scholarly journals A variant of the plane wave least squares method for the time-harmonic Maxwell’s equations

2019 ◽  
Vol 53 (1) ◽  
pp. 85-103 ◽  
Author(s):  
Qiya Hu ◽  
Rongrong Song
Keyword(s):  
Plane Wave ◽  
Error Estimate ◽  
Least Squares ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Least Squares Method ◽  
Three Dimensions ◽  
Plane Wave Method ◽  
Time Harmonic ◽  
Plane Wave Approximation

In this paper we are concerned with the plane wave method for the discretization of time-harmonic Maxwell’s equations in three dimensions. As pointed out in Hiptmair et al. (Math. Comput. 82 (2013) 247–268), it is difficult to derive a satisfactory L2 error estimate of the standard plane wave approximation of the time-harmonic Maxwell’s equations. We propose a variant of the plane wave least squares (PWLS) method and show that the new plane wave approximations yield the desired L2 error estimate. Moreover, the numerical results indicate that the new approximations have sightly smaller L2 errors than the standard plane wave approximations. More importantly, the results are derived for more general models in inhomogeneous media.

scholarly journals Abstract Nonconforming Error Estimates and Application to Boundary Penalty Methods for Diffusion Equations and Time-Harmonic Maxwell’s Equations

2018 ◽  
Vol 18 (3) ◽  
pp. 451-475 ◽  
Author(s):  
Alexandre Ern ◽  
Jean-Luc Guermond
Keyword(s):  
Error Analysis ◽  
Error Estimates ◽  
Material Properties ◽  
Maxwell’S Equations ◽  
Vector Fields ◽  
Maxwell's Equations ◽  
Heterogeneous Material ◽  
Test Functions ◽  
Time Harmonic ◽  
New Formulation

AbstractWe devise a novel framework for the error analysis of finite element approximations to low-regularity solutions in nonconforming settings where the discrete trial and test spaces are not subspaces of their exact counterparts. The key is to use face-to-cell extension operators so as to give a weak meaning to the normal or tangential trace on each mesh face individually for vector fields with minimal regularity and then to prove the consistency of this new formulation by means of some recently-derived mollification operators that commute with the usual derivative operators. We illustrate the technique on Nitsche’s boundary penalty method applied to a scalar diffusion equation and to the time-harmonic Maxwell’s equations. In both cases, the error estimates are robust in the case of heterogeneous material properties. We also revisit the error analysis framework proposed by Gudi where a trimming operator is introduced to map discrete test functions into conforming test functions. This technique also gives error estimates for minimal regularity solutions, but the constants depend on the material properties through contrast factors.

scholarly journals Time-Harmonic Solutions for Maxwell’s Equations in Anisotropic Media and Bochner–Riesz Estimates with Negative Index for Non-elliptic Surfaces

2021 ◽  
Author(s):  
Rainer Mandel ◽  
Robert Schippa
Keyword(s):  
Gaussian Curvature ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Smooth Surfaces ◽  
Negative Index ◽  
Elliptic Surfaces ◽  
Conical Singularities ◽  
Fourier Restriction ◽  
Time Harmonic ◽  
Homogeneous Media

AbstractWe solve time-harmonic Maxwell’s equations in anisotropic, spatially homogeneous media in intersections of $$L^p$$ L p -spaces. The material laws are time-independent. The analysis requires Fourier restriction–extension estimates for perturbations of Fresnel’s wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner–Riesz estimates with negative index for non-elliptic surfaces.

Plane wave ℋ (curl; Ω) conforming finite elements for Maxwell's equations

2003 ◽  
Vol 31 (3) ◽  
pp. 272-283 ◽  
Author(s):  
P. D. Ledger ◽  
K. Morgan ◽  
O. Hassan ◽  
N. P. Weatherill
Keyword(s):  
Finite Elements ◽  
Plane Wave ◽  
Maxwell’S Equations ◽  
Maxwell's Equations

Maxwell fields satisfying Huygens's principle

1968 ◽  
Vol 64 (3) ◽  
pp. 779-785 ◽  
Author(s):  
H. P. Künzle
Keyword(s):  
Wave Equation ◽  
Plane Wave ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Maxwell Fields ◽  
Wave Space

AbstractIt is shown that Huygens's principle holds for the solutions of Maxwell's equations for p-forms of all degrees in a gravitational plane wave space, while the solutions of the wave equation for 1, 2, and 3-forms, however, may have tails.

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