Weighted integral solvers for elastic scattering by open arcs in two dimensions

We present new methodologies for the numerical solution of problems of elastic scattering by open arcs in two dimensions. The algorithms utilize weighted versions of the classical elastic integral operators associated with Dirichlet and Neumann boundary conditions, where the integral weight accounts for (and regularizes) the singularity of the integral-equation solutions at the open-arc endpoints. Crucially, the method also incorporates a certain open-arc elastic Calderon relation introduced in this article, whose validity is demonstrated on the basis of numerical experiments, but whose rigorous mathematical proof is left for future work. (In fact, the aforementioned open-arc elastic Calderon relation generalizes a corresponding elastic Calderon relation for closed surfaces, which is also introduced in this article, and for which a rigorous proof is included.) Using the open-surface Calderon relation in conjunction with spectrally accurate quadrature rules and the Krylov-subspace linear algebra solver GMRES, the proposed overall open-arc elastic solver produces results of high accuracy in small number of iterations, for both low and high frequencies. A variety of numerical examples in this article demonstrate the accuracy and efficiency of the proposed methodology.

Automated summary

New methodologies for solving problems of elastic scattering by open arcs in two dimensions are presented in this article, utilizing weighted versions of classical elastic integral operators, a certain open-arc elastic Calderon relation, spectrally accurate quadrature rules, and the linear algebra solver GMRES. Results of high accuracy are achieved in a small number of iterations for both low and high frequencies, as demonstrated by various numerical examples in the article.