  We present a new type of approximation of a second-order elliptic operator in
a planar domain with a point interaction. It is of a geometric nature, the
approximating family consists of operators with the same symbol and regular
coefficients on the domain with a small hole. At the boundary of it Robin
condition is imposed with the coefficient which depends on the linear size of a
hole. We show that as the hole shrinks to a point and the parameter in the
boundary condition is scaled in a suitable way, nonlinear and singular, the
indicated family converges in the norm-resolvent sense to the operator with the
point interaction. This resolvent convergence is established with respect to
several operator norms and order-sharp estimates of the convergence rates are
provided.
