  We present a Boundary Local Fourier Analysis (BLFA) to optimize the
relaxation parameters of boundary conditions in a multigrid framework. The
method is implemented to solve elliptic equations on curved domains embedded in
a uniform Cartesian mesh, although it is designed to be extended for general
PDEs in curved domains, wherever a multigrid technique can be implemented. The
boundary is implicitly defined by a level-set function and a ghost-point
technique is employed to treat the boundary conditions. Existing strategies in
literature adopt a constant relaxation parameter on the whole boundary. In this
paper, the relaxation parameters are optimized in terms of the distance between
ghost points and boundary, with the goal of smoothing the residual along the
tangential direction. Theoretical results are confirmed by several numerical
tests in 1D, 2D and 3D, showing that the convergence factor associated with the
smoothing on internal equations is not degraded by boundary effects.
