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In this paper, a relaxed Hermitian and skew-Hermitian splitting (RHSS) preconditioner is proposed for saddle point problems from the element-free Galerkin (EFG) discretization method. The EFG method is one of the most widely used meshfree methods for solving partial differential equations. The RHSS preconditioner is constructed much closer to the coefficient matrix than the well-known HSS preconditioner, resulting in a RHSS fixed-point iteration. Convergence of the RHSS iteration is analyzed and an optimal parameter, which minimizes the spectral radius of the iteration matrix is described. Using the RHSS preconditioner to accelerate the convergence of some Krylov subspace methods (like GMRES) is also studied. Theoretical analyses show that the eigenvalues of the RHSS preconditioned matrix are real and located in a positive interval. Eigenvector distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are obtained. A practical parameter is suggested in implementing the RHSS preconditioner. Finally, some numerical experiments are illustrated to show the effectiveness of the new preconditioner.Mathematics subject classification: 65F10.

A smart laminate model developed using the nonlinear piezoelectric model, i.e., electroelastic and electrostrictive effects, and von Kármán type geometric nonlinearity is proposed. On the basis of the proposed model, the analytical solutions for extensional and bending deformations of symmetrically constructed piezoelectric laminates under large electric fields are calculated. A comparison between the present nonlinear extension deformation of a piezoelectric laminated plate and the induced strain model shows that more realistic results are obtained with the present model. Numerical simulation reveals that the calculated results agree well with the experimental data.

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