Numerical Solution to the 3D Static Maxwell Equations in Axisymmetric Singular Domains with Arbitrary Data

Abstract We propose a numerical method to solve the three-dimensional static Maxwell equations in a singular axisymmetric domain, generated by the rotation of a singular polygon around one of its sides. The mathematical tools and an in-depth study of the problem set in the meridian half-plane are exposed in [F. Assous, P. Ciarlet, Jr., S. Labrunie and J. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, J. Comput. Phys. 191 2003, 1, 147–176] and [P. Ciarlet, Jr. and S. Labrunie, Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier singular complement method, Differ. Equ. Appl. 3 2011, 1, 113–155]. Here, we derive a variational formulation and the corresponding approximation method. Numerical experiments are proposed, and show that the approach is able to capture the singular part of the solution. This article can also be viewed as a generalization of the Singular Complement Method to three-dimensional axisymmetric problems.

[1]  P. Ciarlet,et al.  Solution of axisymmetric Maxwell equations , 2003 .

[2]  F. Assous,et al.  Solving Numerically the Static Maxwell Equations in an Axisymmetric Singular Geometry , 2015 .

[3]  Ronald H. W. Hoppe,et al.  Finite element methods for Maxwell's equations , 2005, Math. Comput..

[4]  P. Hansbo,et al.  A finite element method for domain decomposition with non-matching grids , 2003 .

[5]  P. Werner,et al.  A local compactness theorem for Maxwell's equations , 1980 .

[6]  B. Heinrich,et al.  The Fourier-Finite-Element Method for Poisson's Equation in Axisymmetric Domains with Edges , 1996 .

[7]  Qiang Chen,et al.  Introduction to Applications of Numerical Analysis in Time Domain Computational Electromagnetism , 2011 .

[8]  Boniface Nkemzi,et al.  Optimal convergence recovery for the Fourier-finite-element approximation of Maxwell's equations in nonsmooth axisymmetric domains , 2005 .

[9]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[10]  Jun Zou,et al.  Finite element convergence for the Darwin model to Maxwell's equations , 1997 .

[11]  Franck Assous,et al.  Theoretical tools to solve the axisymmetric Maxwell equations , 2002 .

[12]  Franck Assous,et al.  Résolution des équations de Maxwell dans un domaine avec un coin rentrant , 1996 .

[13]  Singular electromagnetic fields: inductive approach , 2005 .

[14]  Franck Assous,et al.  Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method , 2003 .

[15]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[17]  B. Mercier,et al.  Résolution d’un problème aux limites dans un ouvert axisymétrique par éléments finis en $r, z$ et séries de Fourier en $\theta $ , 1982 .

[18]  Jae Ryong Kweon,et al.  The Fourier-finite element method for the Poisson problem on a non-convex polyhedral cylinder , 2009, J. Comput. Appl. Math..

[19]  Franck Assous,et al.  Numerical Solution to the Time-Dependent Maxwell Equations in Two-Dimensional Singular Domains , 2000 .

[20]  Pierre Degond,et al.  On a finite-element method for solving the three-dimensional Maxwell equations , 1993 .

[21]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[22]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[23]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

[24]  Jun Zou,et al.  The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains , 2005, Numerische Mathematik.

[25]  Susanne C. Brenner,et al.  An Adaptive P1 Finite Element Method for Two-Dimensional Maxwell’s Equations , 2013, J. Sci. Comput..

[26]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[27]  J. P. Benque,et al.  A finite element method for Navier-Stokes equations , 1980 .

[28]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[29]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[30]  M. Birman,et al.  L2-Theory of the Maxwell operator in arbitrary domains , 1987 .

[31]  Frédéric Hecht,et al.  A TRUNCATED FOURIER/FINITE ELEMENT DISCRETIZATION OF THE STOKES EQUATIONS IN AN AXISYMMETRIC DOMAIN , 2006 .

[32]  P. Ciarlet,et al.  Numerical solution of Maxwell's equations in axisymmetric domains with the Fourier Singular Complement Method , 2011 .

[33]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[34]  Jun Zou,et al.  The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains , 2006, Numerische Mathematik.

[35]  Franck Assous,et al.  Mathematical Foundations of Computational Electromagnetism , 2018 .

[36]  Franck Assous,et al.  The Singular Complement Method , 2001 .

[37]  Joseph E. Pasciak,et al.  A mixed method for axisymmetric div-curl systems , 2008, Math. Comput..

[38]  M. Costabel,et al.  Singularities of Electromagnetic Fields¶in Polyhedral Domains , 2000 .

[39]  Bernd Heinrich,et al.  Fourier-finite-element approximation of elliptic interface problems in axisymmetric domains , 1996 .

[40]  I. Yousept.  Optimal control of quasilinear H(curl)-elliptic partial differential equations in magnetostatic field problems , 2013 .

[41]  M. Costabel.  A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains , 1990 .

[42]  Monique Dauge,et al.  Spectral Methods for Axisymmetric Domains , 1999 .

[43]  Irwin Yousept,et al.  Optimal Control of Quasilinear $\boldsymbol{H}(\mathbf{curl})$-Elliptic Partial Differential Equations in Magnetostatic Field Problems , 2013, SIAM J. Control. Optim..

[44]  Patrick Ciarlet,et al.  Résolution des équations de Maxwell instationnaires avec charges dans un domaine singulier bidimensionnel , 2000 .

[45]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[46]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .