Iterative solution of large-scale 3D-BEM industrial problems

In the present study we demonstrate that a standard direct boundary integral formulation and the collocation method result, for a wide range of practical applications, in systems of equations having properties that allow efficient use of Krylov iterative solvers with simple diagonal and block-diagonal preconditioning. In practical problems the matrix diagonal dominance, especially in its strong sense, is never observed when unstructured meshes and complex geometry are used. However, from our investigations we obtained that the diagonal dominance is not so important and matrix properties desired for a good convergence are rather associated with eigenvalues distribution. A simple diagonal scaling of a properly structured BEM matrix can improve this distribution dramatically. For industrial BEM packages the data sets are generated via preprocessors. For iterative solvers, in contrast to direct solvers, it is very important to prepare a system of equation in an appropriate form. We show that a correct treatment of boundary conditions and proper ordering of unknowns and equations are essential to obtain convergence. Extensive numerical experiments with preconditioned GMRES(m) and CGS methods for large practical problems have been carried out.