The Mathematical Theory of Time-Harmonic Maxwell's Equations

2015 ◽  
Author(s):  
Andreas Kirsch ◽  
Frank Hettlich
Keyword(s):  
Mathematical Theory ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Time Harmonic

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.

scholarly journals W1,p Regularity of Weak Solutions to Maxwell’s Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Zhihong Chen
Keyword(s):  
Weak Solution ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Coefficient Matrix ◽  
Function Technique ◽  
Open Domain ◽  
Weak Formulation ◽  
Time Harmonic ◽  
Regularity Of Weak Solutions ◽  
Analytical Tools

In this paper, we study the steady-state Maxwell’s equations. The weak solution defined in weak formulation is considered, and the global existence is obtained in general bounded open domain. The interior W1,p2≤p<∞ estimates of the weak solution are obtained, where the coefficient matrix is assumed to be BMO with small seminorm. The main analytical tools are the Vitali covering lemma, the maximal function technique, and the compactness method. We also consider the time-harmonic Maxwell’s equations and obtain the interior W1,p estimates.

scholarly journals EXPONENTIAL CONVERGENCE OF hp-FEM FOR MAXWELL EQUATIONS WITH WEIGHTED REGULARIZATION IN POLYGONAL DOMAINS

2005 ◽  
Vol 15 (04) ◽  
pp. 575-622 ◽  
Author(s):  
MARTIN COSTABEL ◽  
MONIQUE DAUGE ◽  
CHRISTOPH SCHWAB
Keyword(s):  
Finite Element ◽  
Maxwell’S Equations ◽  
Maxwell Equations ◽  
Maxwell's Equations ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Polygonal Domains ◽  
Time Harmonic ◽  
Standard Tool ◽  
Optimal Convergence Rates

The time-harmonic Maxwell equations do not have an elliptic nature by themselves. Their regularization by a divergence term is a standard tool to obtain equivalent elliptic problems. Nodal finite element discretizations of Maxwell's equations obtained from such a regularization converge to wrong solutions in any non-convex polygon. Modification of the regularization term consisting in the introduction of a weight restores the convergence of nodal FEM, providing optimal convergence rates for the h version of finite elements. We prove exponential convergence of hp FEM for the weighted regularization of Maxwell's equations in plane polygonal domains provided the hp-FE spaces satisfy a series of axioms. We verify these axioms for several specific families of hp finite element spaces.

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