Modern engineering applications require efficient methods for solving partial differential equations on complex geometries. A major issue in this context is the construction of the computational grid. Common methods require a mesh that is fitted to the physical domain, which can be very time consuming to be constructed. To eliminate this requirement, one can embed the domain into a larger domain that is simpler to be meshed. This is the philosophy behind the well-known fictitious domain method. In this approach, the physical boundary conditions are imposed weakly by introducing suitable auxiliary variables supported on the embedded boundary. In particular, for Dirichlet problems the main challenge is to design discrete spaces ensuring a uniform discrete inf–sup condition. To this end, we employ the use of element-level bubble functions, originally introduced to stabilize finite element computations. The presentation will show a novel method for solving Dirichlet problems in the spirit of [1], extended to a wide class of regular meshes. This extension is made possible by an enhancement of the classical P1–P0 choice of discrete spaces. In particular, depending on the position of the embedded boundary with respect to the mesh constructed on the larger domain, we enrich the local spaces with two different types of bubble functions. Moreover, we relax the requirement introduced in [1] concerning the ratio between the boundary mesh size and the domain mesh size, while preserving discrete inf–sup stability. This is achieved through the use of a restriction operator based on bubble functions and an algorithm for the construction of the boundary mesh. Classical error bounds will be presented, supported by some numerical results. [1] Girault V., Glowinski R., Error analysis of a fictitious domain method applied to a Dirichlet problem, Japan Journal of Industrial and Applied Mathematics, 12(3), 487-514, 1995.