Volume 3 (2016) Issue 2 Pages 188-205
In an extended element-free Galerkin method (X-EFG), the essential and natural boundary conditions can be imposed by the collocation based method. However, a coefficient matrix of linear systems obtained by X-EFG become asymmetric, although a symmetric structure exists in a part of the coefficient matrix. In fact, the structure of the coefficient matrices almost becomes symmetric when the size of linear systems is large. Hence, efficient effects may be obtained by using preconditioning for symmetric matrices. The purpose of the present study is to investigate effects of preconditioning for symmetric matrices to linear systems obtained by X-EFG. To this end, the incomplete Cholesky factorization (IC) is applied to the linear systems by regarding the coefficient matrix as symmetric one. In numerical experiments, it is found that the linear systems obtained by X-EFG can be efficiently solved by using IC as preconditioning for GMRES(m) and Bi-CGSTAB. In some cases, the efficiency of IC is superior than that of the incomplete LDU factorization as preconditioning for these iterative methods.