Sign-preserving of principal eigenfunctions in P1 finite element approximation of eigenvalue problems of second-order elliptic operators

This paper is concerned with the P1 finite element approximation of the eigenvalue problem of second-order elliptic operators subject to the Dirichlet boundary condition. The focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem. It is shown that when the stiffness matrix is an irreducible M-matrix, the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain. Mesh conditions leading to such a stiffness matrix are also studied. A sufficient condition is that the mesh is simplicial, interiorly connected, and acute when measured in the metric specified by the inverse of the diffusion matrix. The acute requirement can be replaced by the Delaunay condition in two dimensions. Numerical results show that when the stiffness matrix is not an M-matrix, a finite element approximation can be structurally different from the continuous eigenvalue problem: the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue (in modulus) can even be complex for the case with nonsymmetric operators.