An Efficient Algorithm for Solving Eigenproblem

In this paper, we consider second order elliptic ODE eigen problems on general grids. We construct an efficient algorithm for computing the eigen value by using weighted mean combination of the linear finite element method and corresponding 2nd-order finite difference method. We first take the arithmetic mean of the two methods. Then we compute the quasi-optimal combined parameters for different eigen values to improve our efficient algorithm. The algorithm we construct convergence faster and have higher accuracy than the linear finite element method and corresponding 2nd-order finite difference method. Some numerical examples tested on both uniform meshes and nonuniform meshes are given to illustrate the computational cost of different numerical methods for solving eigen value problems. For efficiency, all the matrices use sparse storage in our algorithm.