The numerical solution of partial differential equations is one of the prime tasks in scientific computing. The two most important challenges in this area are the increasing size of the models and the ever-stricter accuracy requirements. To handle these difficult problems in an acceptable amount of time, it is necessary to use powerful computational techniques that utilize the available hardware in an efficient way. Under today’s conditions, this usually includes parallel algorithms.
The Helmholtz equation--a special case of such problems--has important applications in, for example, the modeling of wave propagation and scattering phenomena. For this equation set on the exterior of a bounded domain, Boubendir et al. propose and analyze a new efficient method. The algorithm is based on first truncating the originally unbounded domain where the solution is sought via the introduction of artificial absorbing boundary conditions, and then introducing a nonoverlapping decomposition of the remaining finite domain into a number of subdomains. Up to this point, the method follows the commonly pursued approaches.
One of the key points of the algorithm is the selection of transmission operators that link the individual subdomains. This transmission is based on square roots of suitably chosen regularized surface divergence operators. As these operators are of a nonlocal nature, and hence unsuitable for being handled computationally in an efficient way, they are approximated using Padé approximation techniques, thus removing the nonlocality. For the solution of the resulting linear problems, numerical experiments then indicate that the generalized minimal residual method (GMRES) performs in a highly satisfactory way. The authors’ detailed analysis demonstrates the very good convergence properties of the complete algorithm. Detailed numerical examples support these theoretical results.
The paper is likely to be of interest to researchers dealing with the simulation of problems modeled by the Helmholtz equation. It could also provide useful information for the development of efficient algorithms for related problems.