Finite element analysis of a coupling eigenvalue problem on overlapping domains

In this paper, we consider a nonstandard elliptic eigenvalue problem on a rectangular domain, consisting of two overlapping rectangles, where the interaction between the subdomains is expressed through an integral coupling condition on their intersection. For this problem we set up finite element (FE) approximations, without and with numerical quadrature. The involved error analysis is affected by the nonlocal coupling condition, which requires the introduction and error estimation of a suitably modified vector Lagrange interpolant on the overall FE mesh. As a consequence, the resulting error estimates are sub-optimal, as compared to the ones established, e.g., in Vanmaele and van Keer (RAIRO – Math. Mod. Num. Anal 29(3) (1995) 339–365) for classical eigenvalue problems with local boundary or transition conditions.