Flexible membranes which adhere to a substrate surface along striped surface domains were theoretically studied using a "superposition" approach and Monte Carlo (MC) simulations. In the "superposition" approach, the fluctuation-induced repulsion between the membrane and the nonadhesive surface domains was approximately taken into account as a local potential. A detailed comparison of this superposition approach and the MC data shows that the strength of the fluctuation-induced potential depends both on the boundary conditions and on the physical quantity under consideration and, thus, must be viewed as a fit parameter. The shape of the adhering membrane which consists of unbound membrane arches between bound membrane segments is accessible to experiments using, for example, reflection interference microscopy.