A preconditioning strategy for the solution of linear boundary element systems using the GMRES method
The original system is \(Hx=C\) and the authors try to find a nonsingular upper triangular matrix \(A\) such that \(AH= \overline{D}+ \overline{L}+ \overline{V}\), where \(\overline{D}\) and \(\overline{L}\) are diagonal and strictly lower triangular parts of \(AH\) and the purpose is to minimize \(|\overline{V}+ \overline{D}- D|_\infty\) or, in practice, the Frobenius norm \(|\overline{V}+ \overline{D}-D|_F\). This means that the preconditioned system is taken in the form \[ M^{-1}AHx= M^{-1}AC, \qquad M\equiv D+\overline{L}; \] \[ |M^{-1}AH-I|\leq|(I+D^{-1} \overline{L})^{-1}|\cdot|D^{-1}(\overline{V}+ \overline{D}-D)|. \] Numerical experiments associated with the GMRES method are presented; a 20-fold reduction in time was obtained in comparison with the Gauss method for systems with approximately 1300 unknowns.
scientific article; zbMATH DE number 1036643
scientific article