A new numerical method, called Robin boundary condition iteration (RBCI), is proposed for the finite-element (FE) solution of electromagnetic scattering problems in open boundary domains. The unbounded domain is truncated to a bounded one by means of a fictitious boundary that contains the scatterer and on which a suitable nonhomogeneous Robin (mixed) boundary condition is assumed for the Helmholtz equation in the bounded domain. The Robin condition is expressed by means of an integral formula (the second Green identity) in terms of the field in the interior of the bounded domain, with the integration surface being a surface strictly enclosed by the truncation boundary. The discretized differential and integral equations are then coupled together to solve the problem. The formulation is completely immune from the well-known interior resonance problems. A simple and effective iterative solving scheme is described. Examples are also provided to validate RBCI and compare it with other methods. Index Terms-Finite-element (FE) methods, integral equations, scattering.
SUMMARYThree algebraic multigrid (AMG) methods for solving generalized eigenvalue problems are presented. The first method combines modern AMG techniques with a non-linear multigrid approach and nested iteration strategy. The second method is a preconditioned inverse iteration with linear AMG preconditioner. The third method is an enhancement of the previous one, namely the locally optimal block preconditioned conjugate gradient. Efficiency and accuracy of solutions computed by these AMG eigensolvers are validated on standard benchmarks where part of the spectrum is known. In particular, the problem of isospectral drums is addressed.
An iterative procedure is described for the finite-element solution of scalar scattering problems in unbounded domains. The scattering objects may have multiple connectivity, may be of different materials or with different boundary conditions. A fictitious boundary enclosing all the objects involved is introduced. An appropriate Robin (mixed) condition is initially guessed on this boundary and is iteratively improved making use of Green's formula. It will be seen that the best choice for the Robin boundary condition is an absorbing-like one. A theorem about the theoretical convergence of the procedure is demonstrated. An analytical study of the special case of a circular cylindrical scatterer is made. Comparisons are made with other methods. Some numerical examples are provided in order to illustrate and validate the procedure and to show its applicability whatever the frequency of the incident wave. Although particular emphasis is laid in the paper on electromagnetic problems, the procedure is fully applicable to other kinds of physical phenomena such as acoustic ones.1998 John Wiley & Sons, Ltd.
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