Cell-by-cell approximate Schur complement technique in preconditioning of meshfree discretized piezoelectric equations

The radial point interpolation meshfree discretization is a very efficient numerical framework for the analysis of piezoelectricity, in which the fundamental electrostatic equations governing piezoelectric media are solved without mesh generation. Due to the mechanical-electrical coupling property and the piezoelectric constant, the discrete linear system is sparse, of generalized saddle point form and often very ill conditioned. In this work, we propose a technique for constructing a family of cell-by-cell approximate Schur complement matrices, to be used in preconditioning to accelerate the convergence of Krylov subspace iteration methods for such problems. The approximate Schur complement matrices are simply and cheaply constructed in the process of the meshfree discretization and have a sparse structure. It is proved that the so-constructed approximate Schur complement matrices are spectrally equivalent to the exact Schur complement matrix, which leads to very fast convergence when used in preconditioning. In addition, nondimensionalization of the piezoelectric equations is considered to make the computations more stable. The robustness and the efficiency of the proposed preconditioners is illustrated numerically on two test problems, arising from a piezoelectric strip shear deformation problem and a piezoelectric strip bending problem. Numerical results show that the number of iterations to achieve a given tolerance is independent of the number of degrees of freedom as well as of the various problem parameters.