二阶椭圆算子特征值问题P1有限元近似中主特征函数的符号保持性
Sign-preserving of principal eigenfunctions in P1 finite element approximation of eigenvalue problems of second-order elliptic operators
AI总结：
本文研究了二阶椭圆算子在Dirichlet边界条件下的P1有限元近似，探讨了主特征值及特征函数的基本性质保持问题，提出当刚度矩阵为不可约M矩阵时，代数特征值问题保持最小特征值实简单性和特征函数在物理域内符号一致性的条件。
摘要：
本文研究了二阶椭圆算子在Dirichlet边界条件下的P1有限元近似，探讨了主特征值及特征函数的基本性质保持问题。证明当刚度矩阵为不可约M矩阵时，代数特征值问题保持最小特征值为实简单值，对应的特征函数在物理域内为正或负。研究了导致此类刚度矩阵的网格条件，提出网格应为单纯形、在扩散矩阵逆指定的度量下为锐角，并且内部连通。二维情况下锐角要求可被Delaunay条件替代。数值结果用于验证理论发现。
英文摘要：
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 basic properties of the principal eigenvalue and eigenfunctions of continuous problems. It is shown that when the stiffness matrix is an irreducible $M$-matrix, the algebraic eigenvalue problem maintains those 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, acute when measured in the metric specified by the inverse of the diffusion matrix, and interiorly connected. The acute requirement can be replaced by the Delaunay condition in two dimensions. Numerical results are presented to verify the theoretical findings.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
