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This paper presents novel methodologies for the numerical simulation ofscattering of elastic waves by both closed and open surfaces inthree-dimensional space. The proposed approach utilizes new integralformulations as well as an extension to the elastic context of the efficienthigh-order singular-integration methods~\cite{BG18} introduced recently for theacoustic case. In order to obtain formulations leading to iterative solvers(GMRES) which converge in small numbers of iterations we investigate,theoretically and computationally, the character of the spectra of variousoperators associated with the elastic-wave Calder\'on relation---including someof their possible compositions and combinations. In particular, by relying onthe fact that the eigenvalues of the composite operator $NS$ are bounded awayfrom zero and infinity, new uniquely-solvable, low-GMRES-iteration integralformulation for the closed-surface case are presented. The introduction ofcorresponding low-GMRES-iteration equations for the open-surface equationsadditionally requires, for both spectral quality as well as accuracy andefficiency, use of weighted versions of the classical integral operators tomatch the singularity of the unknown density at edges. Several numericalexamples demonstrate the accuracy and efficiency of the proposed methodology.

Regularized integral equation methods for elastic scattering problems in three dimensions