SUMMARYThis paper is concerned with the structure of the singular and regular parts of the solution of timeharmonic Maxwell's equations in polygonal plane domains and their e ective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H 2 when the boundary of the domain has non-acute angles. A splitting of the solution into a regular part belonging to the space H 2 , and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a ÿnite element discretization which combines local mesh grading and the singular ÿeld methods and derive a priori error estimates that show optimal convergence as known for the classical ÿnite element method for problems with regular solutions.
AbstractIn [Nkemzi and Jung, 2013] explicit extraction formulas for the computation of the edge flux intensity functions for the Laplacian at axisymmetric edges are presented. The present paper proposes a new adaptation for the Fourier-finite-element method for efficient numerical treatment of boundary value problems for the Poisson equation in axisymmetric domains Ω̂ ⊂ ℝ3 with edges. The novelty of the method is the use of the explicit extraction formulas for the edge flux intensity functions to define a postprocessing procedure of the finite element solutions of the reduced boundary value problems on the two-dimensional meridian of Ω̂. A priori error estimates show that the postprocessing finite element strategy exhibits optimal rate of convergence on regular meshes. Numerical experiments that validate the theoretical results are presented.
In this paper we present the basic mathematical tools for treating boundary value problems for the Maxwell equations in three‐dimensional axisymmetric domains with reentrant edges using the Fourier‐finite‐element method. We consider both the classical and the regularized time‐harmonic Maxwell equations subject to perfect conductor boundary conditions. The partial Fourier decomposition reduces the three‐dimensional boundary value problem into an infinite sequence of two‐dimensional boundary value problems in the plane meridian domain of the axsiymmetric domain. Here, suitable weighted Sobolev spaces that characterize the solutions of the reduced problems are given, and their trace properties on the rotational axis are proved. In these spaces, it is proved that the reduced problems are well posed, and the asymptotic behavior of the solutions near reentrant corners of the meridian domain is explicitly described by suitable singularity functions. Finally, a finite number of the two‐dimensional problems is considered and treated using H1‐conforming finite elements. An approximation of the solution of the three‐dimensional problem is obtained by Fourier synthesis. For domains with reentrant edges, the singular field method is employed to compensate the singular behavior of the solutions of the reduced problems. Emphases are given to convergence analysis of the combined approximations in H1 under different regularity assumptions on the solution.
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