Hierarchical bases on the standard and dual graph for stable solutions of the EFIE operator

This paper presents a dual generalized Haar basis for the electric field integral equation (EFIE) that regularizes the vector and the scalar potential on structured and unstructured meshes. Hierarchical preconditioners that regularize both potentials of the EFIE operator have been developed for structured meshes (for example obtained by a dyadic mesh refinement), but not for unstructured ones. In this contribution, we leverage graph Laplacians to transform the scalar potential into a single layer potential, while the vector potential is first transformed into the hypersingular operator and then into an operator that is equivalent to the single layer potential up to a compact perturbation by using the inverse Laplace-Beltrami operator. Then generalized Haar bases constructed from graph Laplacians of the primal and dual mesh are applied. Notice that the new preconditioner maintains the leading complexity set by fast matrix-vector multiplication methods. The presented results demonstrated the validity and effectiveness of the proposed approach and highlight the necessity thereof.