The use of thermal waves in photothermal techniques for non-destructive testing of composite materials is taken as an example for the time-harmonic diffusion problem that is discussed. The problem requires to take into account materials with a constant conductivity and density in the exterior of a bounded region and non-homogeneous media of the interior one. The mathematical formulation results in a Helmholtz transmission problem with absorption, i.e., the wave numbers are complex, and a region where the coefficients of the equation are not constant.
To get a numerical solution a boundary element method (BEM) finite element method (FEM) discretization is desribed consisting of a coupled formulation with a system of boundary integral equations for the exterior area and a variational form of the interior one. Piecewise constant functions and lowest order Raviart-Thomas elements are used for the scalar field and the flux, respectively, based on a triangular grid. Spaces of periodic splines are taken for the boundary unknowns to approximate easily the weakly singular boundary integrals by quadrature rules.
The BEM and FEM routines work independently. The coupling is done by a proper location of the exterior nodes with respect to the boundary grid. To prove the well-posedness the variational problem is transformed into an equivalent operator equation. Stability and convergence are proved for the algorithm comparing it with an auxiliary BEM-FEM method where curved triangles are used. The method is illustrated by academic examples.