Corpus ID: 212747782

Convergence of Lagrange finite elements for the Maxwell Eigenvalue Problem in 2D

@article{Boffi2020ConvergenceOL,
  title={Convergence of Lagrange finite elements for the Maxwell Eigenvalue Problem in 2D},
  author={Daniele Boffi and Johnny Guzm{\'a}n and Michael Neilan},
  journal={ArXiv},
  year={2020},
  volume={abs/2003.08381}
}
We consider finite element approximations of the Maxwell eigenvalue problem in two dimensions. We prove, in certain settings, convergence of the discrete eigenvalues using Lagrange finite elements. In particular, we prove convergence in three scenarios: piecewise linear elements on Powell--Sabin triangulations, piecewise quadratic elements on Clough--Tocher triangulations, and piecewise quartics (and higher) elements on general shape-regular triangulations. We provide numerical experiments that… Expand

References

SHOWING 1-10 OF 25 REFERENCES
New Mixed Elements for Maxwell Equations
TLDR
A general theory of stability and error estimates is developed and it is shown that the proposed mixed elements provide spectral-correct, spurious-free approximations for eigenvalues and for both singular and smooth solutions. Expand
Finite element approximation of eigenvalue problems
  • D. Boffi
  • Mathematics, Computer Science
  • Acta Numerica
  • 2010
TLDR
The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations. Expand
Maxwell and Lamé eigenvalues on polyhedra
In a convex polyhedron, a part of the Lame eigenvalues with hard simple support boundary conditions does not depend on the Lame coefficients and coincides with the Maxwell eigenvalues. The otherExpand
Finite element interpolation of nonsmooth functions satisfying boundary conditions
In this paper, we propose a modified Lagrange type interpolation operator to approximate functions in Sobolev spaces by continuous piecewise polynomials. In order to define interpolators for "rough"Expand
Computational Models of Electromagnetic Resonators: Analysis of Edge Element Approximation
The purpose of this paper is to address some difficulties which arise in computing the eigenvalues of Maxwell's system by a finite element method. Depending on the method used, the spectrum may beExpand
Finite element exterior calculus: From hodge theory to numerical stability
This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it weExpand
Combined finite element-modal solution of three-dimensional eddy current problems
The reliability of finite-element methods for modal analysis of two- and three-dimensional eddy-current problems is addressed. Separation of variables is used to convert transient-eddy-currentExpand
On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form
TLDR
This paper presents examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues, and proves that bad behavior is proved analytically and demonstrated in numerical experiments. Expand
Stokes Complexes and the Construction of Stable Finite Elements with Pointwise Mass Conservation
Two families of conforming finite elements for the two-dimensional Stokes problem are developed, guided by two discrete smoothed de Rham complexes, which we coin “Stokes complexes.” We show that theExpand
Mixed Finite Elements
This chapter explains the need for mixed finite element methods and the algorithmic ingredients of this discretization approach. Various Diffpack tools for easy programming of mixed methods onExpand
...
1
2
3
...