability to handle large-scale geo-electrical models, and incorporate conductivity anisotropy which should be a common characteristic in realistic subsurface environments. However, existing tools in the geo-electromagnetic community often fall short of these three demands. Addressing this gap, our study introduces a scalable and parallel anisotropic inversion technique for CSEM data, capitalizing on the potential of unstructured tetrahedral grids. We first apply the tetrahedral longest-edge bisection method to create a refined dense, heterogeneous forward modelling grid from a coarse inversion grid. This refinement, focused on areas around transmitters and receivers, is seamlessly integrated within the coarser inversion grid's topology, enabling precise conductivity mapping and preserving electromagnetic response accuracy during model updates. We further innovate with a source-mesh double-level parallel strategy, utilizing the message passing interface technique for parallel handling of independent CSEM data sets and large-scale geo-electrical models. Externally, we dedicate a processor for inversion model updates employing the Limited-memory Broyden–Fletcher–Goldfarb–Shanno optimization algorithm and divide other processors into groups, each associated with specific transmitting sources and frequencies. Internally, in each group, we employ a domain-decomposition-based scalable and robust iterative solvers using the Auxiliary-Space Maxwell pre-conditioner to parallel quickly calculate the electromagnetic responses from its assigned source-frequency set. Additionally, recognizing the potential for electrical conductivity anisotropy in field data, we incorporate the case of vertical transverse isotropy. We validate the effectiveness of our method through examples, including an isotropic land model with undulating topography, an anisotropic marine model and a real-field data case. Results from both synthetic and field data inversions underscore our method's significant advancements in efficiency and practicality, particularly in addressing large-scale 3-D CSEM data sets inversion challenges in realistic geological environments. [ABSTRACT FROM AUTHOR]

We propose to use a hybridizable discontinuous Galerkin (HDG) method combined with the continuous Galerkin (CG) method to approximate Maxwell's equations. We make two contributions in this paper. First, even though there are many papers using HDG methods to approximate Maxwell's equations, to our knowledge they all assume that the coefficients are smooth (or constant). Here, we derive optimal convergence estimates for our HDG-CG approximation when the electromagnetic coefficients are piecewise W 1 , ∞ . This requires new techniques of analysis. Second, we use CG elements to approximate the Lagrange multiplier used to enforce the divergence condition and we obtain a discrete system in which we can decouple the discrete Lagrange multiplier. Because we are using a continuous Lagrange multiplier space, the number of degrees of freedom devoted to this are less than for other HDG methods. We present numerical experiments to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

This paper aims to obtain solutions in terms of the complex potential structure for the plain strain problem of an elastic micropolar and isotropic body with pores. The constitutive equations on which the method is applied and is useful are the well-known equations of the elasticity theory for the above-mentioned body. We intend to solve the Kirch problem using the procedure of complex variables. [ABSTRACT FROM AUTHOR]

Nonlinear problems, which often arise in various scientific and engineering disciplines, typically involve nonlinear equations or functions with multiple solutions. Analytical solutions to these problems are often impossible to obtain, necessitating the use of numerical techniques. This research proposes an efficient and stable Caputo-type inverse numerical fractional scheme for simultaneously approximating all roots of nonlinear equations, with a convergence order of 2 ψ + 2 . The scheme is applied to various nonlinear problems, utilizing dynamical analysis to determine efficient initial values for a single root-finding Caputo-type fractional scheme, which is further employed in inverse fractional parallel schemes to accelerate convergence rates. Several sets of random initial vectors demonstrate the global convergence behavior of the proposed method. The newly developed scheme outperforms existing methods in terms of accuracy, consistency, validation, computational CPU time, residual error, and stability. [ABSTRACT FROM AUTHOR]

Objective. Recent studies have indicated that repetitive transcranial magnetic stimulation (rTMS) could enhance cognition in Alzheimer's Disease (AD) patients, but to now the molecular-level interaction mechanisms driving this effect remain poorly understood. While cognitive scores have been the primary measure of rTMS effectiveness, employing molecular-based approaches could offer more precise treatment predictions and prognoses. To reach this goal, it is fundamental to assess the electric field (E-field) and the induced current densities (J) within the stimulated brain areas and to translate these values to in vitro systems specifically devoted in investigating molecular-based interactions of this stimulation. Approach. This paper offers a methodological procedure to guide dosimetric assessment to translate the E-field induced in humans (in a specific pilot study) into in vitro settings. Electromagnetic simulations on patients' head models and cellular holders were conducted to characterize exposure conditions and determine necessary adjustments for in vitro replication of the same dose delivered in humans using the same stimulating coil. Main results. Our study highlighted the levels of E-field and J induced in the target brain region and showed that the computed E-field and J were different among patients that underwent the treatment, so to replicate the exposure to the in vitro system, we have to consider a range of electric quantities as reference. To match the E-field to the levels calculated in patients' brains, an increase of at least the 25% in the coil feeding current is necessary when in vitro stimulations are performed. Conversely, to equalize current densities, modifications in the cells culture medium conductivity have to be implemented reducing it to one fifth of its value. Significance. This dosimetric assessment and subsequent experimental adjustments are essential to achieve controlled in vitro experiments to better understand rTMS effects on AD cognition. Dosimetry is a fundamental step for comparing the cognitive effects with those obtained by stimulating a cellular model at an equal dose rigorously evaluated. [ABSTRACT FROM AUTHOR]

This paper aims at passive noise control for vibroacoustic problems, which are analyzed by finite and boundary element techniques. The author distinguishes interior and exterior problems mainly because of the quantities used as the objective function to assess the acoustic quality. For interior problems, it is common to use local quantities such as the sound pressure at a field point or, in rare cases, energy density at a field point. The situation is different for exterior problems where the radiated sound power accounts for a suitable and global quantity to assess the emission from a vibrating structure. For most engineering purposes, the assessment requires frequency sweeps in which the problem needs to be solved at many discrete frequencies. In vibroacoustic optimization and in sampling based uncertainty quantification, it is very common that structural parameters are varied, while the acoustic field remains the same throughout the entire process. We will review concepts and recent developments of efficient frequency sweeps and repeated analysis with unmodified fluid domain. For many practical cases, the situation for interior problems is rather simple to survey. Either the authors have applied a modal analysis and used a modal superposition for the frequency sweep and repeated analysis, or the concept of unmodified acoustic transfer vectors is applied. Both concepts are quite successful as long as certain conditions are fulfilled. For exterior problems, a modal superposition is possible but, so far, only for a limited number of cases practically applicable as discussed herein. The concept of using acoustic transfer vectors becomes inefficient since the evaluation of the radiated sound power as an integral over a closed enveloping surface would require an excessively high amount of storage capacity. Therefore, other concepts are being followed. For frequency sweeps, a number of methods are using a frequency interpolation based on a limited number of discrete sample frequencies. Often, these techniques are used together with Krylov-subspace model order reduction techniques. Further recently published approaches investigate low-rank approximations, greedy algorithms, and deflated Krylov subspace techniques. A completely different kind of approach is based on multi-fidelity models and Gaussian processes. The field of efficient repeated analysis shows some interesting developments, which can be easily applied to sampling based uncertainty quantification but do not seem to be easily and generally applied to optimization. [ABSTRACT FROM AUTHOR]

Considering positive H (curl , Ω) problems, we propose a substructured version of the Hiptmair-Xu preconditioner based on a new formula that expresses the inverse of Schur systems in terms of the inverse matrix of the global volume problem. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive review of research conducted on the Method of Auxiliary Sources (MAS) over a period of the last 22 years, i.e., since the last up-to-date survey was published. MAS is a very attractive numerical technique due to its simple algorithmic structure and the generally low computational cost it requires in terms of memory and CPU time; this is why it has been applied to a vast variety of cases, as concluded by the long citations list included. After a short introduction summarizing the fundamental concepts of the method, references since 2002 are categorized, briefly described, and commented on. This work is intended to assist every researcher who is involved in MAS computations, providing an exhaustive, to the best of the authors' knowledge, list of related publications. [ABSTRACT FROM AUTHOR]

In this article, we study general properties of distributed shape derivatives admitting a volumetric tensor representation of order two. We obtain a general result providing a range of expressions for the shape derivative, with the distributed shape derivative at one end of the range and the standard Hadamard formula at the other end. We further apply this result to a cost functional depending on the solution of a fourth-order elliptic equation, and obtain the distributed shape derivative in the case of open sets, and the Hadamard formula for sets of class C4. We also consider the case of polygons, for which a description of the weak singularities of the solution appearing in the neighbourhood of the vertices is required to obtain the Hadamard formula. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [ABSTRACT FROM AUTHOR]

We present new second-kind integral-equation formulations of the interior and exterior Dirichlet problems for Laplace's equation. The operators in these formulations are both continuous and coercive on general Lipschitz domains in R d , d ≥ 2 , in the space L 2 (Γ) , where Γ denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (1) the Galerkin method converges when applied to these formulations; and (2) the Galerkin matrices are well-conditioned as the discretisation is refined, without the need for operator preconditioning (and we prove a corresponding result about the convergence of GMRES). The main significance of these results is that it was recently proved (see Chandler-Wilde and Spence in Numer Math 150(2):299–371, 2022) that there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which the operators in the standard second-kind integral-equation formulations for Laplace's equation (involving the double-layer potential and its adjoint) cannot be written as the sum of a coercive operator and a compact operator in the space L 2 (Γ) . Therefore there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which Galerkin methods in L 2 (Γ) do not converge when applied to the standard second-kind formulations, but do converge for the new formulations. [ABSTRACT FROM AUTHOR]

We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension d. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber k with a number of degrees of freedom scaling as k d - 1 / 2 , which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as k d to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation. [ABSTRACT FROM AUTHOR]

We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The permutations of vertex indices correspond to the symmetry group of the simplex. That symmetry group is represented on simplicial finite element spaces by the pullback action. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djoković and Malzan's classification of monomial irreducible representations of the symmetric group and show new symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Explicit invariant bases with complex coefficients are constructed in dimensions two and three for different spaces of finite element differential forms. [ABSTRACT FROM AUTHOR]

One potential application of metamaterials is for designing invisibility cloaks. In this paper, we are interested in a rotation cloak model. Here we carry out the mathematical analysis of this model for the first time. Through a careful analysis, we reformulate a new system of governing partial differential equations by reducing one unknown variable from the originally developed modeling equations in Yang et al. (Commun Comput Phys 25:135–154, 2019). Then some novel finite element schemes are proposed and their stability and optimal error estimate are proved. Numerical simulations are presented to demonstrate that the new schemes for the reduced modeling equations can effectively reproduce the rotation cloaking phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyze a second order unconditionally convergent mixed finite element method for solving incompressible magnetohydrodynamic problem based on magnetic vector potential. We utilize the second-order backward difference formula within the framework of the mixed finite element method and linearize the nonlinear terms by using the extrapolated technique. The finite element approximation is proposed with the fluid equations are discretized by Taylor–Hood type elements and the magnetic vector potential equation by edge elements. As a result, the divergence-free condition on the magnetic induction preserves exactly on the discrete level. Unconditional error estimates for the velocity and magnetic vector potential are established in the sense that no any restrictions are imposed on the relationship between the time-step size and the spatial size. Finally, several three-dimensional numerical simulations are carried out to illustrate the developed scheme, including convergence tests and some benchmark problems, such as the driven cavity problems and hydromagnetic Kelvin–Helmholtz instability [ABSTRACT FROM AUTHOR]

Multiple scattering of waves is eminent in a wide range of applications and extensive research is being undertaken into multiple scattering by ever more complicated structures, with emphasis on the design of metamaterial structures that manipulate waves in a desired fashion. Ongoing research investigates the design of structures and new solution methods for the governing partial differential equations. There is a pressing need for easy-to-use software that empowers rapid prototyping of designs and for validating other solution methods. We develop a general formulation of the multiple scattering problem that facilitates efficient application of the multipole-based method. The shape and morphology of the scatterers is not restricted, provided their T-matrices are available. The multipole method is implemented in the Tmatsolver software package, which uses our general formulation and the T-matrix methodology to simulate accurately multiple scattering by complex configurations with a large number of identical or non-identical scatterers that can have complex shapes and/or morphologies. This article provides a mathematical description of the algorithm and demonstrates application of the software to four contemporary metamaterial problems. It concludes with a brief overview of the object-oriented structure of the Tmatsolver code. [ABSTRACT FROM AUTHOR]

In this paper, we study the improved block splitting (IBS) iteration method and its accelerated variant, the accelerated improved block splitting (AIBS) iteration method, for solving linear systems of equations stemming from the discretization of the complex Helmholtz equation. We conduct a comprehensive convergence analysis and derive optimal iteration parameters aimed at minimizing the spectral radius of the iteration matrix. Through numerical experiments, we validate the efficiency of both iteration methods. [ABSTRACT FROM AUTHOR]

We propose and analyze a hybridizable discontinuous Galerkin (HDG) method for solving a mixed magnetic advection–diffusion problem within a more general Friedrichs system framework. With carefully constructed numerical traces, we introduce two distinct stabilization parameters: τ t for the tangential trace and τ n for the normal trace. These parameters are tailored to satisfy different requirements, ensuring the stability and convergence of the method. Furthermore, we incorporate a weight function to facilitate the establishment of stability conditions. We also investigate an elementwise postprocessing technique that proves to be effective for both two-dimensional and three-dimensional problems in terms of broken H (curl) semi-norm accuracy improvement. Extensive numerical examples are presented to showcase the performance and effectiveness of the HDG method and the postprocessing techniques. [ABSTRACT FROM AUTHOR]

The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen et al. (Numer Math 149(2):273–304, 2021) require a parameter C st , 1 that is found not robust as the polynomial degree p increases. This is related to the H 1 stability bound of the L 2 projection onto polynomials of degree at most p and its growth C st, 1 ∝ (p + 1) 1 / 2 as p → ∞ . A similar estimate for the Galerkin projection holds with a p-robust constant C st , 2 and C st , 2 ≤ 2 for right-isosceles triangles. This paper utilizes the new inequality with the constant C st , 2 to design a modified hybrid high-order eigensolver that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in the simplicial mesh. A key advance is a p-robust parameter selection. The analysis of the new method with a different fine-tuned volume stabilization allows for a priori quasi-best approximation and improved L 2 error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs superior in computer benchmarks with striking numerical evidence for optimal higher empirical convergence rates. [ABSTRACT FROM AUTHOR]

We introduce a new discretization based on a polynomial Trefftz-DG method for solving the Stokes equations. Discrete solutions of this method fulfill the Stokes equations pointwise within each element and yield element-wise divergence-free solutions. Compared to standard DG methods, a strong reduction of the degrees of freedom is achieved, especially for higher polynomial degrees. In addition, in contrast to many other Trefftz-DG methods, our approach allows us to easily incorporate inhomogeneous right-hand sides (driving forces) by using the concept of the embedded Trefftz-DG method. On top of a detailed a priori error analysis, we further compare our approach to other (hybrid) discontinuous Galerkin Stokes discretizations and present numerical examples. [ABSTRACT FROM AUTHOR]

This work develops optimal preconditioners for the discrete H(curl) and H(div) problems on two-dimensional surfaces by nodal auxiliary space preconditioning (Hiptmair and Xu in SIAM J Numer Anal 45:2483–2509, 2007). In particular, on unstructured triangulated surfaces, we develop fast and user-friendly preconditioners for the edge and face element discretizations of curl–curl and grad–div problems based on inverting several discrete surface Laplacians. The proposed preconditioners lead to efficient iterative methods for computing harmonic tangential vector fields on discrete surfaces. Numerical experiments on two- and three-dimensional hypersurfaces are presented to test the performance of those surface preconditioners. [ABSTRACT FROM AUTHOR]

Conductivity reconstruction in an inverse eddy current problem is considered in the present paper. With the electric field measurement on part of domain boundary, we formulate the reconstruction problem as a constrained optimization problem with total variation regularization. The existence and stability are proved for the solution to the optimization problem. The finite element method is employed to discretize the optimization problem. The gradient Lipschitz property of the objective functional is established for the discrete optimization problem. We propose a novel modification to the traditional alternating direction method of multipliers, and prove the convergence of the modified algorithm. Finally, we show some numerical experiments to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Being able to characterise objects at low frequencies, but in situations where the modelling error in the eddy current approximation of the Maxwell system becomes large, is important for improving current metal detection technologies. Importantly, the modelling error becomes large as the frequency increases, but the accuracy of the eddy current model also depends on the object topology and on its materials, with the error being much larger for certain geometries compared to others of the same size and materials. Additionally, the eddy current model breaks down at much smaller frequencies for highly magnetic conducting materials compared to non-permeable objects (with similar conductivities, sizes and shapes) and, hence, characterising small magnetic objects made of permeable materials using the eddy current at typical frequencies of operation for a metal detector is not always possible. To address this, we derive a new asymptotic expansion for permeable highly conducting objects that is valid for small objects and holds not only for frequencies where the eddy current model is valid but also for situations where the eddy current modelling error becomes large and applying the eddy approximation would be invalid. The leading-order term we derive leads to new forms of object characterisations in terms of polarizability tensor object descriptions where the coefficients can be obtained from solving vectorial transmission problems. We expect these new characterisations to be important when considering objects at greater stand-off distance from the coils, which is important for safety critical applications, such as the identification of landmines, unexploded ordnance and concealed weapons. We also expect our results to be important when characterising artefacts of archaeological and forensic significance at greater depths than the eddy current model allows and to have further applications parking sensors and improving the detection of hidden, out-of-sight, metallic objects. [ABSTRACT FROM AUTHOR]

Sound-soft fractal screens can scatter acoustic waves even when they have zero surface measure. To solve such scattering problems we make what appears to be the first application of the boundary element method (BEM) where each BEM basis function is supported in a fractal set, and the integration involved in the formation of the BEM matrix is with respect to a non-integer order Hausdorff measure rather than the usual (Lebesgue) surface measure. Using recent results on function spaces on fractals, we prove convergence of the Galerkin formulation of this "Hausdorff BEM" for acoustic scattering in R n + 1 ( n = 1 , 2 ) when the scatterer, assumed to be a compact subset of R n × { 0 } , is a d-set for some d ∈ (n - 1 , n ] , so that, in particular, the scatterer has Hausdorff dimension d. For a class of fractals that are attractors of iterated function systems, we prove convergence rates for the Hausdorff BEM and superconvergence for smooth antilinear functionals, under certain natural regularity assumptions on the solution of the underlying boundary integral equation. We also propose numerical quadrature routines for the implementation of our Hausdorff BEM, along with a fully discrete convergence analysis, via numerical (Hausdorff measure) integration estimates and inverse estimates on fractals, estimating the discrete condition numbers. Finally, we show numerical experiments that support the sharpness of our theoretical results, and our solution regularity assumptions, including results for scattering in R 2 by Cantor sets, and in R 3 by Cantor dusts. [ABSTRACT FROM AUTHOR]

This work presents the multiharmonic analysis and derivation of functional type a posteriori estimates of a distributed eddy current optimal control problem and its state equation in a time-periodic setting. The existence and uniqueness of the solution of a weak space-time variational formulation for the optimality system and the forward problem are proved by deriving inf-sup and sup-sup conditions. Using the inf-sup and sup-sup conditions, we derive guaranteed, sharp and fully computable bounds of the approximation error for the optimal control problem and the forward problem in the functional type a posteriori estimation framework. We present here the first computational results on the derived estimates. [ABSTRACT FROM AUTHOR]

The impact of Artificial Intelligence (AI) is rapidly expanding, revolutionizing both science and society. It is applied to practically all areas of life, science, and technology, including materials science, which continuously requires novel tools for effective materials characterization. One of the widely used techniques is scanning probe microscopy (SPM). SPM has fundamentally changed materials engineering, biology, and chemistry by providing tools for atomic-precision surface mapping. Despite its many advantages, it also has some drawbacks, such as long scanning times or the possibility of damaging soft-surface materials. In this paper, we focus on the potential for supporting SPM-based measurements, with an emphasis on the application of AI-based algorithms, especially Machine Learning-based algorithms, as well as quantum computing (QC). It has been found that AI can be helpful in automating experimental processes in routine operations, algorithmically searching for optimal sample regions, and elucidating structure-property relationships. Thus, it contributes to increasing the efficiency and accuracy of optical nanoscopy scanning probes. Moreover, the combination of AI-based algorithms and QC may have enormous potential to enhance the practical application of SPM. The limitations of the AI-QC-based approach were also discussed. Finally, we outline a research path for improving AI-QC-powered SPM. RESEARCH HIGHLIGHTS: Artificial intelligence and quantum computing as support for scanning probe microscopy. The analysis indicates a research gap in the field of scanning probe microscopy. The research aims to shed light into ai-qc-powered scanning probe microscopy., (© 2024 Wiley Periodicals LLC.)

In this paper, we consider the numerical solution of a large complex linear system with a saddle-point form obtained by the discretization of the time-harmonic eddy-current optimal control problem. A new Schur complement is proposed for this algebraic system, extending it to both the block-triangular preconditioner and the structured preconditioner. A theoretical analysis proves that the eigenvalues of block-triangular and structured preconditioned matrices are located in the interval [1/2, 1]. Numerical simulations show that two new preconditioners coupled with a Krylov subspace acceleration have good feasibility and effectiveness and are superior to some existing efficient algorithms. [ABSTRACT FROM AUTHOR]

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates. [ABSTRACT FROM AUTHOR]

This review paper addresses the development of numerical modeling of electromagnetic fields in geophysics with a focus on recent finite element simulation. It discusses ways of estimating errors of our solutions for a perfectly matched modeling domain and the problems that arise from its insufficient representation. After a brief outline of early methods and modeling approaches, the paper mainly discusses the capabilities of the finite element method formulated on unstructured grids and the advantages of local h-refinement allowing for both a flexible and largely accurate representation of the geometries of the multi-scale geomaterial and an accurate evaluation of the underlying functions representing the physical fields. In summary, the accuracy of the solution depends on the geometric mapping, the choice of the mathematical model, and the spatial discretization. Although the available error estimators do not necessarily provide reliable error bounds for our complex geomodels, they are still useful to guide grid refinement. Therefore, an overview of the most common a posteriori error estimators is given. It will be shown that the sensitivity is the most important function in both guiding the geometric mapping and the local refinement. [ABSTRACT FROM AUTHOR]

This work deals with the spectral element discretization of the time-dependent Stokes problem in two- and three-dimensional domains. The boundary condition is defined on the normal component of the velocity and the tangential components of the vorticity. The discretization related to the time variable is processed by a Backward Euler method. We prove through a detailed numerical analysis the well-posedness of the full discrete problem. [ABSTRACT FROM AUTHOR]

With the aim to continue developing a hybridizable discontinuous Galerkin (HDG) method for problems arisen from photovoltaic cells modeling, in this manuscript we consider the time harmonic Maxwell’s equations in an inhomogeneous bounded bi-periodic domain with quasi-periodic conditions on part of the boundary. We propose an HDG scheme where quasi-periodic boundary conditions are imposed on the numerical trace space. Under regularity assumptions and a proper choice of the stabilization parameter, we prove that the approximations of the electric and magnetic fields converge, in the L 2 -norm, to the exact solution with order h k + 1 and h k + 1 / 2 , resp., where h is the meshsize and k the polynomial degree of the discrete spaces. Although, numerical evidence suggests optimal order of convergence for both variables. An a posteriori error estimator for an energy norm is also proposed. We show that it is reliable and locally efficient under certain conditions. Numerical examples are provided to illustrate the performance of the quasi-periodic HDG method and the adaptive scheme based on the proposed error indicator. [ABSTRACT FROM AUTHOR]

We propose a new residual-based a posteriori error estimator for discontinuous Galerkin discretizations of time-harmonic Maxwell's equations in first-order form. We establish that the estimator is reliable and efficient, and the dependency of the reliability and efficiency constants on the frequency is analyzed and discussed. The proposed estimates generalize similar results previously obtained for the Helmholtz equation and conforming finite element discretizations of Maxwell's equations. In addition, for the discontinuous Galerkin scheme considered here, we also show that the proposed estimator is asymptotically constant-free for smooth solutions. [ABSTRACT FROM AUTHOR]

This paper presents efficient data structures for the implementation of matrix-free finite element methods on block-structured, hybrid tetrahedral grids. It provides a complete categorization of all geometric sub-objects that emerge from the regular refinement of the unstructured, tetrahedral coarse grid and describes efficient iteration patterns and analytical linearization functions for the mapping of coefficients to memory addresses. This foundation enables the implementation of fast, extreme-scalable, matrix-free, iterative solvers, and in particular geometric multigrid methods by design. Their application to the variable-coefficient Stokes system subject to an enriched Galerkin discretization and to the curl-curl problem discretized with Nédélec edge elements showcases the flexibility of the implementation. Finally, the solution of a curl-curl problem with $ 1.6\cdot 10^{11} $ 1.6 ⋅ 10 11 (more than one hundred billion) unknowns on more than 32,000 processes with a matrix-free full multigrid solver demonstrates its extreme-scalability. [ABSTRACT FROM AUTHOR]

An hp-version interior penalty discontinuous Galerkin method under nonconforming meshes is proposed to solve the quad-curl eigenvalue problem. We prove well-posedness of the numerical scheme for the quad-curl equation and then derive an error estimate in a mesh-dependent norm, which is optimal with respect to h but has different p-version error bounds under conforming and nonconforming tetrahedron meshes. The hp-version discrete compactness of the DG space is established for the convergence proof. The performance of the method is demonstrated by numerical experiments using conforming/nonconforming meshes and h-version/p-version refinement. The optimal h-version convergence rate and the exponential p-version convergence rate are observed. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a new structure-preserving finite element method for the stationary magnetohydrodynamic equations with magnetic-current formulation on Lipschitz domains. Using a mixed finite element approach, we discretize the hydrodynamic unknowns by inf-sup stable velocity-pressure finite element pairs, and the current density, the induced electric field and the magnetic field by using the edge-edge-face elements from a discrete de-Rham complex pair. To deal with the divergence-free condition of the magnetic field, we introduce an augmented term to the discrete scheme rather than a Lagrange multiplier in the existing schemes. Thanks to discrete differential forms and finite element exterior calculus, the proposed scheme preserves the divergence-free property exactly for the magnetic induction on the discrete level. The well-posedness of the discrete problem is further proved under the small data condition. Under weak regularity assumptions, we rigorously establish the error estimates of the finite element schemes. Numerical results are provided to illustrate the theoretical results and demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper deals with a high-order H (curl) -conforming Bernstein–Bézier finite element method (BBFEM) to accurately solve time-harmonic Maxwell short wave problems on unstructured triangular mesh grids. We suggest enhanced basis functions, defined on the reference triangle and tetrahedron, aiming to reduce the condition number of the resulting global matrix. Moreover, element-level static condensation of the interior degrees of freedom is performed in order to reduce memory requirements. The performance of BBFEM is assessed using several benchmark tests. A preliminary analysis is first conducted to highlight the advantage of the suggested basis functions in improving the conditioning. Numerical results dealing with the electromagnetic scattering from a perfect electric conductor demonstrate the effectiveness of BBFEM in mitigating the pollution effect and its efficiency in capturing high-order evanescent wave modes. Electromagnetic wave scattering by a circular dielectric, with high wave speed contrast, is also investigated. The interior curved interface between layers is accurately described based on a linear blending map to avoid numerical errors due to geometry description. The achieved results support our expectations for highly accurate and efficient BBFEM for time harmonic wave problems. [ABSTRACT FROM AUTHOR]

In two and three dimensions, we design and analyze a posteriori error estimators for the mixed Stokes eigenvalue problem. The unknowns on this mixed formulation are the pseudotress, velocity and pressure. With a lowest order mixed finite element scheme, together with a postprocressing technique, we prove that the proposed estimator is reliable and efficient. We illustrate the results with several numerical tests in two and three dimensions in order to assess the performance of the estimator. [ABSTRACT FROM AUTHOR]

In this article, we propose the weak Galerkin (WG) finite element schemes for H(div;Ω)${\bf H}(\mbox{div}; {\Omega })$‐elliptic problems with and without stabilizers. Optimal orders of convergence are established for the WG approximations in both discrete energy norm and L2 norm. Removing stabilizers from WG finite element methods will simplify the formulations, reduce programming complexity, and may also speed up the computation time. More precisely, for sufficiently smooth solutions, we have proved the supercloseness of order two for the stabilizer free weak Galerkin finite element solution. Several numerical tests are presented to demonstrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. Analysis for approximations of Schur complements is included. Some numerical experiments validate our analytical findings. [ABSTRACT FROM AUTHOR]

We present a framework for the structure-preserving approximation of partial differential equations on mapped multipatch domains, extending the classical theory of finite element exterior calculus (FEEC) to discrete de Rham sequences which are broken, i.e., fully discontinuous across the patch interfaces. Following the Conforming/Nonconforming Galerkin (CONGA) schemes developed in Campos Pinto and Sonnendrücker (Math Comput 85:2651–2685, 2016) and Campos Pinto and Güçlü (Broken-FEEC discretizations and Hodge Laplace problems. , 2022), our approach is based on: (i) the identification of a conforming discrete de Rham sequence with stable commuting projection operators, (ii) the relaxation of the continuity constraints between patches, and (iii) the construction of conforming projections mapping back to the conforming subspaces, allowing to define discrete differentials on the broken sequence. This framework combines the advantages of conforming FEEC discretizations (e.g. commuting projections, discrete duality and Hodge–Helmholtz decompositions) with the data locality and implementation simplicity of interior penalty methods for discontinuous Galerkin discretizations. We apply it to several initial- and boundary-value problems, as well as eigenvalue problems arising in electromagnetics. In each case our formulations are shown to be well posed thanks to an appropriate stabilization of the jumps across the interfaces, and the solutions are extremely robust with respect to the stabilization parameter. Finally we describe a construction using tensor-product splines on mapped cartesian patches, and we detail the associated matrix operators. Our numerical experiments confirm the accuracy and stability of this discrete framework, and they allow us to verify that expected structure-preserving properties such as divergence or harmonic constraints are respected to floating-point accuracy. [ABSTRACT FROM AUTHOR]

We introduce two a posteriori error estimators for Nédélec finite element discretizations of the curl-curl problem. These estimators pertain to a new Prager–Synge identity and an associated equilibration procedure. They are reliable and efficient, and the error estimates are polynomial-degree-robust. In addition, when the domain is convex, the reliability constants are fully computable. The proposed error estimators are also cheap and easy to implement, as they are computed by solving divergence-constrained minimization problems over edge patches. Numerical examples highlight our key findings, and show that both estimators are suited to drive adaptive refinement algorithms. Besides, these examples seem to indicate that guaranteed upper bounds can be achieved even in non-convex domains. [ABSTRACT FROM AUTHOR]

Geomagnetic depth sounding (GDS) is a geophysical electromagnetic (EM) method that studies the deep structure and composition of the Earth by using long-period EM signals from geomagnetic observatories and satellites. In this paper, a 3-D anisotropic GDS modelling algorithm is developed. The curl–curl equation is discretized using the edge-based finite-element method on unstructured tetrahedral grids. In order to solve the computationally demanding problem of EM modelling on a global scale, the complex linear system is first separated into the equivalent real linear systems and then the real system is iteratively solved by the flexible generalized minimum residual method with a block diagonal pre-conditioner. This will greatly reduce the condition number of the linear system and thus speed up the solution process. We verify the accuracy of the proposed algorithm by comparing our results with the existing methods. After that, we design a subduction zone model to simulate the EM responses under isotropic and anisotropic environments, respectively. The numerical results show the high efficiency of the proposed algorithm and the response differences between isotropic and anisotropic models. This research can provide theoretical and technical support for the high-accuracy and efficient inversion of GDS data for the geo-dynamic study. [ABSTRACT FROM AUTHOR]

Purpose: The ability to characterise highly conducting objects, that may also be highly magnetic, by the complex symmetric rank–2 magnetic polarizability tensor (MPT) is important for metal detection applications including discriminating between threat and non-threat objects in security screening, identifying unexploded anti-personnel landmines and ordnance and identifying metals of high commercial value in scrap sorting. Many everyday non-threat items have both a large electrical conductivity and a magnetic behaviour, which, for sufficiently weak fields and the frequencies of interest, can be modelled by a high relative magnetic permeability. This paper aims to discuss the aforementioned idea. Design/methodology/approach: The numerical simulation of the MPT for everyday non-threat highly conducting magnetic objects over a broad range of frequencies is challenging due to the resulting thin skin depths. The authors address this by employing higher order edge finite element discretisations based on unstructured meshes of tetrahedral elements with the addition of thin layers of prismatic elements. Furthermore, computer aided design (CAD) geometrical models of the non-threat and threat object are often not available and, instead, the authors extract the geometrical features of an object from an imaging procedure. Findings: The authors obtain accurate numerical MPT characterisations that are in close agreement with experimental measurements for realistic physical objects. The assessment of uncertainty shows the impact of geometrical and material parameter uncertainties on the computational results. Originality/value: The authors present novel computations and measurements of MPT characterisations of realistic objects made of magnetic materials. A novel assessment of uncertainty in the numerical predictions of MPT characterisations for uncertain geometry and material parameters is included. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical analysis of a finite element method for the time-dependent Ginzburg–Landau equations under the Coulomb gauge. The main challenge is that the magnetic potential A is divergence-free and satisfies a Stokes-like structure under the Coulomb gauge. The proposed method uses linear Lagrange element P 1 to solve for the order parameter ψ , the lowest order Nédélec edge element N D 1 and linear Lagrange element P 1 to approximate the magnetic potential A and the electric potential ϕ , respectively. In particular, the proposed method preserves a weakly divergence-free property for A in the discrete level. The main aim of this work is to establish the second order spatial convergence of the most important variable ψ , though the numerical solutions of A are only O(h) in space. Our analysis is based on a nonstandard quasi-projection for ψ and the corresponding H - 1 -norm estimates for Maxwell projection. With the quasi-projection, we prove that the lower-order approximation to A does not pollute the accuracy of ψ h . An effective one step recovery is also proposed to obtain second order numerical solution for A . Our numerical experiments confirm the optimal second order convergence of ψ h and the efficiency of the recovery step. [ABSTRACT FROM AUTHOR]

For the incompressible MHD equations, we present a fully discrete second-order-in-time scheme based on a blended BDF and extrapolation treatments for nonlinear terms. The proposed scheme is more accurate than the two-step BDF with additional nominal computational time and is still A-stable. Then, unconditional stability, long-time stability, and optimal convergence rate of the scheme are presented. At last, the numerical findings are verified by comparison with the two-step BDF scheme in several 2D and 3D numerical experiments. [ABSTRACT FROM AUTHOR]