The present paper aims at analyzing the existence and convergence of approximate solutions in shape optimization. Motivated by illustrative examples, an abstract setting of the underlying shape optimization problem is suggested, taking into account the so-called two norm discrepancy. A Ritz-Galerkin-type method is applied to solve the associated necessary condition. Existence and convergence of approximate solutions are proved, provided that the infinite dimensional shape problem admits a stable second order optimizer. The rate of convergence is confirmed by numerical results.