On the numerical approximation of some types of nonstandard second-order eigenvalue problems for vector valued functions

✍ Scribed by Hennie De Schepper; Roger Van Keer

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 147 KB
• Volume: 115
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(99)00122-3

✦ Synopsis

In this paper we consider some types of second-order elliptic eigenvalue problems (EVPs) for vector valued functions on a convex polygonal domain in the plane, with nonstandard boundary conditions (BCs) of nonlocal type. The aim of the paper is twofold. First, we pass to a variational form of the EVP, which is shown to be formally equivalent to the di erential EVP and which is proved to ÿt into the well-known general framework of abstract elliptic EVPs for bilinear forms in Hilbert spaces, treated, e.g., in Raviart, Thomas, Introduction  a l'analyse numà erique des à equations aux dà erivà ees partielles, 3rd Edition, Masson, Paris, 1992. This implies the existence of exact eigenpairs with suitable properties. Next, we study ÿnite element approximation methods for this problem. We argue that similar convergence results and error estimates hold as those established, e.g., in Dautray, Lions, Analyse numà erique et calcul numà erique pour les sciences et les techniques, tome 2, Masson, Paris, 1985, Chapitre 12, Raviart, Thomas, Introduction  a l'analyse numà erique des à equations aux dà erivà ees partielles, 3rd Edition, Masson, Paris, 1992 or Strang, Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cli s, NJ, 1973 for elliptic EVPs for a scalar function, with classical local BCs of Dirichlet, Neumann or Robin type.

Here the nonlocal character of the BCs constitutes a major di culty in the analysis, requiring the introduction and error estimation of a new, suitably modiÿed (vector) Lagrange interpolant on the FE-mesh. The theoretical error estimate for the eigenvalues is conÿrmed by an illustrative numerical example.