Two-dimensional curl-conforming singular elements for FEM solutions of dielectric waveguiding structures

Abstract

This paper proposes a curl-conforming singular element for modeling electromagnetic fields around singular points. Similar to the Nedelec types of regular vector elements, the space of the proposed singular elements consists of gradient and rotational subspaces. The proposed singular elements have arbitrary singularity orders that are precomputed analytically according to local geometry and material properties. The singularity orders of the gradient bases depend on the electric-field behavior; the rotational bases on magnetic-field behavior. Assigning integer singularity orders transforms singular elements into regular elements. Since the gradient subspace is properly modeled, the proposed singular elements are free from contamination by spurious modes. By including the singular elements in the solution space, deterioration of convergence rates often encountered with waveguides containing singular corners is avoided. Validation of the proposed singular elements is provided both theoretically in terms of the de Rham diagram and numerically by solving canonical singular dielectric waveguiding structures.