  We consider systems with slab geometry of finite thickness L that undergo
second order phase transitions in the bulk limit and belong to the universality
class of O(n)-symmetric systems with short-range interactions. In these systems
the critical fluctuations at the bulk critical temperature T_c induce a
long-range effective force called the "thermodynamic Casimir force". We
describe the systems in the framework of the O(n)-symmetric phi^4-model,
restricting us to the large-n limit n->infty. In this limit the physically
relevant case of three space dimensions d=3 can be treated analytically in
systems with translational symmetry as, e.g., in the bulk or slabs with
periodic or antiperiodic boundary conditions. We consider Dirichlet and open
boundary conditions at the surfaces that break the translational invariance
along the axis perpendicular to the slab. From the broken translational
invariance we conclude the necessity to solve the systems numerically. We
evaluate the Casimir amplitudes for Dirichlet and open boundary conditions on
both surfaces and for Dirichlet on one and open on the other surface. Belonging
to the same surface universality class we find the expected asymptotic
equivalence of Dirichlet and open boundary conditions. To test the quality of
our method we confirm the analytical results for periodic and antiperiodic
boundary conditions.
