Addressing partial differential equations (PDEs) in complex-shaped domains is critical for numerous applications but poses significant computational challenges, particularly in accurately handling boundaries and their associated conditions. This talk focuses on developing high-accuracy discretization techniques for boundary conditions to improve computational precision in the framework of unfitted boundary methods. We introduce advanced high-order boundary condition discretization methods using the ghost point approach. In this framework, the computational grid is extended with external ghost points, whose values are determined through carefully designed discretization strategies that enforce the boundary conditions. Traditional methods often rely on simplistic extrapolation, assigning ghost point values independently from nearby internal grid points. In contrast, we adopt a more sophisticated, coupled strategy, where ghost point values are computed in conjunction with neighboring internal and other ghost points. This results in an augmented linear system that simultaneously incorporates equations for both internal and ghost points, leading to greater accuracy and stability. The effectiveness of this method is demonstrated through numerical experiments on elliptic equations and applications in simulating incompressible fluid dynamics, particularly in scenarios involving moving objects, such as oscillating bubbles. Although the method is presented in the context of finite difference schemes, its principles are generalizable to finite volume and finite element methods. To efficiently solve the augmented linear systems arising from these discretizations, we implement a customized multigrid solver tailored for curved boundaries. By relaxing the boundary conditions and optimizing relaxation parameters for each ghost point, we enhance the smoothing effect of relaxations along directions tangential to the boundary. This optimization is guided by a Boundary Local Fourier Analysis (BLFA), ensuring computational performance comparable to that achieved in simpler geometries, such as rectangular domains.