1,766 research outputs found
A nodal-based finite element approximation of the Maxwell problem suitable for singular solutions
A new mixed finite element approximation of Maxwell’s problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous
problem and the introduction of a mesh dependent stabilizing term, which yields a very weak control on the divergence of the unknown. The method is shown to be stable and convergent in the natural H (curl; Ω) norm for this unknown. In particular, convergence also applies to singular solutions, for which classical nodal
based interpolations are known to suffer from spurious convergence upon mesh
refinement
A nodal-based finite element approximation of the Maxwell problem suitable for singular solutions
A new mixed finite element approximation of Maxwell’s problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh
dependent stabilizing term, which yields a very weak control on the divergence of the unknown. The method is shown to be stable and convergent in the natural H(curl; ) norm for this unknown. In particular, convergence
also applies to singular solutions, for which classical nodal based interpolations are known to suffer from spurious convergence upon mesh refinement.Postprint (published version
A combined nodal continuous-discontinuous finite element formulation for the Maxwell problem
Continuous Galerkin formulations are appealing due to their low computational cost, whereas discontinuous Galerkin formulation facilitate adaptative mesh refinement and are more accurate in regions with jumps of physical parameters. Since many electromagnetic problems involve materials with different physical properties, this last point is very important. For this reason, in this article we have developed a combined cG–dG formulation for Maxwell’s problem that allows arbitrary finite element spaces with functins continuous in patches of finite elements and discontinuous on the interfaces of these patches. In particular, the second formulation we propose comes from a novel continuous Galerkin formulation that reduces the amount of stabilization introduced in the numerical system. In all cases, we have performed stability and convergence analyses of the methods. The outcome of this work is a new approach that keeps the low CPU cost of recent nodal continuous formulations with the ability to deal with coefficient jumps and adaptivity of discontinuous ones. All these methods have been tested using a problem with singular solution and another one with different materials, in order to prove that in fact the resulting formulations can properly deal with these problems
Finite element eigenvalue enclosures for the Maxwell operator
We propose employing the extension of the Lehmann-Maehly-Goerisch method
developed by Zimmermann and Mertins, as a highly effective tool for the
pollution-free finite element computation of the eigenfrequencies of the
resonant cavity problem on a bounded region. This method gives complementary
bounds for the eigenfrequencies which are adjacent to a given real parameter.
We present a concrete numerical scheme which provides certified enclosures in a
suitable asymptotic regime. We illustrate the applicability of this scheme by
means of some numerical experiments on benchmark data using Lagrange elements
and unstructured meshes.Comment: arXiv admin note: substantial text overlap with arXiv:1306.535
On a general implementation of - and -adaptive curl-conforming finite elements
Edge (or N\'ed\'elec) finite elements are theoretically sound and widely used
by the computational electromagnetics community. However, its implementation,
specially for high order methods, is not trivial, since it involves many
technicalities that are not properly described in the literature. To fill this
gap, we provide a comprehensive description of a general implementation of edge
elements of first kind within the scientific software project FEMPAR. We cover
into detail how to implement arbitrary order (i.e., -adaptive) elements on
hexahedral and tetrahedral meshes. First, we set the three classical
ingredients of the finite element definition by Ciarlet, both in the reference
and the physical space: cell topologies, polynomial spaces and moments. With
these ingredients, shape functions are automatically implemented by defining a
judiciously chosen polynomial pre-basis that spans the local finite element
space combined with a change of basis to automatically obtain a canonical basis
with respect to the moments at hand. Next, we discuss global finite element
spaces putting emphasis on the construction of global shape functions through
oriented meshes, appropriate geometrical mappings, and equivalence classes of
moments, in order to preserve the inter-element continuity of tangential
components of the magnetic field. Finally, we extend the proposed methodology
to generate global curl-conforming spaces on non-conforming hierarchically
refined (i.e., -adaptive) meshes with arbitrary order finite elements.
Numerical results include experimental convergence rates to test the proposed
implementation
Analysis of an unconditionally convergent stabilized finite element formulation for incompressible magnetohydrodynamics
In this work, we analyze a recently proposed stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation with respect to existing ones is the fact that it always converges to the physical solution, even when it is singular. We have performed a detailed stability and convergence analysis of the formulation in a simplified setting. From the convergence analysis, we infer that a particular type of meshes with a macro-element structure is needed, which can be easily obtained after a straight modification of any original mesh.Preprin
On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics
In this work, we propose a new stabilized finite element formulation for the approximation of the
resistive magnetohydrodynamics equations. The novelty of this formulation with respect to existing
ones is the fact that it always converges to the physical solution, even for singular ones. We have performed
a detailed stability and convergence analysis of the formulation in a simplified setting. From
the convergence analysis, we infer that a particular type of meshes with a macro-element structure is
needed, which can be easily obtained after a straight modification of any original mesh. A detailed
set of numerical experiments have been performed in order to validate our approach.Peer ReviewedPreprin
Stokes, Maxwell and Darcy: A single finite element approximation for three model problems
In this work we propose stabilized finite element methods for Stokesʼ, Maxwellʼs and Darcyʼs problems that accommodate any interpolation of velocities and pressures. We briefly review the formulations we have proposed for these three problems independently in a unified manner, stressing the advantages of our approach. In particular, for Darcyʼs problem we are able to design stabilized methods that yield optimal convergence both for the primal and the dual problems. In the case of Maxwellʼs problem, the formulation we propose allows one to use continuous finite element interpolations that converge optimally to the continuous solution even if it is non-smooth. Once the formulation is presented for the three model problems independently, we also show how it can be used for a problem that combines all the operators of the independent problems. Stability and convergence is achieved regardless of the fact that any of these operators dominates the others, a feature not possible for the methods of which we are aware
Approximation of singular solutions and singular data for Maxwell’s equations by Lagrange elements
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