The generalized finite element method for Helmholtz equation. Part II: Effect of choice of handbook functions, error due to absorbing boundary conditions and its assessment
In Part I [T. Strouboulis, I. Babuka, R. Hidajat, The generalized finite element method for Helmholtz equation: theory, computation, and open problems, Comput. Methods Appl. Mech. Engrg. 195 (2006) 47114731] we introduced the q-version of the generalized finite element method (GFEM) for the Helmholtz equation and we addressed its: (a) pollution error due to the wave number; (b) exponential q-convergence; (c) robustness to perturbations of the mesh, the roundoff and numerical quadrature errors; and (d) a-posteriori error estimation. Here we continue the development of the GFEM for Helmholtz and we address the effects of: (a) alternative handbook functions and mesh types; (b) the error due to the artificial truncation boundary conditions and its assessment. The conclusions are: (1) the employment of plane-wave, wave-band, and Vekua handbook functions lead to equivalent results; and (2) for high q, the most significant component of error may be the one due to the artificial truncation boundary conditions. A rather straightforward approach for assessing this error is proposed.
helmholtz equation handbook functions generalized finite element method (gfem) plane-wave versus wave-band functions error due to artificial truncation boundary condition multiple scatterers