参数化泊松方程的模型降阶预处理OA

This paper aims at the fast numerical solution of parameterized Poisson equation.A preconditioned conjugate gradient(PCG)method based on the model order reduction method is proposed to speed up the so-lution of the obtained discrete linear system.First,the general formulation of preconditioning matrix based on model order reduction is designed,and the symmetry and positive definition of the matrix are proved.In the off-line stage of the method,by using very few solution data,the PCG algorithm combined with the proper orthogonal decomposition(POD)method are adopted to generate a set of dynamic preconditioning matrices.In the on-line stage,the MPCG algorithm is proposed by combining the PCG algorithm and these dynamic preconditioning matrices.To verify the performance of the method,parameterized Poisson equations on the unit rectangular and L-shaped domains are numerically solved.It is shown that,in comparison with the stan-dard CG algorithm,the average computation time is speeded up by more than 43 times with the same calcula-tion accuracy.