Graded meshes for the BEM approximation of 2D mixed elastodynamic problems

In this contribution we take into account the numerical solution of elastodynamic wave propagation problems in a bounded polygonal domain with mixed boundary conditions. The mathematical model is based on the resolution of the Navier-Lamè equation, whose unknown represents the displacement field of the elastic body. This field exhibits singularities at both corners and points where the boundary conditions change [1,2], hence, asymptotic behavior is theoretically investigated, deriving it in particular from a detailed study of the Dirichlet trace of the solution and the traction at the boundary. The analysis leads to quasi-optimal error estimates for piecewise polynomial approximations. The numerical solution of the overall problem is obtained by a Boundary Element Method, based specifically on the representation of the relevant physical fields in terms of layer potentials. The resulting Boundary Integral Equations (BIEs) involve the single layer, double layer and hypersingular boundary integrals operators and provide a convenient reduction of the dimensional complexity. The BIEs are weakly reformulated through energy arguments and numerically solved in virtue of a space-time discretization of Galerkin type (E-BEM). The linear system is characterized by a block Toeplitz structure with symmetric blocks, arising from a time discretization with Lagrange basis functions. The matrix entries are computed using numerical quadrature strategies for the computation of the well-known space integrals for the BIE system. Accuracy and long-time stability of the energetic Galerkin approach is proved for 2D interior problems [3] and 2D contact problems [4], making the use of boundary meshes strategically graded towards points of singularity the step-forward of the presented work, leading to quasi-optimal numerical approximations that improve the slow convergence and demanding computational costs on uniform meshes. Numerical examples, performed in virtue of the E-BEM approach, prove the theoretical results for 2D polygonal geometries, giving a numerical confirmation of the quasi-optimal convergence rates.