The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators

Abstract

This paper discusses the order-preserving convergence for spectral approximation of the self-adjoint completely continuous operator T. Under the condition that the approximate operator Th converges to T in norm, it is proven that the k-th eigenvalue of Th converges to the k-th eigenvalue of T. (We sorted the positive eigenvalues in decreasing order and negative eigenvalues in increasing order.) Then we apply this result to conforming elements, nonconforming elements and mixed elements of self-adjoint elliptic differential operators eigenvalue problems, and prove that the k-th approximate eigenvalue obtained by these methods converges to the k-th exact eigenvalue.