Abstract
The overall efficiency and accuracy of standard finite element methods may be severely reduced if the solution of the boundary value problem entails singularities. In the particular case of time-harmonic Maxwell’s equations in nonconvex polygonal domains Ω, H1-conforming nodal finite element methods may even fail to converge to the physical solution. In this paper, we present a new nodal finite element adaptation for solving time-harmonic Maxwell’s equations with perfectly conducting electric boundary condition in general polygonal domains. The originality of the present algorithm lies in the use of explicit extraction formulas for the coefficients of the singularities to define an iterative procedure for the improvement of the finite element solutions. A priori error estimates in the energy norm and in the L2 norm show that the new algorithm exhibits the same convergence properties as it is known for problems with regular solutions in the Sobolev space H2Ω2 in convex and nonconvex domains without the use of graded mesh refinements or any other modification of the bilinear form or the finite element spaces. Numerical experiments that validate the theoretical results are presented.
Highlights
Refer [20, 21] for some other recent adaptive finite element approaches. e common future of all the abovementioned methods is that they are applied to time-harmonic Maxwell’s equations on two-dimensional domains with reentrant corners in order to enforce the convergence of the nodal finite element approximations to the physical solution, but the accuracy of the computed solutions is still not optimal as one will expect for problems with regular solutions
E main purpose of the present paper is fourfold: firstly, to propose extraction formulas for the computation of the coefficients of the singularities for time-harmonic Maxwell’s equations in general polygonal domains; secondly, to present an adaptation of the standard nodal linear finite element approximation of the solution on quasiuniform meshes; thirdly, to show by means of a priori error estimates that the presented algorithm is efficient and the rate of convergence is optimal as it is known for problems with solutions u ∈ H2(Ω)2; lastly, to present numerical experiments that validate the theoretical results
We have presented extraction formulas for the coefficients of the singularities of solutions of time-harmonic Maxwell’s equations in two-dimensional domains with corners
Summary
2.1. e Model Problem. e classical time-harmonic Maxwell equations in a bounded and connected polygonal domain Ω ⊂ R2 with boundary Γ ≔ zΩ, representing a homogeneous isotropic medium and satisfying perfectly conducting electric boundary condition, are (cf. [1, 2, 14, 16, 26,27,28]). We note that if ωj > π, the functions sj from (11) belong to the space H(curl, div, Ω) but not to the space H1(Ω)2 It follows immediately from (4) and (12) and the fact that the functions sjl are harmonic that w, the regular part of u, is the unique solution of the boundary value problem: yj ΓJ = Γ0. We observe from (14) that if the coefficients cjl were known, the solution w ∈ HN(Ω) ∩ H2(Ω) can be optimally approximated by means of the standard H1-conforming nodal finite element methods, irrespective of the singular nature of the exact solution u ∈ H0(curl, div, Ω) of (7) In this case, an approximation of u is obtained using relation (12). If higher order polynomials are to be employed, this type of singularity reduces the accuracy
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