This paper presents a study of efficiencies of a few preconditioned GMRES solvers for large dense nonsymmetric linear system of equations. Generalized minimal residual (GMRES) has been known as the most efficient technique by minimizing the residual vector in the Krylov subspace. The matrix generated by the higher-order boundary element method (HOBEM) is usually very dense and nonsymmetric. In order to make GMRES more efficient, Jacob, SSOR, Incomplete Cholesky factorization and Neumann polynomial are selected as preconditioners. Numerical experiments are carried out and the results are compared with Gauss elimination method. It has been found that Jacob type preconditioner of GMRES can save large amount of CPU time, especially for large nonsymmetric system of equations. The performance can be further improved dramatically by optimization of the code. Optimal Jac-GMRES can be 234 times faster than Gauss elimination without optimization when the matrix size is 1024. It is our objective to find the best preconditioning scheme for GMRES for efficient use of HOBEM and HOBEM-based numerical wave tank simulation.
The boundary element method (BEM) has been becoming increasingly popular for the analysis of marine hydrodynamic problems, during the last few years. Since higher-order boundary element method (HOBEM) uses much fewer elements and much less CPU time with higher accuracy than constant panel method (CPM) (Lieu et al, 1990, applications of HOBEM are becoming more significant (Lieu et al, 1992, 1993 A, B; Xu et al, 1992; Lee et al, 1994; Boo et al, 1994). One of the major drawbacks of BEM is that one has to solve nonsymmetric and very dense matrix equation. The matrices produced by the finite difference method (FDM) and finite element method (FEM) possess a kind of regular structure e.g. symmetric positive definite (SPD) and are very sparse. Thus the solutions for these problems are well developed and highly efficient. But this is not the case for the BEM.
