Construction of a Finite Element Basis of the First de Rham Cohomology Group and Numerical Solution of 3D Magnetostatic Problems

Rodríguez, Ana Alonso; Bertolazzi, Enrico; Ghiloni, Riccardo; Valli, Alberto

doi:10.1137/120890648

Cited by 37 publications

(59 citation statements)

References 60 publications

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“…The approximation of harmonic functions is out of the aims of this paper. We point the reader to possible approaches for the approximation of H: a direct discretization of the space has been proposed in [1]; an adaptive algorithm has been presented in [25]; another indirect approach may be the use of the following alternative mixed formulation known as Kikuchi formulation (see [31,6]): find λ ∈ R such that for u ∈ H 0 (curl; Ω) and p ∈ H 1 0 (Ω), with u = 0, it holds…”

Section: Maxwell's Eigenvalue Problem and Its Finite Element Discretimentioning

confidence: 99%

Adaptive Finite Element Method for the Maxwell Eigenvalue Problem

Boffi¹,

Gastaldi²

2019

SIAM J. Numer. Anal.

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Section: Maxwell's Eigenvalue Problem and Its Finite Element Discretimentioning

confidence: 99%

Adaptive Finite Element Method for the Maxwell Eigenvalue Problem

Boffi¹,

Gastaldi²

2019

SIAM J. Numer. Anal.

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“…From Theorem 3 in Alonso Rodríguez et al [5] we find b i = 0 and c n = 0. We have thus obtained m C j=1 a j w h,j|Ω C = 0, and consequently a j = 0.…”

Section: It Is Well-known Thatmentioning

confidence: 65%

“…In particular, in the sequel we consider the RaviartThomas interpolant of J e|Ω I , and we set J I e,h = Π RT h J e|Ω I . We replace H e with H e,h ∈ N h defined in the following way: having constructed as in Alonso Rodríguez et al [5] …”

Section: It Is Well-known Thatmentioning

confidence: 99%

“…, n v , the Lagrange nodal basis function for each node v i inside Ω I or on its boundary ∂Ω I = Γ ∪ ∂Ω. We also consider the Nédélec finite element basis T I 0,n of the first de Rham cohomology group of Ω I constructed in Alonso Rodríguez et al [5], and denote by T 0,n the extension of T I 0,n by setting value 0 at all the degrees of freedom associated to the edges strictly inside Ω C .…”

Section: It Is Well-known Thatmentioning

confidence: 99%

“…In particular, we can use the maximal set of independent loop fields provided for any bounded domain Ω I by the general algorithm derived in Alonso Rodríguez et al [5]. It is worth noting that this algorithm determines the loop fields using only the knowledge of the vertices and the tetrahedra of the mesh, and of the mesh induced on the boundary of Ω I .…”

Section: Introductionmentioning

confidence: 99%

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Finite element simulation of eddy current problems using magnetic scalar potentials

Rodríguez

Bertolazzi

Ghiloni

et al. 2015

Journal of Computational Physics

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We propose a new implementation of the finite element approximation of eddy current problems using as principal unknown the magnetic field. In the non-conducting region a scalar magnetic potential is introduced. The method can deal automatically with any topological configuration of the conducting region and, being based on the search of a scalar magnetic potential in the non-conducting region, has the advantage of making use of a reduced number of unknowns. Several numerical tests are presented for illustrating the performance of the proposed method; in particular, the numerical simulation of a new type of transformer of complicated topological shape is shown.

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Construction of a spanning tree for high‐order edge elements

Santos

Rodríguez

Rapetti

2022

Int J Numerical Modelling

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The well-known tree-cotree gauging method for low-order edge finite elements is extended to high-order approximations within the first family of Nédélec finite element spaces. The key point in this method is the identification of degrees of freedom for edge and nodal finite element spaces such that the matrix of the gradient operator is the transposed of the all-node incidence matrix of a directed graph. This is straightforward for low-order finite elements and it can be proved that it is still possible in the high-order case using either moments or weights as degrees of freedom. In the case of weights, the geometrical realization of the graph associated with the gradient operator is very natural. We recall in details the definition of weights and present an algorithm for the construction of a spanning tree of this graph. The starting point of the algorithm is a spanning tree of the graph given by vertices and edges of the mesh (the so-called global spanning tree, that is the one used in the low-order case). This global step, interpreted in the high-order sense, is enriched locally, with a loop over the elements of the mesh, with arcs corresponding to edge, face and volume degrees of freedom required in the high-order case.

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Construction of a Finite Element Basis of the First de Rham Cohomology Group and Numerical Solution of 3D Magnetostatic Problems

Cited by 37 publications

References 60 publications

Adaptive Finite Element Method for the Maxwell Eigenvalue Problem

Adaptive Finite Element Method for the Maxwell Eigenvalue Problem

Finite element simulation of eddy current problems using magnetic scalar potentials

Construction of a spanning tree for high‐order edge elements

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