The parameter dependent Brinkman equation can be used to model viscous and porous flow. The equation covers a family of problems, ranging from the Stokes problem to the Darcy problem. We apply the stabilization methods; the pressure stabilized petrov--galerkin (PSPG) method and the continuous interior penalty (CIP) method, on the Brinkman equation with weakly imposed boundary conditions by the Nitsche method. An a priori error estimate is proved for the CIP stabilization with the Nitsche method. We look at operator preconditioning for these stabilized problems with appropriate norms. We also explore a fictitious domain method and stabilize the cut elements with ghost-penalties.