On the generalized Calderón formulas for closed- and open-surface elastic scattering problems

Abstract

The Calderón formulas (i.e., the combination of single-layer and hyper-singular boundary integral operators) have been widely utilized in the process of constructing valid boundary integral equation systems which could possess highly favorable spectral properties. This work is devoted to studying the theoretical properties of elastodynamic Calderón formulas which provide us with a solid basis for the design of fast boundary integral equation methods solving elastic wave problems defined on a close-surface or an open-surface in two dimensions. For the closed-surface case, it is proved that the Calderón formula is a Fredholm operator of second-kind except for certain circumstances.Regarding to the open-surface case, we investigate weighted integral operators instead of the original integral operators which are resulted from dealing with edge singularities of potentials corresponding to the elastic scattering problems by open-surfaces, and show that the Calderón formula is a compact perturbation of a bounded and invertible operator. To complete the proof, we need to use the wellposedness result of the elastic scattering problem, the analysis of the zero-frequency integral operators defined on the straight arc, the singularity decompositions of the kernels of integral operators, and a new representation formula of the hyper-singular operator. Moreover, it can be demonstrated that the accumulation point of the spectrum of the invertible operator is the same as that of the eigenvalues of the Calderón formula in the closed-surface case.

Spectral Galerkin method for solving elastic wave scattering problems with multiple open arcs

“…Then, one requires suitable preconditioners or regularized BIEs to construct more efficient numerical solvers. This topic has received attention in recent years [3,9,11,12] and some attempts have been carried out to tackle elastic wave problems [4][5][6]29] by considering the composition of the weakly-and hyper-singular boundary integral operators (BIOs). Indeed, including preconditioning techniques into the presented numerical method is relevant but for the sake for brevity will be left as future work.…”