An adaptive Dirichlet-to-Neumann finite element method for the thermoelastic scattering problem

Abstract

This paper presents the analysis and computation of an adaptive Dirichlet-to-Neumann (DtN) finite element method for solving the two-dimensional thermoelastic wave scattering problem. Using the Helmholtz decomposition, the vectorial coupled governing equations of thermoelastic waves are transformed into three Helmholtz equations for scalar potentials with distinct wavenumbers. The DtN map and the corresponding transparent boundary condition are derived through Fourier series expansions of the potentials. Well-posedness results are established for both the variational problem and its truncated formulation, which accounts for the truncation of the DtN map. Both a priori and a posteriori error estimates are established, accounting for the truncation of the DtN operator and the finite element discretization. Numerical experiments are conducted to validate the theoretical findings.