Summary: This paper presents a numerical study of a recent technique that consists in modeling embedded geometries by a level-set representation in combination with local anisotropic mesh refinement. The local anisotropic mesh procedure is suitable for various orders \(p\) of finite element approximations. This method proves beneficial in simulations involving complex geometries, as it suppresses the need for the tedious process of body-fitted mesh generation, without altering the finite element formulation nor the prescription of boundary conditions. The first part of the study deals with a simple Laplace problem featuring a planar interface on which a Dirichlet boundary condition is imposed. It is shown that the appropriate amount of local isotropic refinement yields the optimal convergence rate for various finite element orders \(p\), unlike uniform refinement. Anisotropic refinement further ensures geometric convergence and limits the growth of the number of unknowns. Then, we explain how to use metric-based anisotropic adaptation to obtain nearly body-fitted meshes with arbitrary geometries. The optimal rate of convergence, both for the solution and the geometry, is demonstrated on 2D and 3D academic Laplace problems involving curved boundaries. Finally, applications in the field of fluid dynamics and material science are presented. The results for these simulations successfully converge towards data from the literature, analytical solutions or values obtained with body-fitted meshes.