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It has been recognised that numerical computations of magnetic
fields by the finite-element method may require new types of elements,
whose degrees of freedom are not field values at mesh nodes, but other
field-related quantities like, e.g. circulations along edges of the
mesh. A rationale for the use of these special `mixed' elements can be
obtained if one expresses basic equations in terms of differential
forms, instead of vector fields. The authors gives an elementary
introduction to this point of view, presents Whitney forms (the mixed
finite elements alluded to), and sketches two numerical methods (dual,
in some sense), for eddy-current studies, based on these elements

Content uploaded by Alain Bossavit

Author content

All content in this area was uploaded by Alain Bossavit on Mar 02, 2013

Content may be subject to copyright.

... The geometrical structure of the equations is often better understood when written in the language of differential geometry, using differential forms to rewrite various objects of the Maxwell equations [13,56], or the diffeomorphism group for fluid evolution [6]. Much effort has been put into the investigation of discretizations that preserve those underlying differential structures [12,62,36]. ...

... The construction of discretizations that preserve vector calculus identities such as curl grad = 0 and div curl = 0, or the more general exterior calculus identity d • d = 0 [46,47] allows the preservation of some important physical invariants. Among those discretizations, the FEEC framework [4,5], which provides high order discretization of the de Rham complex, has been used for a variety of application such as the Poisson equation in mixed form [1], the Maxwell equations [13,46] or the Vlasov-Maxwell system [50]. This framework was recently extended to broken spaces [18,17,39], leading to discrete operators with better locality properties and potentially more efficient discretizations. ...

In this article we propose two finite element schemes for the Navier-Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the pointwise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization which preserve these invariants at the fully discrete level and we analyse its well-posedness in terms of a CFL condition. Numerical test cases performed with spline finite elements allow us to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.

... In the 3 case, and are discretized using edge and facet elements [19]. The Degrees of Freedom (DoF) of are the circulations on the edges of the mesh. ...

The Proper Generalized Decomposition has shown its efficiency to solve parameterized problems in nonlinear system events when it is combined with the Discrete Empirical Interpolation Method. Nevertheless, the solution of finite element model with the Proper Generalized Decomposition framework requires to have access to matrices and vectors of the discretized problem, which makes the method highly intrusive. In this context, based on a set of finite element solutions for a set of input parameters, a surrogate model can be developed applying a non-intrusive Proper Generalized Decomposition approach. The proposed non-intrusive approach is based on a canonical decomposition of the finite element solutions combined with an interpolation method. We then obtain a surrogate model approximating the finite element solutions for a wide range of parameters. The surrogate model, given its evaluation speed, can be used for real-time applications. In this paper, the proposed non-intrusive Proper Generalized Decomposition approach is employed to approximate a nonlinear magnetostatic problem and is applied to a single phase standard transformer and to a three-phase inductance.

... Discrete H(div; )and L 2 ( )-conforming Finite Element spaces mimicking the exactness property of the rightmost portion of the de Rham complex have been used since the early 80s to prove the stability of mixed formulations of scalar diffusion problems [23,24]. The first use of the full de Rham complex, on the other hand, was made a few years later to devise stable approximations of problems in computational electromagnetism [10]. In recent years, the study of compatible Finite Elements has gravitated towards generalisations based on the formalism of exterior calculus (see, e.g., [1] and references therein). ...

In this work we prove that, for a general polyhedral domain of $$\mathbb {R}^3$$ R 3 , the cohomology spaces of the discrete de Rham complex of Di Pietro and Droniou (Found Comput Math 23:85–164, 2023, https://doi.org/10.1007/s10208-021-09542-8 ) are isomorphic to those of the continuous de Rham complex. This is, to the best of our knowledge, the first result of this kind for a fully computable arbitrary-order complex built from a general polyhedral mesh.

... Mixed finite element spaces which preserve the de Rham structure offer a flexible and powerful framework for the approximation of partial differential equations. This discretization paradigm has been extensively studied in the scope of electromagnetic modelling [7,8,22] and has given rise to an elegant body of theoretical work which guarantees that compatible spaces of nodal, egde, face and volume type lead to stable and accurate approximations to various differential operators in domains with non-smooth or non-connected boundaries [21,2,5,6]. ...

We present stable commuting projection operators on de Rham sequences of two-dimensional multipatch spaces with local tensor-product parametrization and non-matching interfaces. Our construction covers the case of shape-regular patches with different mappings and locally refined patches, under the assumption that neighbouring patches have nested resolutions and that interior vertices are shared by exactly four patches. Following a broken-FEEC approach we first apply a tensor-product construction on the single-patch de Rham sequences and modify the resulting patch-wise commuting projections to enforce their conformity while preserving their commuting, projection, and L2 stability properties. The resulting operators are local and stable in L2, with constants independent of both the size and the inner resolution of the individual patches.

This paper deals with a high-order H(curl)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H({\text {curl}})$$\end{document}-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.

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.

In this Research, some problems associated with numerical weather prediction are discussed. we have been able to simulate some finite difference schemes to predict weather trends of Abuja. By analyzing the results from these schemes, it has shown that the best scheme in the finite difference method that gives a close accurate weather forecast is the trapezoidal scheme when comparing with sunshine, Rainfall, and windspeed. We use the trapezoidal scheme to stimulate the numerical weather data obtained from the federal Airports Authority. Finally using Matlab (2021a) to acquire subsequent numerical tendency.

We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex.

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