On the Numerical Evaluation of Invertible Preconditioners Based on High-Order Discretizations of the Laplace Operator for Linear Systems

To alleviate the ill-posed condition of linear systems, the discrete Laplace operator is often used as a preconditioner incorporated with iterative methods. However, with a traditional lower-order(typically, second-order) approximation of the Laplace operator it is often difficult to achieve the optimal effect. For this reason, we construct preconditioners based on high-order finite difference discretizations to further develop the potential of the discrete Laplace operator and evaluate their numerical efficiency. The sparse band structure and symmetric property of such high-order preconditioners are derived from the theoretical aspect , revealing the cheap computing cost of the corresponding preconditioning processes for each problem dimension. Numerical experiments are implemented to confirm our analysis and the computing results show advantages of the proposed preconditioned iterative methods compared with the classical methods.